**Authors**
Christoph Schiller

**License**
CC-BY-NC-ND-3.0

Christoph Schiller MOTION MOUNTAIN the adventure of physics – vol.i fall, flow and heat www.motionmountain.net Christoph Schiller Motion Mountain The Adventure of Physics Volume I Fall, Flow and Heat Edition 31, available as free pdf with films at www.motionmountain.net Editio trigesima prima. Proprietas scriptoris © Chrestophori Schiller primo anno Olympiadis trigesimae secundae. Omnia proprietatis iura reservantur et vindicantur. Imitatio prohibita sine auctoris permissione. Non licet pecuniam expetere pro aliqua, quae partem horum verborum continet; liber pro omnibus semper gratuitus erat et manet. Thirty-first edition. Copyright © 1990–2023 by Christoph Schiller, from the third year of the 24th Olympiad to the first year of the 32nd Olympiad. This pdf file is licensed under the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Germany Licence, whose full text can be found on the website creativecommons.org/licenses/by-nc-nd/3.0/de, with the additional restriction that reproduction, distribution and use, in whole or in part, in any product or service, be it commercial or not, is not allowed without the written consent of the copyright owner. The pdf file was and remains free for everybody to read, store and print for personal use, and to distribute electronically, but only in unmodified form and only at no charge. To Britta, Esther and Justus Aaron τῷ ἐμοὶ δαὶμονι Die Menschen stärken, die Sachen klären. PR E FAC E “ ” Primum movere, deinde docere.* Antiquity T his book series is for anybody who is curious about motion in nature. How do hings, people, animals, images and empty space move? The answer leads o many adventures; this volume presents the best ones about everyday mo- Motion Mountain – The Adventure of Physics tion. Carefully observing everyday motion allows us to deduce six essential statements: everyday motion is continuous, conserved, relative, reversible, mirror-invariant – and lazy. Yes, nature is indeed lazy: in every motion, it minimizes change. This text explores how these six results are deduced and how they fit with all those observations that seem to contradict them. In the structure of modern physics, shown in Figure 1, the results on everyday motion form the major part of the starting point at the bottom. The present volume is the first of a six-volume overview of physics. It resulted from a threefold aim I have pursued since 1990: to present motion in a way that is simple, up to date and captivating. In order to be simple, the text focuses on concepts, while keeping mathematics to the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net necessary minimum. Understanding the concepts of physics is given precedence over using formulae in calculations. The whole text is within the reach of an undergraduate. In order to be up to date, the text is enriched by the many gems – both theoretical and empirical – that are scattered throughout the scientific literature. In order to be captivating, the text tries to startle the reader as much as possible. Read- ing a book on general physics should be like going to a magic show. We watch, we are astonished, we do not believe our eyes, we think, and finally we understand the trick. When we look at nature, we often have the same experience. Indeed, every page presents at least one surprise or provocation for the reader to think about. Numerous interesting challenges are proposed. The motto of the text, die Menschen stärken, die Sachen klären, a famous statement on pedagogy, translates as: ‘To fortify people, to clarify things.’ Clarifying things – and adhering only to the truth – requires courage, as changing the habits of thought produces fear, often hidden by anger. But by overcoming our fears we grow in strength. And we experience intense and beautiful emotions. All great adventures in life allow this, and exploring motion is one of them. Enjoy it. Christoph Schiller * ‘First move, then teach.’ In modern languages, the mentioned type of moving (the heart) is called motiv- ating; both terms go back to the same Latin root. 8 preface Final, unified description of motion: upper limit c4/4Ghbar Adventures: describing precisely all motion, understanding the origin of colours, space -time and particles, enjoying extreme thinking, calculating masses and couplings, catching a further, tiny glimpse of bliss (vol. VI). PHYSICS: An arrow indicates an Describing motion with precision, increase in precision by i.e., using the least action principle. adding a motion limit. upper limit: Quantum theory General relativity: 1/4G hbar with classical gravity Quantum field theory upper limit c4/4G Adventures: bouncing (the ‘standard model’): Adventures: the neutrons, under- upper limit c/hbar night sky, measu- standing tree Adventures: building Motion Mountain – The Adventure of Physics ring curved and growth (vol. V). accelerators, under- wobbling space, standing quarks, stars, exploring black bombs and the basis of holes and the life, matter & radiation universe, space (vol. V). and time (vol. II). Classical gravity: upper limit: c Special relativity Quantum theory: upper limit 1/4G Adventures: light, upper limit 1/hbar Adventures: magnetism, length Adventures: biology, climbing, skiing, c contraction, time birth, love, death, space travel, limits dilation and chemistry, evolution, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the wonders of G fast E0 = mc2 h, e, k enjoying colours, art, astronomy and limits motion (vol. II). limit paradoxes, medicine geology (vol. I). uniform tiny and high-tech business motion motion (vol. IV and vol. V). Galilean physics, heat and electricity: no limits The world of everyday motion: human scale, slow and weak. Adventures: sport, music, sailing, cooking, describing beauty and understanding its origin (vol. I); using electricity, light and computers, understanding the brain and people (vol. III). F I G U R E 1 A complete map of physics, the science of motion, as ﬁrst proposed by Matvei Bronshtein (b. 1907 Vinnytsia, d. 1938 Leningrad). The Bronshtein cube starts at the bottom with everyday motion, and shows the connections to the ﬁelds of modern physics. Each connection increases the precision of the description and is due to a limit to motion that is taken into account. The limits are given for uniform motion by the gravitational constant G, for fast motion by the speed of light c, and for tiny motion by the Planck constant h, the elementary charge e and the Boltzmann constant k. preface 9 Using this b o ok Marginal notes refer to bibliographic references, to other pages or to challenge solutions. In the colour edition, marginal notes, pointers to footnotes and links to websites are typeset in green. Over time, links on the internet tend to disappear. Most links can be recovered via www.archive.org, which keeps a copy of old internet pages. In the free pdf edition of this book, available at www.motionmountain.net, all green pointers and links are clickable. The pdf edition also contains all films; they can be watched directly in Adobe Reader. Solutions and hints for challenges are given in the appendix. Challenges are classified as easy (e), standard student level (s), difficult (d) and research level (r). Challenges for which no solution has yet been included in the book are marked (ny). Advice for learners Learning allows us to discover what kind of person we can be. Learning widens know- ledge, improves intelligence and provides a sense of achievement. Therefore, learning Motion Mountain – The Adventure of Physics from a book, especially one about nature, should be efficient and enjoyable. Avoid bad learning methods like the plague! Do not use a marker, a pen or a pencil to highlight or underline text on paper. It is a waste of time, provides false comfort and makes the text unreadable. Add notes and comments instead! And do not learn from a screen. In particular, do not learn from videos, from games or from a smartphone. All games and almost all videos are drugs for the brain. Smartphones are drug dispensers that make people addicted and prevent learning. Learn from paper – at your speed, and allow your mind to wander! Nobody marking paper or looking at a screen is learning efficiently. In my experience as a pupil and teacher, one learning method never failed to trans- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net form unsuccessful pupils into successful ones: if you read a text for study, summarize every section you read, in your own words and images, aloud. If you are unable to do so, read the section again. Repeat this until you can clearly summarize what you read in your own words and images, aloud. And enjoy the telling aloud! You can do this alone or with friends, in a room or while walking. If you do this with everything you read, you will reduce your learning and reading time significantly; you will enjoy learning from good texts much more and hate bad texts much less. Masters of the method can use it even while listening to a lecture, in a low voice, thus avoiding to ever take notes. Advice for teachers A teacher likes pupils and likes to lead them into exploring the field he or she chose. His or her enthusiasm is the key to job satisfaction. If you are a teacher, before the start of a lesson, picture, feel and tell yourself how you enjoy the topic of the lesson; then picture, feel and tell yourself how you will lead each of your pupils into enjoying that topic as much as you do. Do this exercise consciously, every day. You will minimize trouble in your class and maximize your teaching success. This book is not written with exams in mind; it is written to make teachers and stu- dents understand and enjoy physics, the science of motion. 10 preface Feedback The latest pdf edition of this text is and will remain free to download from the internet. I would be delighted to receive an email from you at fb@motionmountain.net, especially on the following issues: Challenge 1 s — What was unclear and should be improved? — What story, topic, riddle, picture or film did you miss? Also help on the specific points listed on the www.motionmountain.net/help.html web page is welcome. All feedback will be used to improve the next edition. You are welcome to send feedback by mail or by sending in a pdf with added yellow notes, to provide illustrations or photographs, or to contribute to the errata wiki on the website. If you would like to translate a chapter of the book in your language, please let me know. On behalf of all readers, thank you in advance for your input. For a particularly useful contribution you will be mentioned – if you want – in the acknowledgements, receive a reward, or both. Motion Mountain – The Adventure of Physics Support Your donation to the charitable, tax-exempt non-profit organisation that produces, trans- lates and publishes this book series is welcome. For details, see the web page www. motionmountain.net/donation.html. The German tax office checks the proper use of your donation. If you want, your name will be included in the sponsor list. Thank you in advance for your help, on behalf of all readers across the world. The paper edition of this book is available, either in colour or in black and white, from www.amazon.com, in English and in certain other languages. And now, enjoy the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net reading. C ON T E N T S 7 Preface Using this book 9 • Advice for learners 9 • Advice for teachers 9 • Feedback 10 • Support 10 11 Contents 15 1 Why should we care about motion? Does motion exist? 16 • How should we talk about motion? 18 • What are Motion Mountain – The Adventure of Physics the types of motion? 20 • Perception, permanence and change 25 • Does the world need states? 27 • Galilean physics in six interesting statements 29 • Curiosities and fun challenges about motion 30 • First summary on motion 33 34 2 From motion measurement to continuity What is velocity? 35 • What is time? 40 • Clocks 44 • Why do clocks go clockwise? 48 • Does time flow? 48 • What is space? 49 • Are space and time absolute or relative? 52 • Size – why length and area exist, but volume does not 52 • What is straight? 59 • A hollow Earth? 60 • Curiosities and fun challenges about everyday space and time 61 • Summary about everyday space copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net and time 74 75 3 How to describe motion – kinematics Throwing, jumping and shooting 78 • Enjoying vectors 80 • What is rest? What is velocity? 82 • Acceleration 85 • From objects to point particles 85 • Legs and wheels 89 • Curiosities and fun challenges about kinematics 92 • Summary of kinematics 96 98 4 From objects and images to conservation Motion and contact 99 • What is mass? 100 • Momentum and mass 102 • Is motion eternal? – Conservation of momentum 108 • More conservation – en- ergy 110 • The cross product, or vector product 115 • Rotation and angular mo- mentum 116 • Rolling wheels 121 • How do we walk and run? 122 • Curiosities and fun challenges about mass, conservation and rotation 123 • Summary on conservation in motion 134 135 5 From the rotation of the earth to the relativity of motion How does the Earth rotate? 145 • Does the Earth move? 150 • Is velocity absolute? – The theory of everyday relativity 156 • Is rotation relative? 158 • Curiosities and fun challenges about rotation and relativity 158 • Legs or wheels? – Again 168 • Summary on Galilean relativity 172 173 6 Motion due to gravitation Gravitation as a limit to uniform motion 173 • Gravitation in the sky 174 • Grav- itation on Earth 178 • Properties of gravitation: 𝐺 and 𝑔 182 • The gravitational 12 contents potential 186 • The shape of the Earth 188 • Dynamics – how do things move in various dimensions? 189 • The Moon 190 • Orbits – conic sections and more 192 • Tides 197 • Can light fall? 201 • Mass: inertial and gravitational 202 • Curios- ities and fun challenges about gravitation 204 • Summary on gravitation 225 226 7 Classical mechanics, force and the predictability of motion Should one use force? Power? 227 • Forces, surfaces and conservation 231 • Friction and motion 232 • Friction, sport, machines and predictability 234 • Complete states – initial conditions 237 • Do surprises exist? Is the future de- termined? 238 • Free will 241 • Summary on predictability 242 • From predictability to global descriptions of motion 242 248 8 Measuring change with action The principle of least action 253 • Lagrangians and motion 256 • Why is motion so often bounded? 257 • Curiosities and fun challenges about Lagrangians 261 • Summary on action 264 266 9 Motion and symmetry Why can we think and talk about the world? 270 • Viewpoints 271 • Sym- Motion Mountain – The Adventure of Physics metries and groups 272 • Multiplets 273 • Representations 275 • The symmet- ries and vocabulary of motion 276 • Reproducibility, conservation and Noether’s theorem 280 • Parity inversion and motion reversal 284 • Interaction symmet- ries 285 • Curiosities and fun challenges about symmetry 285 • Summary on sym- metry 286 288 10 Simple motions of extended bodies – oscillations and waves Oscillations 288 • Damping 289 • Resonance 291 • Waves: general and har- monic 293 • Water waves 295 • Waves and their motion 300 • Why can we talk to each other? – Huygens’ principle 304 • Wave equations 305 • Why are music and copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net singing voices so beautiful? 307 • Measuring sound 310 • Is ultrasound imaging safe for babies? 313 • Signals 314 • Solitary waves and solitons 316 • Curiosities and fun challenges about waves and oscillation 318 • Summary on waves and oscillations 331 333 11 Do extended bodies exist? – Limits of continuity Mountains and fractals 333 • Can a chocolate bar last forever? 333 • The case of Galileo Galilei 335 • How high can animals jump? 337 • Felling trees 338 • Little hard balls 339 • The sound of silence 340 • How to count what cannot be seen 340 • Experiencing atoms 342 • Seeing atoms 344 • Curiosities and fun challenges about solids and atoms 345 • Summary on atoms 351 354 12 Fluids and their motion What can move in nature? – Flows of all kinds 354 • The state of a fluid 357 • Laminar and turbulent flow 361 • The atmosphere 364 • The physics of blood and breath 367 • Curiosities and fun challenges about fluids 370 • Summary on flu- ids 382 383 13 On heat and motion reversal invariance Temperature 383 • Thermal energy 387 • Why do balloons take up space? – The end of continuity 389 • Brownian motion 391 • Why stones can be neither smooth nor fractal, nor made of little hard balls 394 • Entropy 395 • Entropy from particles 398 • The characteristic entropy of nature – the quantum of informa- tion 399 • Is everything made of particles? 400 • The second principle of contents 13 thermodynamics 402 • Why can’t we remember the future? 404 • Flow of en- tropy 404 • Do isolated systems exist? 405 • Curiosities and fun challenges about reversibility and heat 405 • Summary on heat and time-invariance 413 415 14 Self-organization and chaos – the simplicity of complexity Appearance of order 418 • Self-organization in sand 420 • Self-organization of spheres 422 • Conditions for the appearance of order 422 • The mathematics of order appearance 423 • Chaos 424 • Emergence 426 • Curiosities and fun challenges about self-organization 427 • Summary on self-organization and chaos 434 435 15 From the limitations of physics to the limits of motion Research topics in classical dynamics 435 • What is contact? 436 • What determ- ines precision and accuracy? 437 • Can all of nature be described in a book? 437 • Something is wrong about our description of motion 438 • Why is measure- ment possible? 439 • Is motion unlimited? 439 441 a Notation and conventions The Latin alphabet 441 • The Greek alphabet 443 • The Hebrew alphabet and Motion Mountain – The Adventure of Physics other scripts 445 • Numbers and the Indian digits 446 • The symbols used in the text 447 • Calendars 449 • People Names 451 • Abbreviations and eponyms or concepts? 451 453 b Units, measurements and constants SI units 453 • The meaning of measurement 456 • Curiosities and fun challenges about units 456 • Precision and accuracy of measurements 459 • Limits to preci- sion 460 • Physical constants 460 • Useful numbers 468 469 c Sources of information on motion 475 Challenge hints and solutions copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net 523 Bibliography 556 Credits Acknowledgements 556 • Film credits 557 • Image credits 557 564 Name index 577 Subject index Fall, Flow and Heat In our quest to learn how things move, the experience of hiking and other motion leads us to introduce the concepts of velocity, time, length, mass and temperature. We learn to use them to measure change and find that nature minimizes it. We discover how to float in free space, why we have legs instead of wheels, why disorder can never be eliminated, and why one of the most difficult open issues in science is the flow of water through a tube. Chapter 1 W H Y SHOU L D W E C A R E A B OU T MOT ION ? “ ” All motion is an illusion. Zeno of Elea** W ham! The lightning striking the tree nearby violently disrupts our quiet forest alk and causes our hearts to suddenly beat faster. In the top of the tree Motion Mountain – The Adventure of Physics e see the fire start and fade again. The gentle wind moving the leaves around us helps to restore the calmness of the place. Nearby, the water in a small river follows its complicated way down the valley, reflecting on its surface the ever-changing shapes of the clouds. Motion is everywhere: friendly and threatening, terrible and beautiful. It is funda- mental to our human existence. We need motion for growing, for learning, for thinking, for remaining healthy and for enjoying life. We use motion for walking through a forest, for listening to its noises and for talking about all this. Like all animals, we rely on motion to get food and to survive dangers. Like all living beings, we need motion to reproduce, to breathe and to digest. Like all objects, motion keeps us warm. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Motion is the most fundamental observation about nature at large. It turns out that everything that happens in the world is some type of motion. There are no exceptions. Motion is such a basic part of our observations that even the origin of the word is lost in the darkness of Indo-European linguistic history. The fascination of motion has always made it a favourite object of curiosity. By the fifth century b ce in ancient Greece, its Ref. 1 study had been given a name: physics. Motion is also important to the human condition. What can we know? Where does the world come from? Who are we? Where do we come from? What will we do? What should we do? What will the future bring? What is death? Where does life lead? All these questions are about motion. And the study of motion provides answers that are both deep and surprising. Ref. 2 Motion is mysterious. Though found everywhere – in the stars, in the tides, in our eyelids – neither the ancient thinkers nor myriads of others in the 25 centuries since then have been able to shed light on the central mystery: what is motion? We shall discover that the standard reply, ‘motion is the change of place in time’, is correct, but inadequate. Just recently a full answer has finally been found. This is the story of the way to find it. Motion is a part of human experience. If we imagine human experience as an island, then destiny, symbolized by the waves of the sea, carried us to its shore. Near the centre of ** Zeno of Elea (c. 450 bce), one of the main exponents of the Eleatic school of philosophy. 16 1 why should we care about motion? Astronomy Theory of motion Materials science Quantum Chemistry field theory Geosciences Medicine Quantum Biology theory Motion Electromagnetism Mountain Relativity Thermodynamics Engineering Physics Mechanics Emotion Bay Mathematics Motion Mountain – The Adventure of Physics The humanities Social Sea F I G U R E 2 Experience Island, with Motion Mountain and the trail to be followed. the island an especially high mountain stands out. From its top we can see over the whole landscape and get an impression of the relationships between all human experiences, and copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net in particular between the various examples of motion. This is a guide to the top of what I have called Motion Mountain (see Figure 2; a less symbolic and more exact version is Page 8 given in Figure 1). The hike is one of the most beautiful adventures of the human mind. The first question to ask is: Does motion exist? “ Das Rätsel gibt es nicht. Wenn sich eine Frage überhaupt stellen läßt, so kann sie beantwortet ” werden.* Ludwig Wittgenstein, Tractatus, 6.5 To sharpen the mind for the issue of motion’s existence, have a look at Figure 3 or Fig- Ref. 3 ure 4 and follow the instructions. In all cases the figures seem to rotate. You can exper- ience similar effects if you walk over cobblestone pavement that is arranged in arched patterns or if you look at the numerous motion illusions collected by Kitaoka Akiyoshi Ref. 4 at www.ritsumei.ac.jp/~akitaoka. How can we make sure that real motion is different Challenge 2 s from these or other similar illusions? Many scholars simply argued that motion does not exist at all. Their arguments deeply Ref. 5 influenced the investigation of motion over many centuries. For example, the Greek * ‘The riddle does not exist. If a question can be put at all, it can also be answered.’ 1 why should we care about motion? 17 F I G U R E 3 Illusions of motion: look at the ﬁgure on the left and slightly move the page, or look at the white dot at the centre of the ﬁgure on the right and move your head back and forward. philosopher Parmenides (born c. 515 b ce in Elea, a small town near Naples) argued that since nothing comes from nothing, change cannot exist. He underscored the perman- ence of nature and thus consistently maintained that all change and thus all motion is an illusion. Motion Mountain – The Adventure of Physics Ref. 6 Heraclitus (c. 540 to c. 480 b ce) held the opposite view. Plato describes Heraclitus as making the famous statement πάντα ῥεῖ ‘panta rhei’ or ‘everything flows’.* He saw change as the essence of nature, in contrast to Parmenides. These two equally famous opinions induced many scholars to investigate in more detail whether in nature there are conserved quantities or whether creation is possible. We will uncover the answer later Challenge 3 s on; until then, you might ponder which option you prefer. Parmenides’ collaborator Zeno of Elea (born c. 500 b ce) argued so intensely against motion that some people still worry about it today. In one of his arguments he claims – in simple language – that it is impossible to slap somebody, since the hand first has to copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net travel halfway to the face, then travel through half the distance that remains, then again so, and so on; the hand therefore should never reach the face. Zeno’s argument focuses on the relation between infinity and its opposite, finitude, in the description of motion. Ref. 7 In modern quantum theory, a related issue is a subject of research up to this day. Zeno also stated that by looking at a moving object at a single instant of time, one cannot maintain that it moves. He argued that at a single instant of time, there is no difference between a moving and a resting body. He then deduced that if there is no difference at a single time, there cannot be a difference for longer times. Zeno therefore questioned whether motion can clearly be distinguished from its opposite, rest. Indeed, in the history of physics, thinkers switched back and forward between a positive and a negative answer. It was this very question that led Albert Einstein to the development of general relativity, one of the high points of our journey. In our adventure, we will explore all known differences between motion and rest. Eventually, we will dare to ask whether single instants of time do exist at all. Answering this question is essential for reaching the top of Motion Mountain. When we explore quantum theory, we will discover that motion is indeed – to a cer- tain extent – an illusion, as Parmenides claimed. More precisely, we will show that mo- tion is observed only due to the limitations of the human condition. We will find that we experience motion only because * Appendix A explains how to read Greek text. 18 1 why should we care about motion? F I G U R E 4 Zoom this image to large size or approach it closely in order Motion Mountain – The Adventure of Physics to enjoy its apparent motion (© Michael Bach after the discovery of Kitaoka Akiyoshi). — we have a finite size, — we are made of a large but finite number of atoms, — we have a finite but moderate temperature, — we move much more slowly than the speed of light, — we live in three dimensions, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net — we are large compared with a black hole of our own mass, — we are large compared with our quantum mechanical wavelength, — we are small compared with the universe, — we have a working but limited memory, — we are forced by our brain to approximate space and time as continuous entities, and — we are forced by our brain to approximate nature as made of different parts. If any one of these conditions were not fulfilled, we would not observe motion; motion, then, would not exist! If that were not enough, note that none of the conditions requires human beings; they are equally valid for many animals and machines. Each of these con- ditions can be uncovered most efficiently if we start with the following question: How should we talk ab ou t motion? “ Je hais le mouvement, qui déplace les lignes, ” Et jamais je ne pleure et jamais je ne ris. Charles Baudelaire, La Beauté.* Like any science, the approach of physics is twofold: we advance with precision and with curiosity. Precision is the extent to which our description matches observations. Curios- * Charles Baudelaire (b. 1821 Paris, d. 1867 Paris) Beauty: ‘I hate movement, which changes shapes, and Ref. 8 never do I weep and never do I laugh.’ Beauty. 1 why should we care about motion? 19 Anaximander Empedocles Eudoxus Ctesibius Strabo Frontinus Cleomedes Anaximenes Aristotle Archimedes Varro Maria Artemidor the Jewess Pythagoras Heraclides Konon Athenaius Josephus Sextus Empiricus Almaeon Philolaus Theophrastus Chrysippos Eudoxus Pomponius Dionysius Athenaios Diogenes of Kyz. Mela Periegetes of Nauc. Laertius Heraclitus Zeno Autolycus Eratosthenes Sosigenes Marinus Xenophanes Anthistenes Euclid Dositheus Virgilius Menelaos Philostratus Thales Parmenides Archytas Epicure Biton Polybios Horace Nicomachos Apuleius Alexander Ptolemy II Ptolemy VIII Caesar Nero Trajan 600 BCE 500 400 300 200 100 1 100 200 Socrates Plato Ptolemy I Cicero Seneca Anaxagoras Hicetas Aristarchus Asclepiades Livius Dioscorides Ptolemy Leucippus Pytheas Archimedes Seleukos Vitruvius Geminos Epictetus Protagoras Erasistratus Diocles Manilius Demonax Diophantus Oenopides Aristoxenus Aratos Philo Dionysius Diodorus Valerius Theon Alexander of Byz. Thrax Siculus Maximus of Smyrna of Aphr. Hippocrates Berossos Motion Mountain – The Adventure of Physics Herodotus Herophilus Apollonius Theodosius Plinius Rufus Galen Senior Democritus Straton Hipparchus Lucretius Aetius Arrian Hippasos Speusippos Dikaiarchus Poseidonius Heron Plutarch Lucian Naburimannu Kidinnu F I G U R E 5 A timeline of scientiﬁc and political personalities in antiquity. The last letter of the name is aligned with the year of death. For example, Maria the Jewess is the inventor of the bain-marie process and Thales is the ﬁrst mathematician and scientist known by name. ity is the passion that drives all scientists. Precision makes meaningful communication copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net possible, and curiosity makes it worthwhile. Take an eclipse, a beautiful piece of music Ref. 9 or a feat at the Olympic games: the world is full of fascinating examples of motion. If you ever find yourself talking about motion, whether to understand it more pre- cisely or more deeply, you are taking steps up Motion Mountain. The examples of Fig- ure 6 make the point. An empty bucket hangs vertically. When you fill the bucket with a certain amount of water, it does not hang vertically any more. (Why?) If you continue adding water, it starts to hang vertically again. How much water is necessary for this last Challenge 4 s transition? The second illustration in Figure 6 is for the following puzzle. When you pull a thread from a reel in the way shown, the reel will move either forwards or backwards, depending on the angle at which you pull. What is the limiting angle between the two possibilities? High precision means going into fine details. Being attuned to details actually in- creases the pleasure of the adventure.* Figure 7 shows an example. The higher we get on Motion Mountain, the further we can see and the more our curiosity is rewarded. The views offered are breathtaking, especially from the very top. The path we will follow – one of the many possible routes – starts from the side of biology and directly enters the Ref. 10 forest that lies at the foot of the mountain. Challenge 6 s * Distrust anybody who wants to talk you out of investigating details. He is trying to deceive you. Details are important. Be vigilant also during this journey. 20 1 why should we care about motion? F I G U R E 6 How much water is required to make a bucket hang vertically? At what angle does the reel Challenge 5 s (drawn incorrectly, with too small rims) change direction of motion when pulled along with the thread? (© Luca Gastaldi). Motion Mountain – The Adventure of Physics F I G U R E 7 An example of how precision of observation can lead to the discovery of new effects: the deformation of a tennis ball during the c. 6 ms of a fast bounce (© International Tennis Federation). Intense curiosity drives us to go straight to the limits: understanding motion re- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net quires exploration of the largest distances, the highest velocities, the smallest particles, the strongest forces, the highest precision and the strangest concepts. Let us begin. What are the t ypes of motion? “ ” Every movement is born of a desire for change. Antiquity A good place to obtain a general overview on the types of motion is a large library. Table 1 shows the results. The domains in which motion, movements and moves play a role are indeed varied. Already the earliest researchers in ancient Greece – listed in Figure 5 – had the suspicion that all types of motion, as well as many other types of change, are related. Three categories of change are commonly recognized: 1. Transport. The only type of change we call motion in everyday life is material trans- port, such as a person walking, a leaf falling from a tree, or a musical instrument playing. Transport is the change of position or orientation of objects, fluids included. To a large extent, the behaviour of people also falls into this category. 2. Transformation. Another category of change groups observations such as the dis- solution of salt in water, the formation of ice by freezing, the rotting of wood, the cooking of food, the coagulation of blood, and the melting and alloying of metals. 1 why should we care about motion? 21 TA B L E 1 Content of books about motion found in a public library. Motion topics Motion topics motion pictures and digital effects motion as therapy for cancer, diabetes, acne and depression motion perception Ref. 11 motion sickness motion for fitness and wellness motion for meditation motion control and training in sports and motion ability as health check singing perpetual motion motion in dance, music and other performing arts motion as proof of various gods Ref. 12 motion of planets, stars and angels Ref. 13 economic efficiency of motion the connection between motional and emotional habits motion as help to overcome trauma motion in psychotherapy Ref. 14 locomotion of insects, horses, animals and motion of cells and plants Motion Mountain – The Adventure of Physics robots collisions of atoms, cars, stars and galaxies growth of multicellular beings, mountains, sunspots and galaxies motion of springs, joints, mechanisms, motion of continents, bird flocks, shadows and liquids and gases empty space commotion and violence motion in martial arts motions in parliament movements in art, sciences and politics movements in watches movements in the stock market movement teaching and learning movement development in children Ref. 15 copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net musical movements troop movements Ref. 16 religious movements bowel movements moves in chess cheating moves in casinos Ref. 17 connection between gross national product and citizen mobility These changes of colour, brightness, hardness, temperature and other material prop- erties are all transformations. Transformations are changes not visibly connected with transport. To this category, a few ancient thinkers added the emission and absorption of light. In the twentieth century, these two effects were proven to be special cases of transformations, as were the newly discovered appearance and disappearance of mat- ter, as observed in the Sun and in radioactivity. Mind change, such as change of mood, Ref. 18 of health, of education and of character, is also (mostly) a type of transformation. Ref. 19 3. Growth. This last and especially important category of change, is observed for an- imals, plants, bacteria, crystals, mountains, planets, stars and even galaxies. In the nineteenth century, changes in the population of systems, biological evolution, and in the twentieth century, changes in the size of the universe, cosmic evolution, were added to this category. Traditionally, these phenomena were studied by separate sci- ences. Independently they all arrived at the conclusion that growth is a combination of transport and transformation. The difference is one of complexity and of time scale. 22 1 why should we care about motion? F I G U R E 8 An example of transport, at Mount Etna (© Marco Fulle). Motion Mountain – The Adventure of Physics At the beginnings of modern science during the Renaissance, only the study of transport was seen as the topic of physics. Motion was equated to transport. The other two domains were neglected by physicists. Despite this restriction, the field of enquiry remains large, Page 16 covering a large part of Experience Island. Early scholars differentiated types of transport by their origin. Movements such as those of the legs when walking were classified as volitional, because they are controlled by one’s will, whereas movements of external objects, such as the fall of a snowflake, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net which cannot be influenced by will-power, were classified as passive. Young humans, es- pecially young male humans, spend considerable time in learning elaborate volitional movements. An example is shown in Figure 10. The complete distinction between passive and volitional motion is made by children by the age of six, and this marks a central step in the development of every human to- wards a precise description of the environment.* From this distinction stems the histor- ical but now outdated definition of physics as the science of motion of non-living things. The advent of machines forced scholars to rethink the distinction between volitional and passive motion. Like living beings, machines are self-moving and thus mimic voli- tional motion. However, careful observation shows that every part in a machine is moved by another, so their motion is in fact passive. Are living beings also machines? Are human actions examples of passive motion as well? The accumulation of observations in the last 100 years made it clear that volitional movement** indeed has the same physical prop- * Failure to pass this stage completely can result in a person having various strange beliefs, such as believing in the ability to influence roulette balls, as found in compulsive players, or in the ability to move other bod- ies by thought, as found in numerous otherwise healthy-looking people. An entertaining and informative account of all the deception and self-deception involved in creating and maintaining these beliefs is given by James R andi, The Faith Healers, Prometheus Books, 1989. A professional magician, he presents many similar topics in several of his other books. See also his www.randi.org website for more details. ** The word ‘movement’ is rather modern; it was imported into English from the old French and became popular only at the end of the eighteenth century. It is never used by Shakespeare. 1 why should we care about motion? 23 F I G U R E 9 Transport, growth and transformation (© Philip Plisson). Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 10 One of the most difﬁcult volitional movements known, performed by Alexander Tsukanov, the ﬁrst man able to do this: jumping from one ultimate wheel to another (© Moscow State Circus). erties as passive movement in non-living systems. A distinction between the two types of motion is thus unnecessary. Of course, from the emotional viewpoint, the differences Ref. 20 are important; for example, grace can only be ascribed to volitional movements. Since passive and volitional motion have the same properties, through the study of motion of non-living objects we can learn something about the human condition. This is most evident when touching the topics of determinism, causality, probability, infinity, time, love and death, to name but a few of the themes we will encounter during our adventure. In the nineteenth and twentieth centuries other classically held beliefs about mo- tion fell by the wayside. Extensive observations showed that all transformations and all growth phenomena, including behaviour change and evolution, are also examples of transport. In other words, over 2 000 years of studies have shown that the ancient classi- 24 1 why should we care about motion? fication of observations was useless: ⊳ All change is transport. And ⊳ Transport and motion are the same. In the middle of the twentieth century the study of motion culminated in the exper- imental confirmation of an even more specific idea, previously articulated in ancient Greece: ⊳ Every type of change is due to the motion of particles. It takes time and work to reach this conclusion, which appears only when we relentlessly pursue higher and higher precision in the description of nature. The first five parts of Motion Mountain – The Adventure of Physics Challenge 7 s this adventure retrace the path to this result. (Do you agree with it?) The last decade of the twentieth century again completely changed the description of motion: the particle idea turns out to be limited and wrong. This recent result, reached through a combination of careful observation and deduction, will be explored in the last part of our adventure. But we still have some way to go before we reach that result, just below the summit of our journey. In summary, history has shown that classifying the various types of motion is not productive. Only by trying to achieve maximum precision can we hope to arrive at the fundamental properties of motion. Precision, not classification, is the path to follow. As copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Ernest Rutherford said jokingly: ‘All science is either physics or stamp collecting.’ In order to achieve precision in our description of motion, we need to select specific examples of motion and study them fully in detail. It is intuitively obvious that the most precise description is achievable for the simplest possible examples. In everyday life, this is the case for the motion of any non-living, solid and rigid body in our environment, such as a stone thrown through the air. Indeed, like all humans, we learned to throw ob- Ref. 21 jects long before we learned to walk. Throwing is one of the first physical experiments we performed by ourselves. The importance of throwing is also seen from the terms de- rived from it: in Latin, words like subject or ‘thrown below’, object or ‘thrown in front’, and interjection or ‘thrown in between’; in Greek, the act of throwing led to terms like symbol or ‘thrown together’, problem or ‘thrown forward’, emblem or ‘thrown into’, and – last but not least – devil or ‘thrown through’. And indeed, during our early childhood, by throwing stones, toys and other objects until our parents feared for every piece of the household, we explored the perception and the properties of motion. We do the same here. “ ” Die Welt ist unabhängig von meinem Willen.* Ludwig Wittgenstein, Tractatus, 6.373 * ‘The world is independent of my will.’ 1 why should we care about motion? 25 Perception, permanence and change “ Only wimps specialize in the general case; real ” scientists pursue examples. Adapted by Michael Berry from a remark by Beresford Parlett Human beings enjoy perceiving. Perception starts before birth, and we continue enjoying it for as long as we can. That is why television or videos, even when devoid of content, are so successful. During our walk through the forest at the foot of Motion Mountain we cannot avoid perceiving. Perception is first of all the ability to distinguish. We use the basic mental act of distinguishing in almost every instant of life; for example, during childhood we first learned to distinguish familiar from unfamiliar observations. This is possible in combination with another basic ability, namely the capacity to memorize ex- periences. Memory gives us the ability to experience, to talk and thus to explore nature. Perceiving, classifying and memorizing together form learning. Without any one of these three abilities, we could not study motion. Children rapidly learn to distinguish permanence from variability. They learn to re- Motion Mountain – The Adventure of Physics cognize human faces, even though a face never looks exactly the same each time it is seen. From recognition of faces, children extend recognition to all other observations. Recog- nition works pretty well in everyday life; it is nice to recognize friends, even at night, and even after many beers (not a challenge). The act of recognition thus always uses a form of generalization. When we observe, we always have some general idea in our mind. Let us specify the main ones. Sitting on the grass in a clearing of the forest at the foot of Motion Mountain, surroun- ded by the trees and the silence typical of such places, a feeling of calmness and tranquil- lity envelops us. We are thinking about the essence of perception. Suddenly, something copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net moves in the bushes; immediately our eyes turn and our attention focuses. The nerve cells that detect motion are part of the most ancient part of our brain, shared with birds Ref. 22 and reptiles: the brain stem. Then the cortex, or modern brain, takes over to analyse the type of motion and to identify its origin. Watching the motion across our field of vision, we observe two invariant entities: the fixed landscape and the moving animal. After we recognize the animal as a deer, we relax again. How did we distinguish, in case of Figure 11, between landscape and deer? Perception involves several processes in the eye and in the brain. An essential part for these pro- cesses is motion, as is best deduced from the flip film shown in the lower left corners Ref. 23 of these pages. Each image shows only a rectangle filled with a mathematically random pattern. But when the pages are scanned in rapid succession, you discern a shape – a square – moving against a fixed background. At any given instant, the square cannot be distinguished from the background; there is no visible object at any given instant of time. Nevertheless it is easy to perceive its motion.* Perception experiments such as this one have been performed in many variations. Such experiments showed that detecting a moving square against a random background is nothing special to humans; flies have the same ability, as do, in fact, all animals that have eyes. * The human eye is rather good at detecting motion. For example, the eye can detect motion of a point of light even if the change of angle is smaller than that which can be distinguished in a fixed image. Details of Ref. 11 this and similar topics for the other senses are the domain of perception research. 26 1 why should we care about motion? F I G U R E 11 How do we distinguish a deer from its environment? (© Tony Rodgers). The flip film in the lower left corner, like many similar experiments, illustrates two Motion Mountain – The Adventure of Physics central attributes of motion. First, motion is perceived only if an object can be distin- guished from a background or environment. Many motion illusions focus on this point.* Second, motion is required to define both the object and the environment, and to dis- tinguish them from each other. In fact, the concept of space is – among others – an abstraction of the idea of background. The background is extended; the moving entity is localized. Does this seem boring? It is not; just wait for a second. We call a localized entity of investigation that can change or move a physical system – or simply a system. A system is a recognizable, thus permanent part of nature. Systems can be objects – also called ‘physical bodies’ – or radiation. Therefore, images, which are made of radiation, are aspects of physical systems, but not themselves physical systems. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 8 s These connections are summarized in Table 2. Now, are holes physical systems? In other words, we call the set of localized aspects that remain invariant or permanent during motion, such as size, shape, colour etc., taken together, a (physical) object or a (physical) body. We will tighten the definition shortly, to distinguish objects from images. We note that to specify permanent moving objects, we need to distinguish them from the environment. In other words, right from the start, we experience motion as a relative process; it is perceived in relation and in opposition to the environment. The conceptual distinction between localized, isolable objects and the extended envir- Challenge 9 s onment is important. True, it has the appearance of a circular definition. (Do you agree?) Page 438 Indeed, this issue will keep us busy later on. On the other hand, we are so used to our ability to isolate local systems from the environment that we take it for granted. How- ever, as we will discover later on in our walk, this distinction turns out to be logically Vol. VI, page 85 and experimentally impossible!** The reason for this impossibility will turn out to be fascinating. To discover the impossibility, we note, as a first step, that apart from mov- * The topic of motion perception is full of interesting aspects. An excellent introduction is chapter 6 of the beautiful text by Donald D. Hoffman, Visual Intelligence – How We Create What We See, W.W. Norton & Co., 1998. His collection of basic motion illusions can be experienced and explored on the associated www.cogsci.uci.edu/~ddhoff website. ** Contrary to what is often read in popular literature, the distinction is possible in quantum theory. It becomes impossible only when quantum theory is unified with general relativity. 1 why should we care about motion? 27 TA B L E 2 Family tree of the basic physical concepts. motion the basic type of change: a system changing position relative to a background parts/systems relations background permanent measurable bounded produce boundaries unbounded have shapes produce shapes extended objects radiation states interactions phase space space-time impenetrable penetrable global local composed simple The corresponding aspects: mass intensity instant source dimension curvature Motion Mountain – The Adventure of Physics size colour position domain distance topology charge image momentum strength volume distance spin appearance energy direction subspaces area etc. etc. etc. etc. etc. etc. world – nature – universe – cosmos the collection of all parts, relations and backgrounds copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ing entities and the permanent background, we also need to describe their relations. The necessary concepts are summarized in Table 2. “ Wisdom is one thing: to understand the thought ” which steers all things through all things. Ref. 24 Heraclitus of Ephesus Does the world need states? “ Das Feste, das Bestehende und der Gegenstand sind Eins. Der Gegenstand ist das Feste, Bestehende; die Konfiguration ist das ” Wechselnde, Unbeständige.* Ludwig Wittgenstein, Tractatus, 2.027 – 2.0271 What distinguishes the various patterns in the lower left corners of this text? In everyday life we would say: the situation or configuration of the involved entities. The situation somehow describes all those aspects that can differ from case to case. It is customary to call the list of all variable aspects of a set of objects their (physical) state of motion, or simply their state. How is the state characterized? * ‘The fixed, the existent and the object are one. The object is the fixed, the existent; the configuration is the changing, the variable.’ 28 1 why should we care about motion? The configurations in the lower left corners differ first of all in time. Time is what makes opposites possible: a child is in a house and the same child is outside the house. Time describes and resolves this type of contradiction. But the state not only distin- guishes situations in time: the state contains all those aspects of a system – i.e., of a group of objects – that set it apart from all similar systems. Two similar objects can differ, at each instant of time, in their — position, — velocity, — orientation, or — angular velocity. These properties determine the state and pinpoint the individuality of a physical system among exact copies of itself. Equivalently, the state describes the relation of an object or a system with respect to its environment. Or, again, equivalently: ⊳ The state describes all aspects of a system that depend on the observer. Motion Mountain – The Adventure of Physics The definition of state is not boring at all – just ponder this: Does the universe have a Challenge 10 s state? And: is the list of state properties just given complete? In addition, physical systems are described by their permanent, intrinsic properties. Some examples are — mass, — shape, — colour, — composition. Intrinsic properties do not depend on the observer and are independent of the state of copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the system. They are permanent – at least for a certain time interval. Intrinsic properties also allow to distinguish physical systems from each other. And again, we can ask: What is the complete list of intrinsic properties in nature? And does the universe have intrinsic Challenge 11 s properties? The various aspects of objects and of their states are called (physical) observables. We will refine this rough, preliminary definition in the following. Describing nature as a collection of permanent entities and changing states is the starting point of the study of motion. Every observation of motion requires the distinc- tion of permanent, intrinsic properties – describing the objects that move – and changing states – describing the way the objects move. Without this distinction, there is no motion. Without this distinction, there is not even a way to talk about motion. Using the terms just introduced, we can say ⊳ Motion is the change of state of permanent objects. The exact separation between those aspects belonging to the object, the permanent in- trinsic properties, and those belonging to the state, the varying state properties, depends on the precision of observation. For example, the length of a piece of wood is not per- manent; wood shrinks and bends with time, due to processes at the molecular level. To be precise, the length of a piece of wood is not an aspect of the object, but an aspect of its state. Precise observations thus shift the distinction between the object and its state; 1 why should we care about motion? 29 the distinction itself does not disappear – at least not in the first five volumes of our adventure. At the end of the twentieth century, neuroscience discovered that the distinction between changing states and permanent objects is not only made by scientists and en- gineers. Also nature makes the distinction. In fact, nature has hard-wired the distinction into the brain! Using the output signals from the visual cortex that processes what the eyes observe, the adjacent parts on the upper side of the human brain – the dorsal stream – process the state of the objects that are seen, such their distance and motion, whereas the adjacent parts on the lower side of the human brain – the ventral stream – process intrinsic properties, such as shape, colours and patterns. In summary, states are indeed required for the description of motion. So are perman- ent, intrinsic properties. In order to proceed and to achieve a complete description of motion, we thus need a complete description of the possible states and a complete de- scription of the intrinsic properties of objects. The first approach that attempts this is called Galilean physics; it starts by specifying our everyday environment and the motion in it as precisely as possible. Motion Mountain – The Adventure of Physics Galilean physics in six interesting statements The study of everyday motion, Galilean physics, is already worthwhile in itself: we will uncover many results that are in contrast with our usual experience. For example, if we recall our own past, we all have experienced how important, delightful or unwelcome surprises can be. Nevertheless, the study of everyday motion shows that there are no sur- prises in nature. Motion, and thus the world, is predictable or deterministic. The main surprise of our exploration of motion is that there are no surprises in nature. Nature is predictable. In fact, we will uncover six aspects of the predictability of everyday copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net motion: 1. Continuity. We know that eyes, cameras and measurement apparatus have a finite resolution. All have a smallest distance they can observe. We know that clocks have a smallest time they can measure. Despite these limitations, in everyday life all move- ments, their states, as well as space and time themselves, are continuous. 2. Conservation. We all observe that people, music and many other things in motion stop moving after a while. The study of motion yields the opposite result: motion never stops. In fact, three aspects of motion do not change, but are conserved: mo- mentum, angular momentum and energy are conserved, separately, in all examples of motion. No exception to these three types of conservation has ever been observed. (In contrast, mass is often, but not always conserved.) In addition, we will discover that conservation implies that motion and its properties are the same at all places and at all times: motion is universal. 3. Relativity. We all know that motion differs from rest. Despite this experience, careful study shows that there is no intrinsic difference between the two. Motion and rest depend on the observer. Motion is relative. And so is rest. This is the first step towards understanding the theory of relativity. 4. Reversibility. We all observe that many processes happen only in one direction. For example, spilled milk never returns into the container by itself. Despite such obser- 30 1 why should we care about motion? F I G U R E 12 A block and tackle and a differential pulley (left) and a farmer (right). Motion Mountain – The Adventure of Physics vations, the study of motion will show us that all everyday motion is reversible. Phys- icists call this the invariance of everyday motion under motion reversal. Sloppily, but incorrectly, one sometimes speaks of ‘time reversal’. 5. Mirror invariance. Most of us find scissors difficult to handle with the left hand, have difficulties to write with the other hand, and have grown with a heart on the left side. Despite such observations, our exploration will show that everyday motion is mirror- invariant (or parity-invariant). Mirror processes are always possible in everyday life. 6. Change minimization. We all are astonished by the many observations that the world copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net offers: colours, shapes, sounds, growth, disasters, happiness, friendship, love. The variation, beauty and complexity of nature is amazing. We will confirm that all ob- servations are due to motion. And despite the appearance of complexity, all motion is simple. Our study will show that all observations can be summarized in a simple way: Nature is lazy. All motion happens in a way that minimizes change. Change can be measured, using a quantity called ‘action’, and nature keeps it to a minimum. Situ- ations – or states, as physicists like to say – evolve by minimizing change. Nature is lazy. These six aspects are essential in understanding motion in sport, in music, in animals, in machines or among the stars. This first volume of our adventure will be an exploration of such movements. In particular, we will confirm, against all appearances of the contrary, the mentioned six key properties in all cases of everyday motion. Curiosities and fun challenges ab ou t motion* In contrast to most animals, sedentary creatures, like plants or sea anemones, have no legs and cannot move much; for their self-defence, they developed poisons. Examples of such plants are the stinging nettle, the tobacco plant, digitalis, belladonna and poppy; * Sections entitled ‘curiosities’ are collections of topics and problems that allow one to check and to expand the usage of concepts already introduced. 1 why should we care about motion? 31 poisons include caffeine, nicotine, and curare. Poisons such as these are at the basis of most medicines. Therefore, most medicines exist essentially because plants have no legs. ∗∗ A man climbs a mountain from 9 a.m. to 1 p.m. He sleeps on the top and comes down the next day, taking again from 9 a.m. to 1 p.m. for the descent. Is there a place on the Challenge 12 s path that he passes at the same time on the two days? ∗∗ Every time a soap bubble bursts, the motion of the surface during the burst is the same, Challenge 13 s even though it is too fast to be seen by the naked eye. Can you imagine the details? ∗∗ Challenge 14 s Is the motion of a ghost an example of motion? ∗∗ Motion Mountain – The Adventure of Physics Challenge 15 s Can something stop moving? How would you show it? ∗∗ Challenge 16 s Does a body moving forever in a straight line show that nature or space is infinite? ∗∗ What is the length of rope one has to pull in order to lift a mass by a height ℎ with a block Challenge 17 s and tackle with four wheels, as shown on the left of Figure 12? Does the farmer on the right of the figure do something sensible? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net In the past, block and tackles were important in many machines. Two particularly Ref. 25 useful versions are the differential block and tackle, also called a differential pulley, which is easy to make, and the Spanish burton, which has the biggest effect with the smallest Challenge 18 e number of wheels. There is even a so-called fool’s tackle. Enjoy their exploration. All these devices are examples of the golden rule of mechanics: what you gain in force, you loose in displacement. Or, equivalently: force times displacement – also called (phys- ical) work – remains unchanged, whatever mechanical device you may use. This is one example of conservation that is observed in everyday motion. ∗∗ Challenge 19 s Can the universe move? ∗∗ To talk about precision with precision, we need to measure precision itself. How would Challenge 20 s you do that? ∗∗ Challenge 21 s Would we observe motion if we had no memory? ∗∗ Challenge 22 s What is the lowest speed you have observed? Is there a lowest speed in nature? 32 1 why should we care about motion? ball v block perfectly flat table F I G U R E 13 What happens? F I G U R E 14 What is the speed of the rollers? Are other roller shapes possible? ∗∗ According to legend, Sissa ben Dahir, the Indian inventor of the game of chaturanga or chess, demanded from King Shirham the following reward for his invention: he wanted Motion Mountain – The Adventure of Physics one grain of wheat for the first square, two for the second, four for the third, eight for the fourth, and so on. How much time would all the wheat fields of the world take to Challenge 23 s produce the necessary grains? ∗∗ When a burning candle is moved, the flame lags behind the candle. How does the flame Challenge 24 s behave if the candle is inside a glass, still burning, and the glass is accelerated? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net A good way to make money is to build motion detectors. A motion detector is a small box with a few wires. The box produces an electrical signal whenever the box moves. What types of motion detectors can you imagine? How cheap can you make such a box? Challenge 25 d How precise? ∗∗ A perfectly frictionless and spherical ball lies near the edge of a perfectly flat and hori- Challenge 26 d zontal table, as shown in Figure 13. What happens? In what time scale? ∗∗ You step into a closed box without windows. The box is moved by outside forces un- Challenge 27 s known to you. Can you determine how you are moving from inside the box? ∗∗ When a block is rolled over the floor over a set of cylinders, as shown in Figure 14, how Challenge 28 s are the speed of the block and that of the cylinders related? ∗∗ Ref. 18 Do you dislike formulae? If you do, use the following three-minute method to change Challenge 29 s the situation. It is worth trying it, as it will make you enjoy this book much more. Life is short; as much of it as possible, like reading this text, should be a pleasure. 1 why should we care about motion? 33 1. Close your eyes and recall an experience that was absolutely marvellous, a situation when you felt excited, curious and positive. 2. Open your eyes for a second or two and look at page 280 – or any other page that contains many formulae. 3. Then close your eyes again and return to your marvellous experience. 4. Repeat the observation of the formulae and the visualization of your memory – steps 2 and 3 – three more times. Then leave the memory, look around yourself to get back into the here and now, and test yourself. Look again at page 280. How do you feel about formulae now? ∗∗ In the sixteenth century, Niccolò Tartaglia* proposed the following problem. Three young couples want to cross a river. Only a small boat that can carry two people is avail- able. The men are extremely jealous, and would never leave their brides with another Challenge 30 s man. How many journeys across the river are necessary? Motion Mountain – The Adventure of Physics ∗∗ Cylinders can be used to roll a flat object over the floor, as shown in Figure 14. The cylin- ders keep the object plane always at the same distance from the floor. What cross-sections other than circular, so-called curves of constant width, can a cylinder have to realize the Challenge 31 s same feat? How many examples can you find? Are objects different than cylinders pos- sible? ∗∗ Hanging pictures on a wall is not easy. First puzzle: what is the best way to hang a picture copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net on one nail? The method must allow you to move the picture in horizontal position after Challenge 32 s the nail is in the wall, in the case that the weight is not equally distributed. Second puzzle: Ref. 26 Can you hang a picture on a wall – this time with a long rope – over two nails in such a Challenge 33 s way that pulling either nail makes the picture fall? And with three nails? And 𝑛 nails? First summary on motion Motion, the change of position of physical systems, is the most fundamental observation in nature. Everyday motion is predictable and deterministic. Predictability is reflected in six aspects of motion: continuity, conservation, reversibility, mirror-invariance, relativity and minimization. Some of these aspects are valid for all motion, and some are valid only Challenge 34 d for everyday motion. Which ones, and why? We explore this now. * Niccolò Fontana Tartaglia (1499–1557), was an important Renaissance mathematician. Chapter 2 F ROM MOT ION M E A SU R E M E N T TO C ON T I N U I T Y “ Physic ist wahrlich das eigentliche Studium des ” Menschen.** Georg Christoph Lichtenberg T he simplest description of motion is the one we all, like cats or monkeys, use hroughout our everyday life: only one thing can be at a given spot at a given time. Motion Mountain – The Adventure of Physics his general description can be separated into three assumptions: matter is impen- etrable and moves, time is made of instants, and space is made of points. Without these Challenge 35 s three assumptions (do you agree with them?) it is not even possible to define velocity. We thus need points embedded in continuous space and time to talk about motion. This description of nature is called Galilean physics, or sometimes Newtonian physics. Galileo Galilei (1564–1642), Tuscan professor of mathematics, was the central founder of modern physics. He became famous for advocating the importance of observations as checks of statements about nature. By requiring and performing these checks through- out his life, he was led to continuously increase the accuracy in the description of mo- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net tion. For example, Galileo studied motion by measuring change of position with a self- constructed stopwatch. Galileo’s experimental aim was to measure all that is measurable about motion. His approach changed the speculative description of ancient Greece into the experimental physics of Renaissance Italy.*** After Galileo, the English alchemist, occultist, theologian, physicist and politician Isaac Newton (1643–1727) continued to explore with vigour the idea that different types of motion have the same properties, and he made important steps in constructing the concepts necessary to demonstrate this idea.**** Above all, the explorations and books by Galileo popularized the fundamental exper- imental statements on the properties of speed, space and time. ** ‘Physics truly is the proper study of man.’ Georg Christoph Lichtenberg (b. 1742 Ober-Ramstadt, d. 1799 Göttingen) was an important physicist and essayist. *** The best and most informative book on the life of Galileo and his times is by Pietro Redondi (see the section on page 335). Galileo was born in the year the pencil was invented. Before his time, it was impossible to do paper and pencil calculations. For the curious, the www.mpiwg-berlin.mpg.de website allows you to read an original manuscript by Galileo. **** Newton was born a year after Galileo died. For most of his life Newton searched for the philosopher’s stone. Newton’s hobby, as head of the English mint, was to supervise personally the hanging of counterfeit- Ref. 27 ers. About Newton’s lifelong infatuation with alchemy, see the books by Dobbs. A misogynist throughout his life, Newton believed himself to be chosen by god; he took his Latin name, Isaacus Neuutonus, and formed the anagram Jeova sanctus unus. About Newton and his importance for classical mechanics, see the Ref. 28 text by Clifford Truesdell. 2 from motion measurement to continuity 35 F I G U R E 15 Galileo Galilei (1564–1642). Motion Mountain – The Adventure of Physics F I G U R E 16 Some speed measurement devices: an anemometer, a tachymeter for inline skates, a sports radar gun and a Pitot–Prandtl tube in an aeroplane (© Fachhochschule Koblenz, Silva, Tracer, Wikimedia). What is velo cit y? “ ” There is nothing else like it. Jochen Rindt* copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Velocity fascinates. To physicists, not only car races are interesting, but any moving en- tity is. Therefore, physicists first measure as many examples as possible. A selection of measured speed values is given in Table 3. The units and prefixes used are explained in Page 453 detail in Appendix B. Some speed measurement devices are shown in Figure 16. Everyday life teaches us a lot about motion: objects can overtake each other, and they can move in different directions. We also observe that velocities can be added or changed smoothly. The precise list of these properties, as given in Table 4, is summarized by math- ematicians in a special term; they say that velocities form a Euclidean vector space.** More Page 81 details about this strange term will be given shortly. For now we just note that in describ- ing nature, mathematical concepts offer the most accurate vehicle. When velocity is assumed to be a Euclidean vector, it is called Galilean velocity. Ve- locity is a profound concept. For example, velocity does not need space and time meas- urements to be defined. Are you able to find a means of measuring velocities without * Jochen Rindt (1942–1970), famous Austrian Formula One racing car driver, speaking about speed. ** It is named after Euclid, or Eukleides, the great Greek mathematician who lived in Alexandria around 300 bce. Euclid wrote a monumental treatise of geometry, the Στοιχεῖα or Elements, which is one of the milestones of human thought. The text presents the whole knowledge of geometry of that time. For the first time, Euclid introduces two approaches that are now in common use: all statements are deduced from a small number of basic axioms and for every statement a proof is given. The book, still in print today, has been the reference geometry text for over 2000 years. On the web, it can be found at aleph0.clarku.edu/ ~djoyce/java/elements/elements.html. 36 2 from motion measurement to continuity TA B L E 3 Some measured velocity values. O b s e r va t i o n Ve l o c i t y Growth of deep sea manganese crust 80 am/s Can you find something slower? Challenge 36 s Stalagmite growth 0.3 pm/s Lichen growth down to 7 pm/s Typical motion of continents 10 mm/a = 0.3 nm/s Human growth during childhood, hair growth 4 nm/s Tree growth up to 30 nm/s Electron drift in metal wire 1 μm/s Sperm motion 60 to 160 μm/s Speed of light at Sun’s centre Ref. 29 1 mm/s Ketchup motion 1 mm/s Slowest speed of light measured in matter on Earth Ref. 30 0.3 m/s Motion Mountain – The Adventure of Physics Speed of snowflakes 0.5 m/s to 1.5 m/s Signal speed in human nerve cells Ref. 31 0.5 m/s to 120 m/s Wind speed at 1 and 12 Beaufort (light air and hurricane) < 1.5 m/s, > 33 m/s Speed of rain drops, depending on radius 2 m/s to 8 m/s Fastest swimming fish, sailfish (Istiophorus platypterus) 22 m/s 2009 Speed sailing record over 500 m (by trimaran Hydroptère) 26.4 m/s Fastest running animal, cheetah (Acinonyx jubatus) 30 m/s Speed of air in throat when sneezing 42 m/s Fastest throw: a cricket ball thrown with baseball technique while running 50 m/s copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Freely falling human, depending on clothing 50 to 90 m/s Fastest bird, diving Falco peregrinus 60 m/s Fastest badminton smash 70 m/s Average speed of oxygen molecule in air at room temperature 280 m/s Speed of sound in dry air at sea level and standard temperature 330 m/s Speed of the equator 434 m/s Cracking whip’s end 750 m/s Speed of a rifle bullet 1 km/s Speed of crack propagation in breaking silicon 5 km/s Highest macroscopic speed achieved by man – the Helios II satellite 70.2 km/s Speed of Earth through universe 370 km/s Average speed (and peak speed) of lightning tip 600 km/s (50 Mm/s) Highest macroscopic speed measured in our galaxy Ref. 32 0.97 ⋅ 108 m/s Speed of electrons inside a colour TV tube 1 ⋅ 108 m/s Speed of radio messages in space 299 792 458 m/s Highest ever measured group velocity of light 10 ⋅ 108 m/s Speed of light spot from a lighthouse when passing over the Moon 2 ⋅ 109 m/s Highest proper velocity ever achieved for electrons by man 7 ⋅ 1013 m/s Highest possible velocity for a light spot or a shadow no limit 2 from motion measurement to continuity 37 TA B L E 4 Properties of everyday – or Galilean – velocity. Ve l o c i t i e s Physical M at h e m at i c a l Definition can propert y name Be distinguished distinguishability element of set Vol. III, page 285 Change gradually continuum real vector space Page 80, Vol. V, page 364 Point somewhere direction vector space, dimensionality Page 80 Be compared measurability metricity Vol. IV, page 236 Be added additivity vector space Page 80 Have defined angles direction Euclidean vector space Page 81 Exceed any limit infinity unboundedness Vol. III, page 286 Challenge 37 d measuring space and time? If so, you probably want to skip to the next volume, jump- ing 2000 years of enquiries. If you cannot do so, consider this: whenever we measure a Motion Mountain – The Adventure of Physics quantity we assume that everybody is able to do so, and that everybody will get the same result. In other words, we define measurement as a comparison with a standard. We thus implicitly assume that such a standard exists, i.e., that an example of a ‘perfect’ velocity can be found. Historically, the study of motion did not investigate this question first, be- cause for many centuries nobody could find such a standard velocity. You are thus in good company. How is velocity measured in everyday life? Animals and people estimate their velo- city in two ways: by estimating the frequency of their own movements, such as their steps, or by using their eyes, ears, sense of touch or sense of vibration to deduce how copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net their own position changes with respect to the environment. But several animals have additional capabilities: certain snakes can determine speeds with their infrared-sensing organs, others with their magnetic field sensing organs. Still other animals emit sounds that create echoes in order to measure speeds to high precision. Other animals use the stars to navigate. A similar range of solutions is used by technical devices. Table 5 gives an overview. Velocity is not always an easy subject. Physicists like to say, provokingly, that what cannot be measured does not exist. Can you measure your own velocity in empty inter- Challenge 38 s stellar space? Velocity is of interest to both engineers and evolution scientist. In general, self- propelled systems are faster the larger they are. As an example, Figure 17 shows how this applies to the cruise speed of flying things. In general, cruise speed scales with the sixth root of the weight, as shown by the trend line drawn in the graph. (Can you find Challenge 39 d out why?) By the way, similar allometric scaling relations hold for many other properties of moving systems, as we will see later on. Some researchers have specialized in the study of the lowest velocities found in nature: Ref. 33 they are called geologists. Do not miss the opportunity to walk across a landscape while Challenge 40 e listening to one of them. Velocity is a profound subject for an additional reason: we will discover that all its seven properties of Table 4 are only approximate; none is actually correct. Improved ex- periments will uncover exceptions for every property of Galilean velocity. The failure of 38 2 from motion measurement to continuity 1 101 102 103 104 Airbus 380 wing load W/A [N/m2] Boeing 747 DC10 Concorde 106 Boeing 727 Boeing 737 Fokker F-28 F-14 Fokker F-27 MIG 23 F-16 105 Learjet 31 Beechcraft King Air Beechcraft Bonanza Beechcraft Baron Piper Warrior 104 Schleicher ASW33B Schleicher ASK23 Ultralight Quicksilver B 103 human-powered plane Skysurfer Motion Mountain – The Adventure of Physics Pteranodon 102 griffon vulture (Gyps fulvus) wandering albatross (Diomedea exulans) white-tailed eagle (Haliaeetus albicilla) whooper swan (Cygnus cygnus) white stork (Ciconia ciconia) graylag goose (Anser anser) black-backed gull (Larus marinus) cormorant (Phalacrocorax carbo) pheasant (Phasianus colchicus) herring gull (Larus argentatus) wild duck (Anas platyrhynchos) 10 peregrine falcon (Falco peregrinus) carrion craw (Corvus corone) coot (Fulica atra) weight W [N] barn owl (Tyto alba) moorhen (Gallinula chloropus) black headed gull (Larus ridibundus) copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net common tern (Sterna hirundo) 1 blackbird (Turdus merula) starling (Sturnus vulgaris) common swift (Apus Apus) ortolan bunting (Emberiza hortulana) sky lark (Alauda arvensis) house sparrow (Passer domesticus) barn swallow (Hirundo rustica) European Robin (Erithacus rubecula) great tit (Parus major) house martin (Delichon urbica) 10-1 winter wren (Troglodytes troglodytes) canary (Serinus canaria) goldcrest (Regulus Regulus) hummingbird (Trochilidae) privet hawkmoth (Sphinx ligustri) stag betle (Lucanus cervus) blue underwing (Catocala fraxini) sawyer beetle (Prionus coriarius) 10-2 yellow-striped dragonfly(S. flaveolum) cockchafer (Melolontha melolontha) eyed hawk-moth (S. ocellata) small stag beetle (Dorcus parallelopipedus) swallowtail (P. machaon) june bug (Amphimallon solstitialis) green dragonfly (Anax junius) garden bumble bee (Bombus hortorum) large white (P. brassicae) 10-3 common wasp (Vespa vulgaris) ant lion (Myrmeleo honey bee (Apis mellifera) formicarius) blowfly (Calliphora vicina) small white (P. rapae) crane fly (Tipulidae) scorpionfly (Panorpidae) damsel fly house fly (Musca domestica) 10-4 (Coenagrionidae) midge (Chironomidae) gnat (Culicidae) mosquito (Culicidae) 10-5 fruit fly (Drosophila melanogaster) 1 2 3 5 7 10 20 30 50 70 100 200 cruise speed at sea level v [m/s] F I G U R E 17 How wing load and sea-level cruise speed scales with weight in ﬂying objects, compared with the general trend line (after a graph © Henk Tennekes). 2 from motion measurement to continuity 39 TA B L E 5 Speed measurement devices in biological and engineered systems. Measurement Device Range Own running speed in insects, leg beat frequency measured 0 to 33 m/s mammals and humans with internal clock Own car speed tachymeter attached to 0 to 150 m/s wheels Predators and hunters measuring prey vision system 0 to 30 m/s speed Police measuring car speed radar or laser gun 0 to 90 m/s Bat measuring own and prey speed at doppler sonar 0 to 20 m/s night Sliding door measuring speed of doppler radar 0 to 3 m/s approaching people Own swimming speed in fish and friction and deformation of 0 to 30 m/s humans skin Motion Mountain – The Adventure of Physics Own swimming speed in dolphins and sonar to sea floor 0 to 20 m/s ships Diving speed in fish, animals, divers pressure change 0 to 5 m/s and submarines Water predators and fishing boats sonar 0 to 20 m/s measuring prey speed Own speed relative to Earth in insects often none (grasshoppers) n.a. Own speed relative to Earth in birds visual system 0 to 60 m/s Own speed relative to Earth in radio goniometry, radar 0 to 8000 m/s copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net aeroplanes or rockets Own speed relative to air in insects filiform hair deflection, 0 to 60 m/s and birds feather deflection Own speed relative to air in aeroplanes Pitot–Prandtl tube 0 to 340 m/s Wind speed measurement in thermal, rotating or 0 to 80 m/s meteorological stations ultrasound anemometers Swallows measuring prey speed visual system 0 to 20 m/s Bats measuring prey speed sonar 0 to 20 m/s Macroscopic motion on Earth Global Positioning System, 0 to 100 m/s Galileo, Glonass Pilots measuring target speed radar 0 to 1000 m/s Motion of stars optical Doppler effect 0 to 1000 km/s Motion of star jets optical Doppler effect 0 to 200 Mm/s the last three properties of Table 4 will lead us to special and general relativity, the failure of the middle two to quantum theory and the failure of the first two properties to the uni- fied description of nature. But for now, we’ll stick with Galilean velocity, and continue with another Galilean concept derived from it: time. 40 2 from motion measurement to continuity F I G U R E 18 A typical path followed by a stone thrown through the air – a parabola – with photographs Motion Mountain – The Adventure of Physics (blurred and stroboscopic) of a table tennis ball rebounding on a table (centre) and a stroboscopic photograph of a water droplet rebounding on a strongly hydrophobic surface (right, © Andrew Davidhazy, Max Groenendijk). “ Without the concepts place, void and time, change cannot be. [...] It is therefore clear [...] that their investigation has to be carried out, by ” studying each of them separately. Aristotle* Physics, Book III, part 1. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net What is time? “ ” Time is an accident of motion. Theophrastus** “ Time does not exist in itself, but only through the perceived objects, from which the concepts ” of past, of present and of future ensue. Lucretius,*** De rerum natura, lib. 1, v. 460 ss. In their first years of life, children spend a lot of time throwing objects around. The term ‘object’ is a Latin word meaning ‘that which has been thrown in front.’ Developmental Ref. 21 psychology has shown experimentally that from this very experience children extract the concepts of time and space. Adult physicists do the same when studying motion at university. When we throw a stone through the air, we can define a sequence of observations. Figure 18 illustrates how. Our memory and our senses give us this ability. The sense of * Aristotle (b. 384/3 Stageira, d. 322 bce Euboea), important Greek philosopher and scientist, founder of the Peripatetic school located at the Lyceum, a gymnasium dedicated to Apollo Lyceus. ** Theophrastus of Eresos (c. 371 – c. 287) was a revered Lesbian philosopher, and successor of Aristoteles at the Lyceum. *** Titus Lucretius Carus (c. 95 to c. 55 bce), Roman scholar and poet. 2 from motion measurement to continuity 41 TA B L E 6 Selected time measurements. O b s e r va t i o n Time Shortest measurable time 10−44 s Shortest time ever measured 10 ys Time for light to cross a typical atom 0.1 to 10 as Shortest laser light pulse produced so far 200 as Period of caesium ground state hyperfine transition 108.782 775 707 78 ps Beat of wings of fruit fly 1 ms Period of pulsar (rotating neutron star) PSR 1913+16 0.059 029 995 271(2) s Human ‘instant’ 20 ms Shortest lifetime of living being 0.3 d Average length of day 400 million years ago 79 200 s Average length of day today 86 400.002(1) s From birth to your 1000 million seconds anniversary 31.7 a Motion Mountain – The Adventure of Physics Age of oldest living tree 4600 a Use of human language 0.2 Ma Age of Himalayas 35 to 55 Ma Age of oldest rocks, found in Isua Belt, Greenland 3.8 Ga and in Porpoise Cove, Hudson Bay Age of Earth 4.6 Ga Age of oldest stars 13.8 Ga Age of most protons in your body 13.8 Ga Lifetime of tantalum nucleus 180𝑚 Ta 1015 a copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Lifetime of bismuth 209 Bi nucleus 1.9(2) ⋅ 1019 a hearing registers the various sounds during the rise, the fall and the landing of the stone. Our eyes track the location of the stone from one point to the next. All observations have their place in a sequence, with some observations preceding them, some observations simultaneous to them, and still others succeeding them. We say that observations are perceived to happen at various instants – also called ‘points in time’ – and we call the sequence of all instants time. An observation that is considered the smallest part of a sequence, i.e., not itself a sequence, is called an event. Events are central to the definition of time; in particular, Challenge 41 s starting or stopping a stopwatch are events. (But do events really exist? Keep this question in the back of your head as we move on.) Sequential phenomena have an additional property known as stretch, extension or duration. Some measured values are given in Table 6.* Duration expresses the idea that sequences take time. We say that a sequence takes time to express that other sequences can take place in parallel with it. How exactly is the concept of time, including sequence and duration, deduced from observations? Many people have looked into this question: astronomers, physicists, * A year is abbreviated a (Latin ‘annus’). 42 2 from motion measurement to continuity watchmakers, psychologists and philosophers. All find: ⊳ Time is deduced by comparing motions. This is even the case for children and animals. Beginning at a very young age, they Ref. 21 develop the concept of ‘time’ from the comparison of motions in their surroundings. Grown-ups take as a standard the motion of the Sun and call the resulting type of time local time. From the Moon they deduce a lunar calendar. If they take a particular village clock on a European island they call it the universal time coordinate (UTC), once known as ‘Greenwich mean time.’*Astronomers use the movements of the stars and call the res- ult ephemeris time (or one of its successors). An observer who uses his personal watch calls the reading his proper time; it is often used in the theory of relativity. Not every movement is a good standard for time. In the year 2000, an Earth rotation Page 456 did not take 86 400 seconds any more, as it did in the year 1900, but 86 400.002 seconds. Can you deduce in which year your birthday will have shifted by a whole day from the Challenge 43 s time predicted with 86 400 seconds? Motion Mountain – The Adventure of Physics All methods for the definition of time are thus based on comparisons of motions. In order to make the concept as precise and as useful as possible, a standard reference motion is chosen, and with it a standard sequence and a standard duration is defined. The device that performs this task is called a clock. We can thus answer the question of the section title: ⊳ Time is what we read from a clock. Note that all definitions of time used in the various branches of physics are equivalent to copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net this one; no ‘deeper’ or more fundamental definition is possible.** Note that the word ‘moment’ is indeed derived from the word ‘movement’. Language follows physics in this case. Astonishingly, the definition of time just given is final; it will never be changed, not even at the top of Motion Mountain. This is surprising at first sight, because many books have been written on the nature of time. Instead, they should investigate the nature of motion! ⊳ Every clock reminds us that in order to understand time, we need to under- stand motion. But this is the aim of our walk anyhow. We are thus set to discover all the secrets of time as a side result of our adventure. Time is not only an aspect of observations, it is also a facet of personal experience. Even in our innermost private life, in our thoughts, feelings and dreams, we experience sequences and durations. Children learn to relate this internal experience of time with * Official UTC is used to determine the phase of the power grid, phone and internet companies’ bit streams and the signal to the GPS system. The latter is used by many navigation systems around the world, especially in ships, aeroplanes and mobile phones. For more information, see the www.gpsworld.com website. The time-keeping infrastructure is also important for other parts of the modern economy. Can you spot the Challenge 42 s most important ones? Ref. 34 ** The oldest clocks are sundials. The science of making them is called gnomonics. 2 from motion measurement to continuity 43 TA B L E 7 Properties of Galilean time. I ns ta nt s o f t i m e Physical M at h e m at i c a l Definition propert y name Can be distinguished distinguishability element of set Vol. III, page 285 Can be put in order sequence order Vol. V, page 364 Define duration measurability metricity Vol. IV, page 236 Can have vanishing duration continuity denseness, completeness Vol. V, page 364 Allow durations to be added additivity metricity Vol. IV, page 236 Don’t harbour surprises translation invariance homogeneity Page 238 Don’t end infinity unboundedness Vol. III, page 286 Are equal for all observers absoluteness uniqueness external observations, and to make use of the sequential property of events in their ac- tions. Studies of the origin of psychological time show that it coincides – apart from its Motion Mountain – The Adventure of Physics lack of accuracy – with clock time.* Every living human necessarily uses in his daily life the concept of time as a combination of sequence and duration; this fact has been checked in numerous investigations. For example, the term ‘when’ exists in all human Ref. 36 languages. Time is a concept necessary to distinguish between observations. In any sequence of observations, we observe that events succeed each other smoothly, apparently without end. In this context, ‘smoothly’ means that observations that are not too distant tend to be not too different. Yet between two instants, as close as we can observe them, there is always room for other events. Durations, or time intervals, measured by different people copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net with different clocks agree in everyday life; moreover, all observers agree on the order of a sequence of events. Time is thus unique in everyday life. One also says that time is absolute in everyday life. Time is necessary to distinguish between observations. For this reason, all observing devices that distinguish between observations, from brains to dictaphones and video cameras, have internal clocks. In particular, all animal brains have internal clocks. These brain clocks allow their users to distinguish between present, recent and past data and observations. When Galileo studied motion in the seventeenth century, there were as yet no stop- watches. He thus had to build one himself, in order to measure times in the range Challenge 44 s between a fraction and a few seconds. Can you imagine how he did it? If we formulate with precision all the properties of time that we experience in our daily life, we are lead to Table 7. This concept of time is called Galilean time. All its prop- erties can be expressed simultaneously by describing time with the help of real numbers. In fact, real numbers have been constructed by mathematicians to have exactly the same Vol. III, page 295 properties as Galilean time, as explained in the chapter on the brain. In the case of Ga- Vol. V, page 42 * The brain contains numerous clocks. The most precise clock for short time intervals, the internal interval timer of the brain, is more accurate than often imagined, especially when trained. For time periods between Ref. 35 a few tenths of a second, as necessary for music, and a few minutes, humans can achieve timing accuracies of a few per cent. 44 2 from motion measurement to continuity lilean time, every instant of time can be described by a real number, often abbreviated 𝑡. The duration of a sequence of events is then given by the difference between the time values of the final and the starting event. We will have quite some fun with Galilean time in this part of our adventure. However, hundreds of years of close scrutiny have shown that every single property of Galilean time listed in Table 7 is approximate, and none is strictly correct. This story is told in the rest of our adventure. Clo cks “ The most valuable thing a man can spend is ” time. Theophrastus A clock is a moving system whose position can be read. There are many types of clocks: stopwatches, twelve-hour clocks, sundials, lunar clocks, seasonal clocks, etc. A few are shown in Figure 19. Most of these clock types are Motion Mountain – The Adventure of Physics Ref. 37 also found in plants and animals, as shown in Table 8. Ref. 38 Interestingly, there is a strict rule in the animal kingdom: large clocks go slow. How this happens is shown in Figure 20, another example of an allometric scaling ‘law’. A clock is a moving system whose position can be read. Of course, a precise clock is a system moving as regularly as possible, with as little outside disturbance as pos- sible. Clockmakers are experts in producing motion that is as regular as possible. We Page 181 will discover some of their tricks below. We will also explore, later on, the limits for the Vol. V, page 45 precision of clocks. Is there a perfect clock in nature? Do clocks exist at all? We will continue to study these questions throughout this work and eventually reach a surprising conclusion. At copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net this point, however, we state a simple intermediate result: since clocks do exist, somehow Challenge 45 s there is in nature an intrinsic, natural and ideal way to measure time. Can you see it? 2 from motion measurement to continuity 45 light from the Sun time read off : 11h00 CEST sub-solar poin t sub-solar point close to Mekk a close to Mekka Sun’s orbit sun's orbi t on May 15th on May 15 th mirror reflects the sunlight winter-spring display screen winter-spring display scree n time scale ring CEST Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 19 Different types of clocks: a high-tech sundial (size c. 30 cm), a naval pocket chronometer (size c. 6 cm), a caesium atomic clock (size c. 4 m), a group of cyanobacteria and the Galilean satellites of Jupiter (© Carlo Heller at www.heliosuhren.de, Anonymous, INMS, Wikimedia, NASA). 46 2 from motion measurement to continuity TA B L E 8 Examples of biological rhythms and clocks. Living being O s c i l l at i n g s ys t e m Period Sand hopper (Talitrus saltator) knows in which direction to flee from circadian the position of the Sun or Moon Human (Homo sapiens) gamma waves in the brain 0.023 to 0.03 s alpha waves in the brain 0.08 to 0.13 s heart beat 0.3 to 1.5 s delta waves in the brain 0.3 to 10 s blood circulation 30 s cellular circhoral rhythms 1 to 2 ks rapid-eye-movement sleep period 5.4 ks nasal cycle 4 to 9 ks growth hormone cycle 11 ks suprachiasmatic nucleus (SCN), 90 ks circadian hormone concentration, Motion Mountain – The Adventure of Physics temperature, etc.; leads to jet lag skin clock circadian monthly period 2.4(4) Ms built-in aging 3.2(3) Gs Common fly (Musca domestica) wing beat 30 ms Fruit fly (Drosophila wing beat for courting 34 ms melanogaster) Most insects (e.g. wasps, fruit winter approach detection (diapause) by yearly flies) length of day measurement; triggers copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net metabolism changes Algae (Acetabularia) Adenosinetriphosphate (ATP) concentration Moulds (e.g. Neurospora crassa) conidia formation circadian Many flowering plants flower opening and closing circadian Tobacco plant flower opening clock; triggered by annual length of days, discovered in 1920 by Garner and Allard Arabidopsis circumnutation circadian growth a few hours Telegraph plant (Desmodium side leaf rotation 200 s gyrans) Forsythia europaea, F. suspensa, Flower petal oscillation, discovered by 5.1 ks F. viridissima, F. spectabilis Van Gooch in 2002 2 from motion measurement to continuity 47 +8 Maximum lifespan of wild birds +7 +6 Reproductive maturity Growth-time in birds +5 Gestation time (max 100 cycles per lifetime) +4 Motion Mountain – The Adventure of Physics Log10 of time/min +3 Metabolism of fat, 0.1% of body mass (max 1 000 000 cycles per lifetime) +2 Sleep cycle Insulin clearance of body plasma volume +1 (max 3 000 000 cycles per lifetime) copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net 0 Circulation of blood volume (max 30 000 000 cycles per lifetime) -1 Respiratory cycles (max 200 000 000 cycles per lifetime) Gut contraction -2 (max 300 000 000 cycles per lifetime) Cardiac cycle (max 1 000 000 000 cycles per lifetime) -3 Fast muscle contraction (max 120 000 000 000 cycles per lifetime) -4 0.001 0.01 0.1 1 10 100 1000 Body mass/kg F I G U R E 20 How biological rhythms scale with size in mammals: all scale more or less with a quarter power of the mass (after data from the EMBO and Enrique Morgado). 48 2 from motion measurement to continuity Why d o clo cks go clo ckwise? “ ” Challenge 46 s What time is it at the North Pole now? Most rotational motions in our society, such as athletic races, horse, bicycle or ice skat- ing races, turn anticlockwise.* Mathematicians call this the positive rotation sense. Every supermarket leads its guests anticlockwise through the hall. Why? Most people are right- handed, and the right hand has more freedom at the outside of a circle. Therefore thou- sands of years ago chariot races in stadia went anticlockwise. As a result, all stadium races still do so to this day, and that is why runners move anticlockwise. For the same reason, helical stairs in castles are built in such a way that defending right-handers, usually from above, have that hand on the outside. On the other hand, the clock imitates the shadow of sundials; obviously, this is true on the northern hemisphere only, and only for sundials on the ground, which were the most common ones. (The old trick to determine south by pointing the hour hand of a horizontal watch to the Sun and halving the angle between it and the direction of 12 Motion Mountain – The Adventure of Physics o’clock does not work on the southern hemisphere – but there you can determine north in this way.) So every clock implicitly continues to state on which hemisphere it was invented. In addition, it also tells us that sundials on walls came in use much later than those on the floor. Does time flow? “ Wir können keinen Vorgang mit dem ‘Ablauf der Zeit’ vergleichen – diesen gibt es nicht –, sondern nur mit einem anderen Vorgang (etwa copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ” dem Gang des Chronometers).** Ludwig Wittgenstein, Tractatus, 6.3611 “ ” Si le temps est un fleuve, quel est son lit?*** The expression ‘the flow of time’ is often used to convey that in nature change follows after change, in a steady and continuous manner. But though the hands of a clock ‘flow’, time itself does not. Time is a concept introduced specially to describe the flow of events around us; it does not itself flow, it describes flow. Time does not advance. Time is neither linear nor cyclic. The idea that time flows is as hindering to understanding nature as is Vol. III, page 90 the idea that mirrors exchange right and left. The misleading use of the incorrect expression ‘flow of time’ was propagated first Ref. 39 by some flawed Greek thinkers and then again by Newton. And it still continues. Ar- istotle, careful to think logically, pointed out its misconception, and many did so after him. Nevertheless, expressions such as ‘time reversal’, the ‘irreversibility of time’, and the much-abused ‘time’s arrow’ are still common. Just read a popular science magazine * Notable exceptions are most, but not all, Formula 1 races. ** ‘We cannot compare any process with ‘the passage of time’ – there is no such thing – but only with another process (say, with the working of a chronometer).’ *** ‘If time is a river, what is his bed?’ 2 from motion measurement to continuity 49 Challenge 47 e chosen at random. The fact is: time cannot be reversed, only motion can, or more pre- cisely, only velocities of objects; time has no arrow, only motion has; it is not the flow of time that humans are unable to stop, but the motion of all the objects in nature. In- Ref. 40 credibly, there are even books written by respected physicists that study different types of ‘time’s arrows’ and compare them with each other. Predictably, no tangible or new result is extracted. ⊳ Time does not flow. Only bodies flow. Time has no direction. Motion has. For the same reason, colloquial expressions such as ‘the start (or end) of time’ should be avoided. A motion expert translates them straight away into ‘the start (or end) of motion’. What is space? “ The introduction of numbers as coordinates [...] ” is an act of violence [...]. Motion Mountain – The Adventure of Physics Hermann Weyl, Philosophie der Mathematik und Naturwissenschaft.* Whenever we distinguish two objects from each other, such as two stars, we first of all distinguish their positions. We distinguish positions with our senses of sight, touch, proprioception and hearing. Position is therefore an important aspect of the physical state of an object. A position is taken by only one object at a time. Positions are limited. The set of all available positions, called (physical) space, acts as both a container and a background. Closely related to space and position is size, the set of positions an object occupies. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Small objects occupy only subsets of the positions occupied by large ones. We will discuss Page 52 size in more detail shortly. How do we deduce space from observations? During childhood, humans (and most higher animals) learn to bring together the various perceptions of space, namely the visual, the tactile, the auditory, the kinaesthetic, the vestibular etc., into one self- consistent set of experiences and description. The result of this learning process is a certain concept of space in the brain. Indeed, the question ‘where?’ can be asked and answered in all languages of the world. Being more precise, adults derive space from dis- tance measurements. The concepts of length, area, volume, angle and solid angle are all deduced with their help. Geometers, surveyors, architects, astronomers, carpet salesmen and producers of metre sticks base their trade on distance measurements. ⊳ Space is formed from all the position and distance relations between objects using metre sticks. Humans developed metre sticks to specify distances, positions and sizes as accurately as possible. * Hermann Weyl (1885–1955) was one of the most important mathematicians of his time, as well as an important theoretical physicist. He was one of the last universalists in both fields, a contributor to quantum theory and relativity, father of the term ‘gauge’ theory, and author of many popular texts. 50 2 from motion measurement to continuity F I G U R E 21 Two proofs that space has three dimensions: the vestibular labyrinth in the inner ear of mammals (here a human) with three canals and a knot (© Northwestern University). Metre sticks work well only if they are straight. But when humans lived in the jungle, there were no straight objects around them. No straight rulers, no straight tools, noth- Motion Mountain – The Adventure of Physics Challenge 48 s ing. Today, a cityscape is essentially a collection of straight lines. Can you describe how humans achieved this? Once humans came out of the jungle with their newly built metre sticks, they collec- ted a wealth of results. The main ones are listed in Table 9; they are easily confirmed by personal experience. In particular, objects can take positions in an apparently continuous manner: there indeed are more positions than can be counted.* Size is captured by de- fining the distance between various positions, called length, or by using the field of view an object takes when touched, called its surface area. Length and area can be measured with the help of a metre stick. (Selected measurement results are given in Table 10; some copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net length measurement devices are shown in Figure 23.) The length of objects is independ- ent of the person measuring it, of the position of the objects and of their orientation. In daily life the sum of angles in any triangle is equal to two right angles. There are no limits to distances, lengths and thus to space. Experience shows us that space has three dimensions; we can define sequences of positions in precisely three independent ways. Indeed, the inner ear of (practically) all vertebrates has three semicircular canals that sense the body’s acceleration in the three dimensions of space, as shown in Figure 21.** Similarly, each human eye is moved by Challenge 49 s three pairs of muscles. (Why three?) Another proof that space has three dimensions is provided by shoelaces: if space had more than three dimensions, shoelaces would not be useful, because knots exist only in three-dimensional space. But why does space have three dimensions? This is one of the most difficult question of physics. We leave it open for the time being. It is often said that thinking in four dimensions is impossible. That is wrong. Just try. Challenge 50 s For example, can you confirm that in four dimensions knots are impossible? Like time intervals, length intervals can be described most precisely with the help * For a definition of uncountability, see page 288 in Volume III. ** Note that saying that space has three dimensions implies that space is continuous; the mathematician and philosopher Luitzen Brouwer (b. 1881 Overschie, d. 1966 Blaricum) showed that dimensionality is only a useful concept for continuous sets. 2 from motion measurement to continuity 51 TA B L E 9 Properties of Galilean space. Points, or Physical M at h e m at i c a l Defini- p o s i t i o n s i n s pa c e propert y name tion Can be distinguished distinguishability element of set Vol. III, page 285 Can be lined up if on one line sequence order Vol. V, page 364 Can form shapes shape topology Vol. V, page 363 Lie along three independent possibility of knots 3-dimensionality Page 81, Vol. IV, directions page 235 Can have vanishing distance continuity denseness, Vol. V, page 364 completeness Define distances measurability metricity Vol. IV, page 236 Allow adding translations additivity metricity Vol. IV, page 236 Define angles scalar product Euclidean space Page 81 Don’t harbour surprises translation invariance homogeneity Can beat any limit infinity unboundedness Vol. III, page 286 Motion Mountain – The Adventure of Physics Defined for all observers absoluteness uniqueness Page 52 of real numbers. In order to simplify communication, standard units are used, so that everybody uses the same numbers for the same length. Units allow us to explore the general properties of Galilean space experimentally: space, the container of objects, is continuous, three-dimensional, isotropic, homogeneous, infinite, Euclidean and unique – or ‘absolute’. In mathematics, a structure or mathematical concept with all the prop- erties just mentioned is called a three-dimensional Euclidean space. Its elements, (math- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ematical) points, are described by three real parameters. They are usually written as (𝑥, 𝑦, 𝑧) (1) and are called coordinates. They specify and order the location of a point in space. (For Page 81 the precise definition of Euclidean spaces, see below.) What is described here in just half a page actually took 2000 years to be worked out, mainly because the concepts of ‘real number’ and ‘coordinate’ had to be discovered first. The first person to describe points of space in this way was the famous mathem- atician and philosopher René Descartes*, after whom the coordinates of expression (1) are named Cartesian. Like time, space is a necessary concept to describe the world. Indeed, space is auto- matically introduced when we describe situations with many objects. For example, when many spheres lie on a billiard table, we cannot avoid using space to describe the relations between them. There is no way to avoid using spatial concepts when talking about nature. Even though we need space to talk about nature, it is still interesting to ask why this is possible. For example, since many length measurement methods do exist – some are * René Descartes or Cartesius (b. 1596 La Haye, d. 1650 Stockholm), mathematician and philosopher, au- thor of the famous statement ‘je pense, donc je suis’, which he translated into ‘cogito ergo sum’ – I think therefore I am. In his view, this is the only statement one can be sure of. 52 2 from motion measurement to continuity F I G U R E 22 René Descartes (1596 –1650). listed in Table 11 – and since they all yield consistent results, there must be a natural or Challenge 51 s ideal way to measure distances, sizes and straightness. Can you find it? As in the case of time, each of the properties of space just listed has to be checked. And again, careful observations will show that each property is an approximation. In simpler and more drastic words, all of them are wrong. This confirms Weyl’s statement at the beginning of this section. In fact, his statement about the violence connected with the Motion Mountain – The Adventure of Physics introduction of numbers is told by every forest in the world. The rest of our adventure will show this. “ ” Μέτρον ἄριστον.* Cleobulus Are space and time absolu te or relative? In everyday life, the concepts of Galilean space and time include two opposing aspects; the contrast has coloured every discussion for several centuries. On the one hand, space copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net and time express something invariant and permanent; they both act like big containers for all the objects and events found in nature. Seen this way, space and time have an ex- istence of their own. In this sense one can say that they are fundamental or absolute. On the other hand, space and time are tools of description that allow us to talk about rela- tions between objects. In this view, they do not have any meaning when separated from objects, and only result from the relations between objects; they are derived, relational Challenge 52 e or relative. Which of these viewpoints do you prefer? The results of physics have altern- ately favoured one viewpoint or the other. We will repeat this alternation throughout our Ref. 41 adventure, until we find the solution. And obviously, it will turn out to be a third option. Size – why length and area exist, bu t volume d oes not A central aspect of objects is their size. As a small child, under school age, every human learns how to use the properties of size and space in their actions. As adults seeking precision, with the definition of distance as the difference between coordinates allows us to define length in a reliable way. It took hundreds of years to discover that this is not the case. Several investigations in physics and mathematics led to complications. The physical issues started with an astonishingly simple question asked by Lewis * ‘Measure is the best (thing).’ Cleobulus (Κλεοβουλος) of Lindos, (c. 620–550 BCE ) was another of the proverbial seven sages. 2 from motion measurement to continuity 53 TA B L E 10 Some measured distance values. O b s e r va t i o n D i s ta nce Galaxy Compton wavelength 10−85 m (calculated only) Planck length, the shortest measurable length 10−35 m Proton diameter 1 fm Electron Compton wavelength 2.426 310 215(18) pm Smallest air oscillation detectable by human ear 11 pm Hydrogen atom size 30 pm Size of small bacterium 0.2 μm Wavelength of visible light 0.4 to 0.8 μm Radius of sharp razor blade 5 μm Point: diameter of smallest object visible with naked eye 20 μm Diameter of human hair (thin to thick) 30 to 80 μm Record diameter of hailstone 20 cm Motion Mountain – The Adventure of Physics Total length of DNA in each human cell 2m Longest human throw with any object, using a boomerang 427 m Highest human-built structure, Burj Khalifa 828 m Largest living thing, the fungus Armillaria ostoyae 3 km Largest spider webs in Mexico c. 5 km Length of Earth’s Equator 40 075 014.8(6) m Total length of human blood vessels (rough estimate) 4𝑡𝑜16 ⋅ 104 km Total length of human nerve cells (rough estimate) 1.5𝑡𝑜8 ⋅ 105 km Average distance to Sun 149 597 870 691(30) m copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Light year 9.5 Pm Distance to typical star at night 10 Em Size of galaxy 1 Zm Distance to Andromeda galaxy 28 Zm Most distant visible object 125 Ym Richardson:* How long is the western coastline of Britain? Following the coastline on a map using an odometer, a device shown in Figure 24, Richardson found that the length 𝑙 of the coastline depends on the scale 𝑠 (say 1 : 10 000 or 1 : 500 000) of the map used: 𝑙 = 𝑙0 𝑠0.25 (2) (Richardson found other exponentials for other coasts.) The number 𝑙0 is the length at scale 1 : 1. The main result is that the larger the map, the longer the coastline. What would happen if the scale of the map were increased even beyond the size of the original? The length would increase beyond all bounds. Can a coastline really have infinite length? Yes, it can. In fact, mathematicians have described many such curves; nowadays, they * Lewis Fray Richardson (1881–1953), English physicist and psychologist. 54 2 from motion measurement to continuity Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 23 Three mechanical (a vernier caliper, a micrometer screw, a moustache) and three optical (the eyes, a laser meter, a light curtain) length and distance measurement devices (© www. medien-werkstatt.de, Naples Zoo, Keyence, and Leica Geosystems). are called fractals. An infinite number of them exist, and Figure 25 shows one example.* * Most of these curves are self-similar, i.e., they follow scaling ‘laws’ similar to the above-mentioned. The term ‘fractal’ is due to the mathematician Benoît Mandelbrot and refers to a strange property: in a certain sense, they have a non-integral number 𝐷 of dimensions, despite being one-dimensional by construction. Mandelbrot saw that the non-integer dimension was related to the exponent 𝑒 of Richardson by 𝐷 = 1 + 𝑒, Ref. 42 thus giving 𝐷 = 1.25 in the example above. The number 𝐷 varies from case to case. Measurements yield a value 𝐷 = 1.14 for the land frontier of Portugal, 𝐷 = 1.13 for the Australian coast and 𝐷 = 1.02 for the South African coast. 2 from motion measurement to continuity 55 TA B L E 11 Length measurement devices in biological and engineered systems. Measurement Device Range Humans Measurement of body shape, e.g. finger muscle sensors 0.3 mm to 2 m distance, eye position, teeth distance Measurement of object distance stereoscopic vision 1 to 100 m Measurement of object distance sound echo effect 0.1 to 1000 m Animals Measurement of hole size moustache up to 0.5 m Measurement of walking distance by desert ants step counter up to 100 m Measurement of flight distance by honey bees eye up to 3 km Measurement of swimming distance by sharks magnetic field map up to 1000 km Measurement of prey distance by snakes infrared sensor up to 2 m Measurement of prey distance by bats, dolphins, sonar up to 100 m Motion Mountain – The Adventure of Physics and hump whales Measurement of prey distance by raptors vision 0.1 to 1000 m Machines Measurement of object distance by laser light reflection 0.1 m to 400 Mm Measurement of object distance by radar radio echo 0.1 to 50 km Measurement of object length interferometer 0.5 μm to 50 km Measurement of star, galaxy or quasar distance intensity decay up to 125 Ym Measurement of particle size accelerator down to 10−18 m copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 24 A curvimeter or odometer (photograph © Frank Müller). n=1 n=2 n=3 n=∞ F I G U R E 25 An example of a fractal: a self-similar curve of inﬁnite length (far right), and its construction. Challenge 53 e Can you construct another? Length has other strange properties. The mathematician Giuseppe Vitali was the first to discover that it is possible to cut a line segment of length 1 into pieces that can be reas- 56 2 from motion measurement to continuity sembled – merely by shifting them in the direction of the segment – into a line segment of length 2. Are you able to find such a division using the hint that it is only possible Challenge 54 d using infinitely many pieces? To sum up ⊳ Length exists. But length is well defined only for lines that are straight or nicely curved, but not for intricate lines, or for lines that can be cut into infinitely many pieces. We therefore avoid fractals and other strangely shaped curves in the following, and we take special care when we talk about infinitely small segments. These are the central as- sumptions in the first five volumes of this adventure, and we should never forget them! We will come back to these assumptions in the last part of our adventure. In fact, all these problems pale when compared with the following problem. Com- monly, area and volume are defined using length. You think that it is easy? You’re wrong, as well as being a victim of prejudices spread by schools around the world. To define area Motion Mountain – The Adventure of Physics and volume with precision, their definitions must have two properties: the values must be additive, i.e., for finite and infinite sets of objects, the total area and volume must be the sum of the areas and volumes of each element of the set; and the values must be ri- gid, i.e., if we cut an area or a volume into pieces and then rearrange the pieces, the value must remain the same. Are such definitions possible? In other words, do such concepts of volume and area exist? For areas in a plane, we proceed in the following standard way: we define the area 𝐴 of a rectangle of sides 𝑎 and 𝑏 as 𝐴 = 𝑎𝑏; since any polygon can be rearranged into a Challenge 55 s rectangle with a finite number of straight cuts, we can then define an area value for all copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net polygons. Subsequently, we can define the area for nicely curved shapes as the limit of Page 251 the sum of infinitely many polygons. This method is called integration; it is introduced in detail in the section on physical action. However, integration does not allow us to define area for arbitrarily bounded regions. Challenge 56 s (Can you imagine such a region?) For a complete definition, more sophisticated tools are needed. They were discovered in 1923 by the famous mathematician Stefan Banach.* He proved that one can indeed define an area for any set of points whatsoever, even if the border is not nicely curved but extremely complicated, such as the fractal curve previ- ously mentioned. Today this generalized concept of area, technically a ‘finitely additive isometrically invariant measure,’ is called a Banach measure in his honour. Mathem- aticians sum up this discussion by saying that since in two dimensions there is a Banach measure, there is a way to define the concept of area – an additive and rigid measure – for any set of points whatsoever.** In short, ⊳ Area exists. Area is well defined for plane and other nicely behaved surfaces, * Stefan Banach (b. 1892 Krakow, d. 1945 Lvov), important mathematician. ** Actually, this is true only for sets on the plane. For curved surfaces, such as the surface of a sphere, there are complications that will not be discussed here. In addition, the problems mentioned in the definition of length of fractals also reappear for area if the surface to be measured is not flat. A typical example is the area of the human lung: depending on the level of details examined, the area values vary from a few up to over a hundred square metres. 2 from motion measurement to continuity 57 dihedral angle F I G U R E 26 A polyhedron with one of its dihedral angles (© Luca Gastaldi). Motion Mountain – The Adventure of Physics but not for intricate shapes. What is the situation in three dimensions, i.e., for volume? We can start in the same way as for area, by defining the volume 𝑉 of a rectangular polyhedron with sides 𝑎, 𝑏, 𝑐 as 𝑉 = 𝑎𝑏𝑐. But then we encounter a first problem: a general polyhedron cannot be cut into a cube by straight cuts! The limitation was discovered in 1900 and 1902 by Max Dehn.* He found that the possibility depends on the values of the edge angles, or dihedral angles, as the mathematicians call them. (They are defined in Figure 26.) If one ascribes to every copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net edge of a general polyhedron a number given by its length 𝑙 times a special function 𝑔(𝛼) of its dihedral angle 𝛼, then Dehn found that the sum of all the numbers for all the edges of a solid does not change under dissection, provided that the function fulfils 𝑔(𝛼 + 𝛽) = 𝑔(𝛼) + 𝑔(𝛽) and 𝑔(π) = 0. An example of such a strange function 𝑔 is the one assigning the value 0 to any rational multiple of π and the value 1 to a basis set of irrational multiples of π. The values for all other dihedral angles of the polyhedron can then be constructed by combination of rational multiples of these basis angles. Using this Challenge 57 s function, you may then deduce for yourself that a cube cannot be dissected into a regular tetrahedron because their respective Dehn invariants are different.** Despite the problems with Dehn invariants, a rigid and additive concept of volume for polyhedra does exist, since for all polyhedra and, in general, for all ‘nicely curved’ shapes, the volume can be defined with the help of integration. Now let us consider general shapes and general cuts in three dimensions, not just the ‘nice’ ones mentioned so far. We then stumble on the famous Banach–Tarski theorem Ref. 43 (or paradox). In 1924, Stefan Banach and Alfred Tarski*** proved that it is possible to * Max Dehn (b. 1878 Hamburg, d. 1952 Black Mountain), mathematician, student of David Hilbert. ** This is also told in the beautiful book by M. Aigler & G. M. Z iegler, Proofs from the Book, Springer Verlag, 1999. The title is due to the famous habit of the great mathematician Paul Erdős to imagine that all beautiful mathematical proofs can be assembled in the ‘book of proofs’. *** Alfred Tarski (b. 1902 Warsaw, d. 1983 Berkeley), influential mathematician. 58 2 from motion measurement to continuity F I G U R E 27 Straight lines found in nature: cerussite (picture width approx. 3 mm, © Stephan Wolfsried) and selenite (picture width approx. 15 m, © Arch. Speleoresearch & Films/La Venta at www.laventa.it and www.naica.com.mx). cut one sphere into five pieces that can be recombined to give two spheres, each the size of the original. This counter-intuitive result is the Banach–Tarski theorem. Even worse, another version of the theorem states: take any two sets not extending to infinity and Motion Mountain – The Adventure of Physics containing a solid sphere each; then it is always possible to dissect one into the other with a finite number of cuts. In particular it is possible to dissect a pea into the Earth, or vice versa. Size does not count!* In short, volume is thus not a useful concept at all! The Banach–Tarski theorem raises two questions: first, can the result be applied to gold or bread? That would solve many problems. Second, can it be applied to empty Challenge 58 s space? In other words, are matter and empty space continuous? Both topics will be ex- plored later in our walk; each issue will have its own, special consequences. For the mo- ment, we eliminate this troubling issue by restricting our interest – again – to smoothly curved shapes (and cutting knives). With this restriction, volumes of matter and of empty copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net space do behave nicely: they are additive and rigid, and show no paradoxes.** Indeed, the cuts required for the Banach–Tarski paradox are not smooth; it is not possible to perform them with an everyday knife, as they require (infinitely many) infinitely sharp bends performed with an infinitely sharp knife. Such a knife does not exist. Nevertheless, we keep in the back of our mind that the size of an object or of a piece of empty space is a tricky quantity – and that we need to be careful whenever we talk about it. In summary, ⊳ Volume only exists as an approximation. Volume is well-defined only for regions with smooth surfaces. Volume does not exist in general, when in- finitely sharp cuts are allowed. We avoid strangely shaped volumes, surfaces and curves in the following, and we take special care when we talk about infinitely small entities. We can talk about length, area and volume only with this restriction. This avoidance is a central assumption in the first * The proof of the result does not need much mathematics; it is explained beautifully by Ian Stewart in Paradox of the spheres, New Scientist, 14 January 1995, pp. 28–31. The proof is based on the axiom of choice, Vol. III, page 286 which is presented later on. The Banach–Tarski paradox also exists in four dimensions, as it does in any Ref. 44 higher dimension. More mathematical detail can be found in the beautiful book by Stan Wagon. ** Mathematicians say that a so-called Lebesgue measure is sufficient in physics. This countably additive isometrically invariant measure provides the most general way to define a volume. 2 from motion measurement to continuity 59 Motion Mountain – The Adventure of Physics F I G U R E 28 A photograph of the Earth – seen from the direction of the Sun (NASA). five volumes of this adventure. Again: we should never forget these restrictions, even though they are not an issue in everyday life. We will come back to the assumptions at copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the end of our adventure. What is straight? When you see a solid object with a straight edge, it is a 99 %-safe bet that it is man- made. Of course, there are exceptions, as shown in Figure 27.* The largest crystals ever Ref. 46 found are 18 m in length. But in general, the contrast between the objects seen in a city – buildings, furniture, cars, electricity poles, boxes, books – and the objects seen in a forest – trees, plants, stones, clouds – is evident: in the forest no object is straight or flat, whereas in the city most objects are. Page 479 Any forest teaches us the origin of straightness; it presents tall tree trunks and rays of daylight entering from above through the leaves. For this reason we call a line straight if it touches either a plumb-line or a light ray along its whole length. In fact, the two defin- Challenge 60 s itions are equivalent. Can you confirm this? Can you find another definition? Obviously, we call a surface flat if for any chosen orientation and position the surface touches a plumb-line or a light ray along its whole extension. Challenge 59 ny * Why do crystals have straight edges? Another example of straight lines in nature, unrelated to atomic Page 429 structures, is the well-known Irish geological formation called the Giant’s Causeway. Other candidates that might come to mind, such as certain bacteria which have (almost) square or (almost) triangular shapes are Ref. 45 not counter-examples, as the shapes are only approximate. 60 2 from motion measurement to continuity Motion Mountain – The Adventure of Physics F I G U R E 29 A model illustrating the hollow Earth theory, showing how day and night appear (© Helmut Diehl). In summary, the concept of straightness – and thus also of flatness – is defined with copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the help of bodies or radiation. In fact, all spatial concepts, like all temporal concepts, require motion for their definition. A hollow E arth? Space and straightness pose subtle challenges. Some strange people maintain that all hu- mans live on the inside of a sphere; they call this the hollow Earth model. They claim that Ref. 47 the Moon, the Sun and the stars are all near the centre of the hollow sphere, as illustrated in Figure 29. They also explain that light follows curved paths in the sky and that when conventional physicists talk about a distance 𝑟 from the centre of the Earth, the real hol- Challenge 61 s low Earth distance is 𝑟he = 𝑅2Earth /𝑟. Can you show that this model is wrong? Roman Sexl* used to ask this question to his students and fellow physicists. The answer is simple: if you think you have an argument to show that the hollow Earth model is wrong, you are mistaken! There is no way of showing that such a view is wrong. It is possible to explain the horizon, the appearance of day and night, as well as the Challenge 62 e satellite photographs of the round Earth, such as Figure 28. To explain what happened during a flight to the Moon is also fun. A consistent hollow Earth view is fully equivalent to the usual picture of an infinitely extended space. We will come back to this problem Vol. II, page 285 in the section on general relativity. * Roman Sexl, (1939–1986), important Austrian physicist, author of several influential textbooks on gravit- ation and relativity. 2 from motion measurement to continuity 61 Curiosities and fun challenges ab ou t everyday space and time How does one measure the speed of a gun bullet with a stopwatch, in a space of 1 m3 , Challenge 63 s without electronics? Hint: the same method can also be used to measure the speed of light. ∗∗ For a striking and interactive way to zoom through all length scales in nature, from the Planck length to the size of the universe, see the website www.htwins.net/scale2/. ∗∗ Challenge 64 s What is faster: an arrow or a motorbike? ∗∗ Challenge 65 s Why are manholes always round? ∗∗ Motion Mountain – The Adventure of Physics Apart from the speed of light, another speed is important in nature: the speed of ear Ref. 48 growth. It was determined in published studies which show that on average, for old men, age 𝑡 and ear circumference 𝑒 are related by 𝑒 = 𝑡0.51 mm/a + 88.1 mm. In the units of the expression, ‘a’, from Latin ‘annus’, is the international abbreviation for ‘year’. ∗∗ Can you show to a child that the sum of the angles in a triangle equals two right angles? Challenge 66 e What about a triangle on a sphere or on a saddle? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ Do you own a glass whose height is larger than its maximum circumference? ∗∗ A gardener wants to plant nine trees in such a way that they form ten straight lines with Challenge 67 e three trees each. How does he do it? ∗∗ How fast does the grim reaper walk? This question is the title of a publication in the Challenge 68 d British Medial Journal from the year 2011. Can you imagine how it is answered? ∗∗ Time measurements require periodic phenomena. Tree rings are traces of the seasons. Glaciers also have such traces, the ogives. Similar traces are found in teeth. Do you know more examples? ∗∗ A man wants to know how many stairs he would have to climb if the escalator in front of him, which is running upwards, were standing still. He walks up the escalator and counts 60 stairs; walking down the same escalator with the same speed he counts 90 stairs. What 62 2 from motion measurement to continuity F I G U R E 30 At what height is a conical glass half full? Challenge 69 s is the answer? ∗∗ Motion Mountain – The Adventure of Physics You have two hourglasses: one needs 4 minutes and one needs 3 minutes. How can you Challenge 70 e use them to determine when 5 minutes are over? ∗∗ You have two fuses of different length that each take one minute to burn. You are not allowed to bend them nor to use a ruler. How do you determine that 45 s have gone by? Challenge 71 e Now the tougher puzzle: How do you determine that 10 s have gone by with a single fuse? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net You have three wine containers: a full one of 8 litres, an empty one of 5 litres, and another Challenge 72 e empty one of 3 litres. How can you use them to divide the wine evenly into two? ∗∗ Challenge 73 s How can you make a hole in a postcard that allows you to step through it? ∗∗ What fraction of the height of a conical glass, shown in Figure 30, must be filled to make Challenge 74 s the glass half full? ∗∗ Challenge 75 s How many pencils are needed to draw a line as long as the Equator of the Earth? ∗∗ Challenge 76 e Can you place five equal coins so that each one touches the other four? Is the stacking of two layers of three coins, each layer in a triangle, a solution for six coins? Why? What is the smallest number of coins that can be laid flat on a table so that every coin Challenge 77 e is touching exactly three other coins? ∗∗ Challenge 78 e Can you find three crossing points on a chessboard that lie on an equilateral triangle? 2 from motion measurement to continuity 63 rubber band F I G U R E 31 Can the snail reach the horse once it starts galloping away? ∗∗ The following bear puzzle is well known: A hunter leaves his home, walks 10 km to the South and 10 km to the West, shoots a bear, walks 10 km to the North, and is back home. Motion Mountain – The Adventure of Physics What colour is the bear? You probably know the answer straight away. Now comes the harder question, useful for winning money in bets. The house could be on several addi- tional spots on the Earth; where are these less obvious spots from which a man can have exactly the same trip (forget the bear now) that was just described and be at home again? Challenge 79 s ∗∗ Imagine a rubber band that is attached to a wall on one end and is attached to a horse at the other end, as shown in Figure 31. On the rubber band, near the wall, there is a snail. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Both the snail and the horse start moving, with typical speeds – with the rubber being Challenge 80 s infinitely stretchable. Can the snail reach the horse? ∗∗ For a mathematician, 1 km is the same as 1000 m. For a physicist the two are different! Indeed, for a physicist, 1 km is a measurement lying between 0.5 km and 1.5 km, whereas 1000 m is a measurement between 999.5 m and 1000.5 m. So be careful when you write down measurement values. The professional way is to write, for example, 1000(8) m to mean 1000 ± 8 m, i.e., a value that lies between 992 and 1008 m with a probability of Page 459 68.3 %. ∗∗ Imagine a black spot on a white surface. What is the colour of the line separating the spot Challenge 81 s from the background? This question is often called Peirce’s puzzle. ∗∗ Also bread is an (approximate) fractal, though an irregular one. The fractal dimension Challenge 82 s of bread is around 2.7. Try to measure it! ∗∗ Challenge 83 e How do you find the centre of a beer mat using paper and pencil? 64 2 from motion measurement to continuity ∗∗ How often in 24 hours do the hour and minute hands of a clock lie on top of each other? Challenge 84 s For clocks that also have a second hand, how often do all three hands lie on top of each other? ∗∗ Challenge 85 s How often in 24 hours do the hour and minute hands of a clock form a right angle? ∗∗ How many times in twelve hours can the two hands of a clock be exchanged with the Challenge 86 s result that the new situation shows a valid time? What happens for clocks that also have a third hand for seconds? ∗∗ Challenge 87 s How many minutes does the Earth rotate in one minute? Motion Mountain – The Adventure of Physics ∗∗ What is the highest speed achieved by throwing (with and without a racket)? What was Challenge 88 s the projectile used? ∗∗ A rope is put around the Earth, on the Equator, as tightly as possible. The rope is Challenge 89 s then lengthened by 1 m. Can a mouse slip under the rope? The original, tight rope is lengthened by 1 mm. Can a child slip under the rope? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Jack was rowing his boat on a river. When he was under a bridge, he dropped a ball into the river. Jack continued to row in the same direction for 10 minutes after he dropped the ball. He then turned around and rowed back. When he reached the ball, the ball had Challenge 90 s floated 600 m from the bridge. How fast was the river flowing? ∗∗ Adam and Bert are brothers. Adam is 18 years old. Bert is twice as old as at the time Challenge 91 e when Adam was the age that Bert is now. How old is Bert? ∗∗ ‘Where am I?’ is a common question; ‘When am I?’ is almost never asked, not even in Challenge 92 s other languages. Why? ∗∗ Challenge 93 s Is there a smallest time interval in nature? A smallest distance? ∗∗ Given that you know what straightness is, how would you characterize or define the Challenge 94 s curvature of a curved line using numbers? And that of a surface? 2 from motion measurement to continuity 65 1 5 10 1 5 10 F I G U R E 32 A 9-to-10 vernier/nonius/clavius and a 19-to-20 version (in fact, a 38-to-40 version) in a caliper (© www.medien-werkstatt.de). ∗∗ Challenge 95 s What is the speed of your eyelid? ∗∗ Motion Mountain – The Adventure of Physics The surface area of the human body is about 400 m2 . Can you say where this large num- Challenge 96 s ber comes from? ∗∗ How does a vernier work? It is called nonius in other languages. The first name is derived from a French military engineer* who did not invent it, the second is a play of words on the Latinized name of the Portuguese inventor of a more elaborate device** and the Latin word for ‘nine’. In fact, the device as we know it today – shown in Figure 32 – was designed around 1600 by Christophonius Clavius,*** the same astronomer whose copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net studies were the basis of the Gregorian calendar reform of 1582. Are you able to design a vernier/nonius/clavius that, instead of increasing the precision tenfold, does so by an Challenge 97 s arbitrary factor? Is there a limit to the attainable precision? ∗∗ Page 55 Fractals in three dimensions bear many surprises. Let us generalize Figure 25 to three dimensions. Take a regular tetrahedron; then glue on every one of its triangular faces a smaller regular tetrahedron, so that the surface of the body is again made up of many equal regular triangles. Repeat the process, glueing still smaller tetrahedrons to these new (more numerous) triangular surfaces. What is the shape of the final fractal, after an Challenge 98 s infinite number of steps? ∗∗ Motoring poses many mathematical problems. A central one is the following parallel parking challenge: what is the shortest distance 𝑑 from the car in front necessary to leave Challenge 99 s a parking spot without using reverse gear? (Assume that you know the geometry of your * Pierre Vernier (1580–1637), French military officer interested in cartography. ** Pedro Nuñes or Peter Nonnius (1502–1578), Portuguese mathematician and cartographer. *** Christophonius Clavius or Schlüssel (1537–1612), Bavarian astronomer, one of the main astronomers of his time. 66 2 from motion measurement to continuity 𝑑 𝑏 𝑤 𝐿 F I G U R E 33 Leaving a parking space. TA B L E 12 The exponential notation: how to write small and large numbers. Number Exponential Number Exponential n o tat i o n n o tat i o n 1 100 Motion Mountain – The Adventure of Physics 0.1 10−1 10 101 0.2 2 ⋅ 10−1 20 2 ⋅ 101 0.0324 3.24 ⋅ 10−2 32.4 3.24 ⋅ 101 0.01 10−2 100 102 0.001 10−3 1000 103 0.000 1 10−4 10 000 104 0.000 056 5.6 ⋅ 10−5 56 000 5.6 ⋅ 104 0.000 01 10−5 etc. 100 000 105 etc. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net car, as shown in Figure 33, and its smallest outer turning radius 𝑅, which is known for every car.) Next question: what is the smallest gap required when you are allowed to Challenge 100 s manoeuvre back and forward as often as you like? Now a problem to which no solution seems to be available in the literature: How does the gap depend on the number, 𝑛, of Challenge 101 s times you use reverse gear? (The author had offered 50 euro for the first well-explained solution; the winning solution by Daniel Hawkins is now found in the appendix.) ∗∗ Scientists use a special way to write large and small numbers, explained in Table 12. ∗∗ Ref. 49 In 1996 the smallest experimentally probed distance was 10−19 m, achieved between quarks at Fermilab. (To savour the distance value, write it down without the exponent.) Challenge 102 s What does this measurement mean for the continuity of space? ∗∗ Zeno, the Greek philosopher, discussed in detail what happens to a moving object at a given instant of time. To discuss with him, you decide to build the fastest possible shutter for a photographic camera that you can imagine. You have all the money you want. What 2 from motion measurement to continuity 67 𝑎 𝑟 𝐴 𝛼 Ω 𝑟 𝑎 𝛼= 𝑟 WF F I G U R E 34 The deﬁnition of plane and solid angles. is the shortest shutter time you would achieve? Motion Mountain – The Adventure of Physics Challenge 103 s ∗∗ Can you prove Pythagoras’ theorem by geometrical means alone, without using Challenge 104 s coordinates? (There are more than 30 possibilities.) ∗∗ Page 59 Why are most planets and moons, including ours, (almost) spherical (see, for example, Challenge 105 s Figure 28)? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ A rubber band connects the tips of the two hands of a clock. What is the path followed Challenge 106 s by the mid-point of the band? ∗∗ There are two important quantities connected to angles. As shown in Figure 34, what is usually called a (plane) angle is defined as the ratio between the lengths of the arc and the radius. A right angle is π/2 radian (or π/2 rad) or 90°. The solid angle is the ratio between area and the square of the radius. An eighth of a sphere is π/2 steradian or π/2 sr. (Mathematicians, of course, would simply leave out the steradian unit.) As a result, a small solid angle shaped like a cone and the angle of the Challenge 107 s cone tip are different. Can you find the relationship? ∗∗ The definition of angle helps to determine the size of a firework display. Measure the time 𝑇, in seconds, between the moment that you see the rocket explode in the sky and the moment you hear the explosion, measure the (plane) angle 𝛼 – pronounced ‘alpha’ – of the ball formed by the firework with your hand. The diameter 𝐷 is 6m 𝐷≈ 𝑇𝛼 . (3) 𝑠° 68 2 from motion measurement to continuity cosec cot cot c se c co cos se tan tan sin sin angle angle sec cos circle of radius 1 circle of radius 1 F I G U R E 35 Two equivalent deﬁnitions of the sine, cosine, tangent, cotangent, secant and cosecant of an angle. Motion Mountain – The Adventure of Physics sky yks copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net horizon noziroh earth htrae F I G U R E 36 How the apparent size of the Moon and the Sun changes during a day. Challenge 108 e Why? For more information about fireworks, see the cc.oulu.fi/~kempmp website. By the way, the angular distance between the knuckles of an extended fist are about 3°, 2° and 3°, the size of an extended hand 20°. Can you determine the other angles related to Challenge 109 s your hand? ∗∗ Using angles, the sine, cosine, tangent, cotangent, secant and cosecant can be defined, as shown in Figure 35. You should remember this from secondary school. Can you confirm Challenge 110 e that sin 15° = (√6 − √2 )/4, sin 18° = (−1 + √5 )/4, sin 36° = √10 − 2√5 /4, sin 54° = 2 from motion measurement to continuity 69 F I G U R E 37 How the size of the Moon actually changes during its orbit (© Anthony Ayiomamitis). Motion Mountain – The Adventure of Physics (1 + √5 )/4 and that sin 72° = √10 + 2√5 /4? Can you show also that sin 𝑥 𝑥 𝑥 𝑥 = cos cos cos ... (4) 𝑥 2 4 8 Challenge 111 e is correct? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Measuring angular size with the eye only is tricky. For example, can you say whether the Moon is larger or smaller than the nail of your thumb at the end of your extended arm? Challenge 112 e Angular size is not an intuitive quantity; it requires measurement instruments. A famous example, shown in Figure 36, illustrates the difficulty of estimating angles. Both the Sun and the Moon seem larger when they are on the horizon. In ancient times, Ptolemy explained this so-called Moon illusion by an unconscious apparent distance Ref. 50 change induced by the human brain. Indeed, the Moon illusion disappears when you look at the Moon through your legs. In fact, the Moon is even further away from the observer when it is just above the horizon, and thus its image is smaller than it was a few Challenge 113 s hours earlier, when it was high in the sky. Can you confirm this? The Moon’s angular size changes even more due to another effect: the orbit of the Moon round the Earth is elliptical. An example of the consequence is shown in Figure 37. ∗∗ Galileo also made mistakes. In his famous book, the Dialogues, he says that the curve formed by a thin chain hanging between two nails is a parabola, i.e., the curve defined Challenge 114 d by 𝑦 = 𝑥2 . That is not correct. What is the correct curve? You can observe the shape (approximately) in the shape of suspension bridges. ∗∗ Draw three circles, of different sizes, that touch each other, as shown in Figure 38. Now 70 2 from motion measurement to continuity A O C B F I G U R E 38 A famous puzzle: how are the F I G U R E 39 What is the area ABC, four radii related? given the other three areas and three right angles at O? Motion Mountain – The Adventure of Physics wall blue ladder of length b ladder of red ladder length l of length r wall wall height h? F I G U R E 40 square Two ladder copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net box of puzzles: a side b height h moderately difﬁcult (left) and a difﬁcult one distance d ? (right). draw a fourth circle in the space between, touching the outer three. What simple relation Challenge 115 s do the inverse radii of the four circles obey? ∗∗ Take a tetrahedron OABC whose triangular sides OAB, OBC and OAC are rectangular in O, as shown in Figure 39. In other words, the edges OA, OB and OC are all perpendicular to each other. In the tetrahedron, the areas of the triangles OAB, OBC and OAC are Challenge 116 s respectively 8, 4 and 1. What is the area of triangle ABC? ∗∗ Ref. 51 There are many puzzles about ladders. Two are illustrated in Figure 40. If a 5 m ladder is put against a wall in such a way that it just touches a box with 1 m height and depth, Challenge 117 s how high does the ladder reach? If two ladders are put against two facing walls, and if 2 from motion measurement to continuity 71 F I G U R E 41 Anticrepuscular rays - where is the Sun in this situation? (© Peggy Peterson) the lengths of the ladders and the height of the crossing point are known, how distant Challenge 118 d are the walls? Motion Mountain – The Adventure of Physics ∗∗ With two rulers, you can add and subtract numbers by lying them side by side. Are you Challenge 119 s able to design rulers that allow you to multiply and divide in the same manner? ∗∗ How many days would a year have if the Earth turned the other way with the same ro- Challenge 120 s tation frequency? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 121 s The Sun is hidden in the spectacular situation shown in Figure 41 Where is it? ∗∗ A slightly different, but equally fascinating situation – and useful for getting used to per- spective drawing – appears when you have a lighthouse in your back. Can you draw the Challenge 122 e rays you see in the sky up to the horizon? ∗∗ Two cylinders of equal radius intersect at a right angle. What is the value of the intersec- Challenge 123 s tion volume? (First make a drawing.) ∗∗ Two sides of a hollow cube with side length 1 dm are removed, to yield a tunnel with square opening. Is it true that a cube with edge length 1.06 dm can be made to pass Challenge 124 s through the hollow cube with side length 1 dm? ∗∗ Ref. 52 Could a two-dimensional universe exist? Alexander Dewdney imagined such a universe in great detail and wrote a well-known book about it. He describes houses, the trans- portation system, digestion, reproduction and much more. Can you explain why a two- 72 2 from motion measurement to continuity F I G U R E 42 Ideal conﬁgurations of ropes made of two, three and four strands. In the ideal conﬁguration, the speciﬁc pitch angle relative to the equatorial plane – 39.4°, 42.8° and 43.8°, respectively – leads to zero-twist structures. In these ideal conﬁgurations, the rope will neither rotate in Motion Mountain – The Adventure of Physics one nor in the other direction under vertical strain (© Jakob Bohr). Challenge 125 d dimensional universe is impossible? ∗∗ Ropes are wonderful structures. They are flexible, they are helically woven, but despite this, they do not unwind or twist, they are almost inextensible, and their geometry de- Ref. 53 pends little on the material used in making them. What is the origin of all these proper- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ties? Laying rope is an old craft; it is based on a purely geometric result: among all possible helices of 𝑛 strands of given length laid around a central structure of fixed radius, there is one helix for which the number of turns is maximal. For purely geometric reasons, ropes with that specific number of turns and the corresponding inner radius have the mentioned properties that make ropes so useful. The geometries of ideal ropes made of two, three and four strands are shown in Figure 42. ∗∗ Challenge 126 s Some researchers are investigating whether time could be two-dimensional. Can this be? ∗∗ Other researchers are investigating whether space could have more than three dimen- Challenge 127 s sions. Can this be? ∗∗ One way to compare speeds of animals and machines is to measure them in ‘body lengths per second’. The click beetle achieves a value of around 2000 during its jump phase, certain Archaea (bacteria-like) cells a value of 500, and certain hummingbirds 380. These are the record-holders so far. Cars, aeroplanes, cheetahs, falcons, crabs, and all other Ref. 54 motorized systems are much slower. 2 from motion measurement to continuity 73 F I G U R E 43 An open research problem: What is the ropelength of a tight knot? (© Piotr Pieranski, from Ref. 55) ∗∗ Challenge 128 e Why is the cross section of a tube usually circular? ∗∗ What are the dimensions of an open rectangular water tank that contains 1 m3 of water Challenge 129 e and uses the smallest amount of wall material? Motion Mountain – The Adventure of Physics ∗∗ Draw a square consisting of four equally long connecting line segments hinged at the vertices. Such a structure may be freely deformed into a rhombus if some force is applied. How many additional line interlinks of the same length must be supplemented to avoid this freedom and to prevent the square from being deformed? The extra line interlinks must be in the same plane as the square and each one may only be pegged to others at Challenge 130 s the endpoints. ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Area measurements can be difficult. In 2014 it became clear that the area of the gastro- intestinal tract of adult health humans is between 30 and 40 m2 . For many years, the mistaken estimate for the area was between 180 and 300 m2 . ∗∗ If you never explored plane geometry, do it once in your life. An excellent introduc- tion is Claudi Alsina & Ro ger B. Nelsen, Icons of Mathematics: An Exploration of Twenty Key Images, MAA Press, 2011. This is a wonderful book with many simple and surprising facts about geometry that are never told in school or university. You will enjoy it. ∗∗ Triangles are full of surprises. Together, Leonhard Euler, Charles Julien Brianchon and Jean Victor Poncelet discovered that in any triangle, nine points lie on the same circle: the midpoints of the sides, the feet of the altitude lines, and the midpoints of the altitude segments connecting each vertex to the orthocenter. Euler also discovered that in every triangle, the orthocenter, the centroid, the circumcenter and the center of the nine-point- circle lie on the same line, now called the Euler line. For the most recent recearch results on plane triangles, see the wonderful Encyclopedia of Triangle Centers, available at faculty.evansville.edu/ck6/encyclopedia/ETC.html. 74 2 from motion measurement to continuity ∗∗ Here is a simple challenge on length that nobody has solved yet. Take a piece of ideal rope: of constant radius, ideally flexible, and completely slippery. Tie a tight knot into it, as shown in Figure 43. By how much did the two ends of the rope come closer together? Challenge 131 r Summary ab ou t everyday space and time Motion defines speed, time and length. Observations of everyday life and precision ex- periments are conveniently and precisely described by describing velocity as a vector, space as three-dimensional set of points, and time as a one-dimensional real line, also made of points. These three definitions form the everyday, or Galilean, description of our environment. Modelling velocity, time and space as continuous quantities is precise and convenient. The modelling works during most of the adventures that follows. However, this common model of space and time cannot be confirmed by experiment. For example, no experi- ment can check distances larger than 1025 m or smaller than 10−25 m; the continuum Motion Mountain – The Adventure of Physics model is likely to be incorrect at smaller scales. We will find out in the last part of our adventure that this is indeed the case. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Chapter 3 HOW TO DE S C R I BE MOT ION – K I N E M AT IC S “ La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi agli occhi (io dico l’universo) ... Egli è scritto in ” lingua matematica.** Galileo Galilei, Il saggiatore VI. Motion Mountain – The Adventure of Physics E xperiments show that the properties of motion, time and space are xtracted from the environment both by children and animals. This xtraction has been confirmed for cats, dogs, rats, mice, ants and fish, among others. They all find the same results. First of all, motion is change of position with time. This description is illustrated by rap- idly flipping the lower left corners of this book, starting at page 242. Each page simulates an instant of time, and the only change that takes place during motion is in the position of the object, say a stone, represented by the dark spot. The other variations from one picture to the next, which are due to the imperfections of printing techniques, can be taken to simulate the inevitable measurement errors. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Stating that ‘motion is the change of position with time’ is neither an explanation nor a definition, since both the concepts of time and position are deduced from motion itself. It is only a description of motion. Still, the statement is useful, because it allows for high precision, as we will find out by exploring gravitation and electrodynamics. After all, pre- cision is our guiding principle during this promenade. Therefore the detailed description of changes in position has a special name: it is called kinematics. The idea of change of positions implies that the object can be followed during its mo- tion. This is not obvious; in the section on quantum theory we will find examples where this is impossible. But in everyday life, objects can always be tracked. The set of all pos- itions taken by an object over time forms its path or trajectory. The origin of this concept Ref. 56 is evident when one watches fireworks or again the flip film in the lower left corners starting at page 242. In everyday life, animals and humans agree on the Euclidean properties of velocity, space and time. In particular, this implies that a trajectory can be described by specify- ing three numbers, three coordinates (𝑥, 𝑦, 𝑧) – one for each dimension – as continuous ** Science is written in this huge book that is continuously open before our eyes (I mean the universe) ... It is written in mathematical language. 76 3 how to describe motion – kinematics collision F I G U R E 44 Two ways to test that the time of free fall does not depend on horizontal velocity. Vol. III, page 289 functions of time 𝑡. (Functions are defined in detail later on.) This is usually written as Motion Mountain – The Adventure of Physics 𝑥 = 𝑥(𝑡) = (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) . (5) For example, already Galileo found, using stopwatch and ruler, that the height 𝑧 of any thrown or falling stone changes as 𝑧(𝑡) = 𝑧0 + 𝑣𝑧0 (𝑡 − 𝑡0 ) − 12 𝑔 (𝑡 − 𝑡0 )2 (6) where 𝑡0 is the time the fall starts, 𝑧0 is the initial height, 𝑣𝑧0 is the initial velocity in the vertical direction and 𝑔 = 9.8 m/s2 is a constant that is found to be the same, within copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net about one part in 300, for all falling bodies on all points of the surface of the Earth. Ref. 57 Where do the value 9.8 m/s2 and its slight variations come from? A preliminary answer will be given shortly, but the complete elucidation will occupy us during the larger part of this hike. The special case with no initial velocity is of great interest. Like a few people before him, Galileo made it clear that 𝑔 is the same for all bodies, if air resistance can be neg- Page 202 lected. He had many arguments for this conclusion; can you find one? And of course, his famous experiment at the leaning tower in Pisa confirmed the statement. (It is a false Ref. 58 urban legend that Galileo never performed the experiment. He did it.) Equation (6) therefore allows us to determine the depth of a well, given the time a Challenge 132 s stone takes to reach its bottom. The equation also gives the speed 𝑣 with which one hits the ground after jumping from a tree, namely 𝑣 = √2𝑔ℎ . (7) A height of 3 m yields a velocity of 27 km/h. The velocity is thus proportional only to the square root of the height. Does this mean that one’s strong fear of falling results from an Challenge 133 s overestimation of its actual effects? Galileo was the first to state an important result about free fall: the motions in the horizontal and vertical directions are independent. He showed that the time it takes for 3 how to describe motion – kinematics 77 space-time configuration hodograph phase space diagrams space graph 𝑧 𝑧 𝑣𝑧 𝑚𝑣𝑧 𝑡 𝑥 v𝑥 𝑧 𝑥 𝑚𝑣𝑥 𝑡 𝑥 Motion Mountain – The Adventure of Physics F I G U R E 45 Various types of graphs describing the same path of a thrown stone. a cannon ball that is shot exactly horizontally to fall is independent of the strength of the gunpowder, as shown in Figure 44. Many great thinkers did not agree with this statement Ref. 59 even after his death: in 1658 the Academia del Cimento even organized an experiment to check this assertion, by comparing the flying cannon ball with one that simply fell Challenge 134 s vertically. Can you imagine how they checked the simultaneity? Figure 44 shows how you can check this at home. In this experiment, whatever the spring load of the cannon, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the two bodies will always collide in mid-air (if the table is high enough), thus proving the assertion. In other words, a flying cannon ball is not accelerated in the horizontal direction. Its horizontal motion is simply unchanging – as long as air resistance is negligible. By extending the description of equation (6) with the two expressions for the horizontal coordinates 𝑥 and 𝑦, namely 𝑥(𝑡) = 𝑥0 + 𝑣x0 (𝑡 − 𝑡0 ) 𝑦(𝑡) = 𝑦0 + 𝑣y0 (𝑡 − 𝑡0 ) , (8) a complete description for the path followed by thrown stones results. A path of this shape Page 40 is called a parabola; it is shown in Figures 18, 44 and 45. (A parabolic shape is also used Challenge 135 s for light reflectors inside pocket lamps or car headlights. Can you show why?) Ref. 60 Physicists enjoy generalizing the idea of a path. As Figure 45 shows, a path is a trace left in a diagram by a moving object. Depending on what diagram is used, these paths have different names. Space-time diagrams are useful to make the theory of relativity ac- cessible. The configuration space is spanned by the coordinates of all particles of a system. For many particles, it has a high number of dimensions and plays an important role in Page 415 self-organization. The difference between chaos and order can be described as a differ- ence in the properties of paths in configuration space. Hodographs, the paths in ‘velocity 78 3 how to describe motion – kinematics space’, are used in weather forecasting. The phase space diagram is also called state space diagram. It plays an essential role in thermodynamics. Throwing , jumping and sho oting The kinematic description of motion is useful for answering a whole range of questions. ∗∗ What is the upper limit for the long jump? The running peak speed world record Ref. 61 in 2019 was over 12.5 m/s ≈ 45 km/h by Usain Bolt, and the 1997 women’s record Ref. 62 was 11 m/s ≈ 40 km/h. However, male long jumpers never run much faster than about 9.5 m/s. How much extra jump distance could they achieve if they could run full speed? How could they achieve that? In addition, long jumpers take off at angles of about 20°, Ref. 63 as they are not able to achieve a higher angle at the speed they are running. How much Challenge 136 s would they gain if they could achieve 45°? Is 45° the optimal angle? ∗∗ Motion Mountain – The Adventure of Physics Why was basketball player Dirk Nowitzki so successful? His trainer Holger Geschwind- ner explained him that a throw is most stable against mistakes when it falls into the basket at around 47 degrees from the horizontal. He further told Nowitzki that the ball flies in a plane, and that therefore the arms should also move in that plane only. And he explained that when the ball leaves the hand, it should roll over the last two fingers like a train moves on rails. Using these criteria to check and to improve Nowitzki’s throws, he made him into one of the best basket ball throwers in the world. ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net What do the athletes Usain Bolt and Michael Johnson, the last two world record holders on the 200 m race at time of this writing, have in common? They were tall, athletic, and had many fast twitch fibres in the muscles. These properties made them good sprinters. A last difference made them world-class sprinters: they had a flattened spine, with almost no S-shape. This abnormal condition saves them a little bit of time at every step, because their spine is not as flexible as in usual people. This allows them to excel at short-distance races. ∗∗ Athletes continuously improve speed records. Racing horses do not. Why? For racing horses, breathing rhythm is related to gait; for humans, it is not. As a result, racing horses cannot change or improve their technique, and the speed of racing horses is essentially the same since it is measured. ∗∗ What is the highest height achieved by a human throw of any object? What is the longest Challenge 137 s distance achieved by a human throw? How would you clarify the rules? Compare the results with the record distance with a crossbow, 1, 871.8 m, achieved in 1988 by Harry Drake, the record distance with a footbow, 1854.4 m, achieved in 1971 also by Harry Drake, and with a hand-held bow, 1, 222.0 m, achieved in 1987 by Don Brown. 3 how to describe motion – kinematics 79 F I G U R E 46 Three superimposed images of a frass pellet shot away by a caterpillar inside a rolled-up leaf (© Stanley Caveney). Motion Mountain – The Adventure of Physics ∗∗ Challenge 138 s How can the speed of falling rain be measured using an umbrella? The answer is import- ant: the same method can also be used to measure the speed of light, as we will find out Vol. II, page 17 later. (Can you guess how?) ∗∗ When a dancer jumps in the air, how many times can he or she rotate around his or her Challenge 139 s vertical axis before arriving back on earth? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ Numerous species of moth and butterfly caterpillars shoot away their frass – to put it Ref. 64 more crudely: their faeces – so that its smell does not help predators to locate them. Stan- ley Caveney and his team took photographs of this process. Figure 46 shows a caterpillar (yellow) of the skipper Calpodes ethlius inside a rolled up green leaf caught in the act. Given that the record distance observed is 1.5 m (though by another species, Epargyreus Challenge 140 s clarus), what is the ejection speed? How do caterpillars achieve it? ∗∗ What is the horizontal distance one can reach by throwing a stone, given the speed and Challenge 141 s the angle from the horizontal at which it is thrown? ∗∗ Challenge 142 s What is the maximum numbers of balls that could be juggled at the same time? ∗∗ Is it true that rain drops would kill if it weren’t for the air resistance of the atmosphere? Challenge 143 s What about hail? ∗∗ 80 3 how to describe motion – kinematics 0.001 0.01 0.1 1 10 antilope cat leopard tiger 20 W/kg lesser dog human horse 1 galago Height of jump [m] locusts and grasshoppers 0.1 fleas standing jumps running jumps 0.01 elephant 0.001 0.01 0.1 1 10 Length of animal [m] Motion Mountain – The Adventure of Physics F I G U R E 47 The height achieved by jumping land animals. Challenge 144 s Are bullets, fired into the air from a gun, dangerous when they fall back down? ∗∗ Police finds a dead human body at the bottom of cliff with a height of 30 m, at a distance Challenge 145 s of 12 m from the cliff. Was it suicide or murder? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ Ref. 65 All land animals, regardless of their size, achieve jumping heights of at most 2.2 m, as Challenge 146 s shown in Figure 47. The explanation of this fact takes only two lines. Can you find it? The last two issues arise because the equation (6) describing free fall does not hold in all cases. For example, leaves or potato crisps do not follow it. As Galileo already knew, this is a consequence of air resistance; we will discuss it shortly. Because of air resistance, the path of a stone is not a parabola. In fact, there are other situations where the path of a falling stone is not a parabola, Challenge 147 s even without air resistance. Can you find one? Enjoying vectors Physical quantities with a defined direction, such as speed, are described with three num- bers, or three components, and are called vectors. Learning to calculate with such multi- component quantities is an important ability for many sciences. Here is a summary. Vectors can be pictured by small arrows. Note that vectors do not have specified points at which they start: two arrows with same direction and the same length are the same vector, even if they start at different points in space. Since vectors behave like arrows, vectors can be added and they can be multiplied by numbers. For example, stretching an arrow 𝑎 = (𝑎𝑥 , 𝑎𝑦 , 𝑎𝑧 ) by a number 𝑐 corresponds, in component notation, to the vector 𝑐𝑎 = (𝑐𝑎𝑥 , 𝑐𝑎𝑦 , 𝑐𝑎𝑧 ). 3 how to describe motion – kinematics 81 In precise, mathematical language, a vector is an element of a set, called vector space, in which the following properties hold for all vectors 𝑎 and 𝑏 and for all numbers 𝑐 and 𝑑: 𝑐(𝑎 + 𝑏) = 𝑐𝑎 + 𝑐𝑏 , (𝑐 + 𝑑)𝑎 = 𝑐𝑎 + 𝑑𝑎 , (𝑐𝑑)𝑎 = 𝑐(𝑑𝑎) and 1𝑎 = 𝑎 . (9) Examples of vector spaces are the set of all positions of an object, or the set of all its Challenge 148 s possible velocities. Does the set of all rotations form a vector space? All vector spaces allow defining a unique null vector and a unique negative vector for Challenge 149 e each vector. In most vector spaces of importance when describing nature the concept of length – specifying the ‘magnitude’ – of a vector can be introduced. This is done via an inter- mediate step, namely the introduction of the scalar product of two vectors. The product is called ‘scalar’ because its result is a scalar; a scalar is a number that is the same for all observers; for example, it is the same for observers with different orientations.* The scalar product between two vectors 𝑎 and 𝑏 is a number that satisfies Motion Mountain – The Adventure of Physics 𝑎𝑎 ⩾ 0 , 𝑎𝑏 = 𝑏𝑎 , (𝑎 + 𝑎 )𝑏 = 𝑎𝑏 + 𝑎 𝑏 , (10) 𝑎(𝑏 + 𝑏 ) = 𝑎𝑏 + 𝑎𝑏 and (𝑐𝑎)𝑏 = 𝑎(𝑐𝑏) = 𝑐(𝑎𝑏) . This definition of a scalar product is not unique; however, there is a standard scalar product. In Cartesian coordinate notation, the standard scalar product is given by copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net 𝑎𝑏 = 𝑎𝑥 𝑏𝑥 + 𝑎𝑦 𝑏𝑦 + 𝑎𝑧 𝑏𝑧 . (11) If the scalar product of two vectors vanishes the two vectors are orthogonal, at a right Challenge 150 e angle to each other. (Show it!) Note that one can write either 𝑎𝑏 or 𝑎 ⋅ 𝑏 with a central dot. The length or magnitude or norm of a vector can then be defined as the square root of the scalar product of a vector with itself: 𝑎 = √𝑎𝑎 . Often, and also in this text, lengths are written in italic letters, whereas vectors are written in bold letters. The magnitude is often written as 𝑎 = √𝑎2 . A vector space with a scalar product is called an Euclidean vector space. The scalar product is especially useful for specifying directions. Indeed, the scalar product between two vectors encodes the angle between them. Can you deduce this im- Challenge 151 s portant relation? * We mention that in mathematics, a scalar is a number; in physics, a scalar is an invariant number, i.e., a number that is the same for all observers. Likewise, in mathematics, a vector is an element of a vector space; in physics, a vector is an invariant element of a vector space, i.e., a quantity whose coordinates, when observed by different observers, change like the components of velocity. 82 3 how to describe motion – kinematics 𝑦 derivative slope: d𝑦/d𝑡 secant slope: Δ𝑦/Δ𝑡 Δ𝑡 Δ𝑦 𝑡 F I G U R E 48 The derivative in a point as the limit of secants. What is rest? What is velo cit y? In the Galilean description of nature, motion and rest are opposites. In other words, a body is at rest when its position, i.e., its coordinates, do not change with time. In other Motion Mountain – The Adventure of Physics words, (Galilean) rest is defined as 𝑥(𝑡) = const . (12) We recall that 𝑥(𝑡) is the abbreviation for the three coordinates (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)). Later we will see that this definition of rest, contrary to first impressions, is not much use and will have to be expanded. Nevertheless, any definition of rest implies that non-resting objects can be distinguished by comparing the rapidity of their displacement. Thus we can define the velocity 𝑣 of an object as the change of its position 𝑥 with time 𝑡. This is usually written as copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net d𝑥 𝑣= . (13) d𝑡 In this expression, valid for each coordinate separately, d/d𝑡 means ‘change with time’. We can thus say that velocity is the derivative of position with respect to time. The speed 𝑣 is the name given to the magnitude of the velocity 𝑣. Thus we have 𝑣 = √𝑣𝑣 . Derivatives are written as fractions in order to remind the reader that they are derived from the idea of slope. The expression d𝑠 Δ𝑠 is meant as an abbreviation of lim , (14) d𝑡 Δ𝑡→0 Δ𝑡 a shorthand for saying that the derivative at a point is the limit of the secant slopes in the neighbourhood of the point, as shown in Figure 48. This definition implies the working Challenge 152 e rules d(𝑠 + 𝑟) d𝑠 d𝑟 d(𝑐𝑠) d𝑠 d d𝑠 d2 𝑠 d(𝑠𝑟) d𝑠 d𝑟 = + , =𝑐 , = , = 𝑟 + 𝑠 , (15) d𝑡 d𝑡 d𝑡 d𝑡 d𝑡 d𝑡 d𝑡 d𝑡2 d𝑡 d𝑡 d𝑡 𝑐 being any number. This is all one ever needs to know about derivatives in physics. Quantities such as d𝑡 and d𝑠, sometimes useful by themselves, are called differentials. 3 how to describe motion – kinematics 83 F I G U R E 49 Gottfried Wilhelm Leibniz (1646–1716). These concepts are due to Gottfried Wilhelm Leibniz.* Derivatives lie at the basis of all calculations based on the continuity of space and time. Leibniz was the person who made it possible to describe and use velocity in physical formulae and, in particular, to use the idea of velocity at a given point in time or space for calculations. The definition of velocity assumes that it makes sense to take the limit Δ𝑡 → 0. In other words, it is assumed that infinitely small time intervals do exist in nature. The Motion Mountain – The Adventure of Physics definition of velocity with derivatives is possible only because both space and time are described by sets which are continuous, or in mathematical language, connected and com- plete. In the rest of our walk we shall not forget that from the beginning of classical phys- ics, infinities are present in its description of nature. The infinitely small is part of our definition of velocity. Indeed, differential calculus can be defined as the study of infin- ity and its uses. We thus discover that the appearance of infinity does not automatically render a description impossible or imprecise. In order to remain precise, physicists use only the smallest two of the various possible types of infinities. Their precise definition Vol. III, page 288 and an overview of other types are introduced later on. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net The appearance of infinity in the usual description of motion was first criticized in Ref. 66 his famous ironical arguments by Zeno of Elea (around 445 b ce), a disciple of Parmen- ides. In his so-called third argument, Zeno explains that since at every instant a given object occupies a part of space corresponding to its size, the notion of velocity at a given instant makes no sense; he provokingly concludes that therefore motion does not ex- ist. Nowadays we would not call this an argument against the existence of motion, but against its usual description, in particular against the use of infinitely divisible space and Challenge 153 e time. (Do you agree?) Nevertheless, the description criticized by Zeno actually works quite well in everyday life. The reason is simple but deep: in daily life, changes are indeed continuous. Large changes in nature are made up of many small changes. This property of nature is not obvious. For example, we note that we have (again) tacitly assumed that the path of an object is not a fractal or some other badly behaved entity. In everyday life this is correct; in other domains of nature it is not. The doubts of Zeno will be partly rehabilitated later Vol. VI, page 65 in our walk, and increasingly so the more we proceed. The rehabilitation is only partial, as the final solution will be different from that which he envisaged; on the other hand, * Gottfried Wilhelm Leibniz (b. 1646 Leipzig, d. 1716 Hannover), lawyer, physicist, mathematician, philo- sopher, diplomat and historian. He was one of the great minds of mankind; he invented the differential calculus (before Newton) and published many influential and successful books in the various fields he ex- plored, among them De arte combinatoria, Hypothesis physica nova, Discours de métaphysique, Nouveaux essais sur l’entendement humain, the Théodicée and the Monadologia. 84 3 how to describe motion – kinematics TA B L E 13 Some measured acceleration values. O b s e r va t i o n A c c e l e r at i o n What is the lowest you can find? Challenge 154 s Back-acceleration of the galaxy M82 by its ejected jet 10 f m/s2 Acceleration of a young star by an ejected jet 10 pm/s2 Fathoumi Acceleration of the Sun in its orbit around the Milky Way 0.2 nm/s2 Deceleration of the Pioneer satellites, due to heat radiation imbalance 0.8 nm/s2 Centrifugal acceleration at Equator due to Earth’s rotation 33 mm/s2 Electron acceleration in household electricity wire due to alternating 50 mm/s2 current Acceleration of fast underground train 1.3 m/s2 Gravitational acceleration on the Moon 1.6 m/s2 Minimum deceleration of a car, by law, on modern dry asphalt 5.5 m/s2 Gravitational acceleration on the Earth’s surface, depending on 9.8 ± 0.3 m/s2 Motion Mountain – The Adventure of Physics location Standard gravitational acceleration 9.806 65 m/s2 Highest acceleration for a car or motorbike with engine-driven wheels 15 m/s2 Space rockets at take-off 20 to 90 m/s2 Acceleration of cheetah 32 m/s2 Gravitational acceleration on Jupiter’s surface 25 m/s2 Flying fly (Musca domestica) c. 100 m/s2 Acceleration of thrown stone c. 120 m/s2 Acceleration that triggers air bags in cars 360 m/s2 copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Fastest leg-powered acceleration (by the froghopper, Philaenus 4 km/s2 spumarius, an insect) Tennis ball against wall 0.1 Mm/s2 Bullet acceleration in rifle 2 Mm/s2 Fastest centrifuges 0.1 Gm/s2 Acceleration of protons in large accelerator 90 Tm/s2 Acceleration of protons inside nucleus 1031 m/s2 Highest possible acceleration in nature 1052 m/s2 the doubts about the idea of ‘velocity at a point’ will turn out to be well-founded. For the moment though, we have no choice: we continue with the basic assumption that in nature changes happen smoothly. Why is velocity necessary as a concept? Aiming for precision in the description of motion, we need to find the complete list of aspects necessary to specify the state of an object. The concept of velocity is obviously on this list. 3 how to describe motion – kinematics 85 Acceleration Continuing along the same line, we call acceleration 𝑎 of a body the change of velocity 𝑣 with time, or d𝑣 d2 𝑥 𝑎= = 2 . (16) d𝑡 d𝑡 Acceleration is what we feel when the Earth trembles, an aeroplane takes off, or a bicycle goes round a corner. More examples are given in Table 13. Acceleration is the time de- rivative of velocity. Like velocity, acceleration has both a magnitude and a direction. In short, acceleration, like velocity, is a vector quantity. As usual, this property is indicated by the use of a bold letter for its abbreviation. In a usual car, or on a motorbike, we can feel being accelerated. (These accelerations are below 1𝑔 and are therefore harmless.) We feel acceleration because some part inside us is moved against some other part: acceleration deforms us. Such a moving part can be, for example, some small part inside our ear, our stomach inside the belly, or simply our limbs against our trunk. All acceleration sensors, including those listed in Table 14 Motion Mountain – The Adventure of Physics or those shown in Figure 50, whether biological or technical, work in this way. Acceleration is felt. Our body is deformed and the sensors in our body detect it, for ex- ample in amusement parks. Higher accelerations can have stronger effects. For example, when accelerating a sitting person in the direction of the head at two or three times the value of usual gravitational acceleration, eyes stop working and the sight is greyed out, because the blood cannot reach the eye any more. Between 3 and 5𝑔 of continuous accele- Ref. 67 ration, or 7 to 9𝑔 of short time acceleration, consciousness is lost, because the brain does not receive enough blood, and blood may leak out of the feet or lower legs. High acce- leration in the direction of the feet of a sitting person can lead to haemorrhagic strokes copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net in the brain. The people most at risk are jet pilots; they have special clothes that send compressed air onto the pilot’s bodies to avoid blood accumulating in the wrong places. Challenge 155 s Can you think of a situation where you are accelerated but do not feel it? Velocity is the time derivative of position. Acceleration is the second time derivative of position. Higher derivatives than acceleration can also be defined, in the same manner. Challenge 156 s They add little to the description of nature, because – as we will show shortly – neither these higher derivatives nor even acceleration itself are useful for the description of the state of motion of a system. From objects to point particles “ Wenn ich den Gegenstand kenne, so kenne ich auch sämtliche Möglichkeiten seines ” Vorkommens in Sachverhalten.* Ludwig Wittgenstein, Tractatus, 2.0123 One aim of the study of motion is to find a complete and precise description of both states and objects. With the help of the concept of space, the description of objects can be refined considerably. In particular, we know from experience that all objects seen in daily Challenge 157 e life have an important property: they can be divided into parts. Often this observation is * ‘If I know an object, then I also know all the possibilities of its occurrence in atomic facts.’ 86 3 how to describe motion – kinematics TA B L E 14 Some acceleration sensors. Measurement Sensor Range Direction of gravity in plants statoliths in cells 0 to 10 m/s2 (roots, trunk, branches, leaves) Direction and value of the utricle and saccule in the inner 0 to 20 m/s2 accelerations in mammals ear (detecting linear accelerations), and the membranes in each semicircular canal (detecting rotational accelerations) Direction and value of acceleration piezoelectric sensors 0 to 20 m/s2 in modern step counters for hikers Direction and value of acceleration airbag sensor using piezoelectric 0 to 2000 m/s2 in car crashes ceramics Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 50 Three accelerometers: a one-axis piezoelectric airbag sensor, a three-axis capacitive accelerometer, and the utricle and saccule near the three semicircular canals inside the human ear (© Bosch, Rieker Electronics, Northwestern University). expressed by saying that all objects, or bodies, have two properties. First, they are made out of matter,* defined as that aspect of an object responsible for its impenetrability, i.e., the property preventing two objects from being in the same place. Secondly, bodies Ref. 68 * Matter is a word derived from the Latin ‘materia’, which originally meant ‘wood’ and was derived via intermediate steps from ‘mater’, meaning ‘mother’. 3 how to describe motion – kinematics 87 α γ Betelgeuse Bellatrix ε δ Mintaka ζ Alnilam Alnitak β κ Rigel Saiph F I G U R E 51 Orion in natural colours (© Matthew Spinelli) and Betelgeuse (ESA, NASA). have a certain form or shape, defined as the precise way in which this impenetrability is Motion Mountain – The Adventure of Physics distributed in space. In order to describe motion as accurately as possible, it is convenient to start with those bodies that are as simple as possible. In general, the smaller a body, the simpler it is. A body that is so small that its parts no longer need to be taken into account is called a particle. (The older term corpuscle has fallen out of fashion.) Particles are thus idealized small stones. The extreme case, a particle whose size is negligible compared with the dimensions of its motion, so that its position is described completely by a single triplet of coordinates, is called a point particle or a point mass or a mass point. In equation (6), the stone was assumed to be such a point particle. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Do point-like objects, i.e., objects smaller than anything one can measure, exist in daily life? Yes and no. The most notable examples are the stars. At present, angular sizes as small as 2 μrad can be measured, a limit given by the fluctuations of the air in the atmosphere. In space, such as for the Hubble telescope orbiting the Earth, the angular limit is due to the diameter of the telescope and is of the order of 10 nrad. Practically all stars seen from Earth are smaller than that, and are thus effectively ‘point-like’, even when seen with the most powerful telescopes. As an exception to the general rule, the size of a few large or nearby stars, mostly of red giant type, can be measured with special instruments.* Betelgeuse, the higher of the two shoulders of Orion shown in Figure 51, Mira in Cetus, Antares in Scorpio, Aldebaran in Taurus and Sirius in Canis Major are examples of stars whose size has been measured; Ref. 69 they are all less than two thousand light years from Earth. For a comparison of dimen- sions, see Figure 52. Of course, like the Sun, also all other stars have a finite size, but one Challenge 158 s cannot prove this by measuring their dimension in photographs. (True?) * The website stars.astro.illinois.edu/sow/sowlist.html gives an introduction to the different types of stars. The www.astro.wisc.edu/~dolan/constellations website provides detailed and interesting information about constellations. For an overview of the planets, see the beautiful book by Kenneth R. L ang & Charles A. Whitney, Vagabonds de l’espace – Exploration et découverte dans le système solaire, Springer Verlag, 1993. Amazingly beautiful pictures of the stars can be found in David Malin, A View of the Universe, Sky Publishing and Cambridge University Press, 1993. 88 3 how to describe motion – kinematics Motion Mountain – The Adventure of Physics F I G U R E 52 A comparison of star sizes (© Dave Jarvis). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 53 Regulus and Mars, photographed with an exposure time of 10 s on 4 June 2010 with a wobbling camera, show the difference between a point-like star that twinkles and an extended planet that does not (© Jürgen Michelberger). The difference between ‘point-like’ and finite-size sources can be seen with the naked Challenge 159 e eye: at night, stars twinkle, but planets do not. (Check it!) A beautiful visualization is shown in Figure 53. This effect is due to the turbulence of air. Turbulence has an effect on the almost point-like stars because it deflects light rays by small amounts. On the other hand, air turbulence is too weak to lead to the twinkling of sources of larger angular size, such as planets or artificial satellites,* because the deflection is averaged out in this case. * A satellite is an object circling a planet, like the Moon; an artificial satellite is a system put into orbit by humans, like the Sputniks. 3 how to describe motion – kinematics 89 An object is point-like for the naked eye if its angular size is smaller than about Challenge 160 s 2 = 0.6 mrad. Can you estimate the size of a ‘point-like’ dust particle? By the way, an object is invisible to the naked eye if it is point-like and if its luminosity, i.e., the intensity of the light from the object reaching the eye, is below some critical value. Can you esti- mate whether there are any man-made objects visible from the Moon, or from the space Challenge 161 s shuttle? The above definition of ‘point-like’ in everyday life is obviously misleading. Do proper, real point particles exist? In fact, is it at all possible to show that a particle has a vanishing size? In the same way, we need to ask and check whether points in space do exist. Our walk will lead us to the astonishing result that all the answers to these ques- Challenge 162 s tions are negative. Can you imagine why? Do not be disappointed if you find this issue difficult; many brilliant minds have had the same problem. However, many particles, such as electrons, quarks or photons are point-like for all practical purposes. Once we know how to describe the motion of point particles, we can also describe the motion of extended bodies, rigid or deformable: we assume that they are made of parts. This is the same approach as describing the motion of an animal as a Motion Mountain – The Adventure of Physics whole by combining the motion of its various body parts. The simplest description, the continuum approximation, describes extended bodies as an infinite collection of point particles. It allows us to understand and to predict the motion of milk and honey, the motion of the air in hurricanes and of perfume in rooms. The motion of fire and all other gaseous bodies, the bending of bamboo in the wind, the shape changes of chewing Ref. 70 gum, and the growth of plants and animals can also be described in this way. All observations so far have confirmed that the motion of large bodies can be de- scribed to full precision as the result of the motion of their parts. All machines that hu- mans ever built are based on this idea. A description that is even more precise than the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Vol. IV, page 15 continuum approximation is given later on. Describing body motion with the motion of body parts will guide us through the first five volumes of our mountain ascent; for example, we will understand life in this way. Only in the final volume will we discover that, at a fundamental scale, this decomposition is impossible. Legs and wheels The parts of a body determine its shape. Shape is an important aspect of bodies: among other things, it tells us how to count them. In particular, living beings are always made of a single body. This is not an empty statement: from this fact we can deduce that animals cannot have large wheels or large propellers, but only legs, fins, or wings. Why? Living beings have only one surface; simply put, they have only one piece of skin. Vol. V, page 365 Mathematically speaking, animals are connected. This is often assumed to be obvious, and Ref. 71 it is often mentioned that the blood supply, the nerves and the lymphatic connections to a rotating part would get tangled up. However, this argument is not so simple, as Figure 54 shows. The figure proves that it is indeed possible to rotate a body continuously against a second one, without tangling up the connections. Three dimensions of space allow tethered rotation. Can you find an example for this kind of motion, often called tethered Challenge 163 s rotation, in your own body? Are you able to see how many cables may be attached to the Challenge 164 s rotating body of the figure without hindering the rotation? Despite the possibility of animals having rotating parts, the method of Figure 54 or 90 3 how to describe motion – kinematics F I G U R E 54 Tethered rotation: How an object can rotate continuously without tangling up the connection to a second object. Motion Mountain – The Adventure of Physics F I G U R E 55 Tethered rotation: copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the continuous rotation of an object attached to its environment (QuickTime ﬁlm © Jason Hise). Figure 55 still cannot be used to make a practical wheel or propeller. Can you see why? Challenge 165 s Therefore, evolution had no choice: it had to avoid animals with (large) parts rotating around axles. That is the reason that propellers and wheels do not exist in nature. Of course, this limitation does not rule out that living bodies move by rotation as a whole: Ref. 72 tumbleweed, seeds from various trees, some insects, several spiders, certain other anim- als, children and dancers occasionally move by rolling or rotating as a whole. Large single bodies, and thus all large living beings, can thus only move through de- Ref. 73 formation of their shape: therefore they are limited to walking, running, jumping, rolling, gliding, crawling, flapping fins, or flapping wings. Moving a leg is a common way to de- form a body. Ref. 74 Extreme examples of leg use in nature are shown in Figure 56 and Figure 57. The most extreme example of rolling spiders – there are several species – are Cebrennus villosus and Ref. 75 live in the sand in Morocco. They use their legs to accelerate the rolling, they can steer the rolling direction and can even roll uphill slopes of 30 % – a feat that humans are 3 how to describe motion – kinematics 91 Motion Mountain – The Adventure of Physics 50 μm F I G U R E 56 Legs and ‘wheels’ in living beings: the red millipede Aphistogoniulus erythrocephalus (15 cm body length), a gecko on a glass pane (15 cm body length), an amoeba Amoeba proteus (1 mm size), the rolling shrimp Nannosquilla decemspinosa (2 cm body length, 1.5 rotations per second, up to 2 m, can even roll slightly uphill slopes) and the rolling caterpillar Pleurotya ruralis (can only roll downhill, to escape predators), (© David Parks, Marcel Berendsen, Antonio Guillén Oterino, Robert Full, John copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Brackenbury / Science Photo Library ). F I G U R E 57 Two of the rare lifeforms that are able to roll uphill also on steep slopes: the desert spider Cebrennus villosus and Homo sapiens (© Ingo Rechenberg, Karva Javi). 92 3 how to describe motion – kinematics unable to perform. Films of the rolling motion can be found at www.bionik.tu-berlin. Vol. V, page 281 de.* Walking on water is shown in Figure 127 on page 170; examples of wings are given Vol. V, page 282 later on, as are the various types of deformations that allow swimming in water. In contrast, systems of several bodies, such as bicycles, pedal boats or other machines, can move without any change of shape of their components, thus enabling the use of axles with wheels, propellers and other rotating devices.** In short, whenever we observe a construction in which some part is turning continu- ously (and without the ‘wiring’ of Figure 54) we know immediately that it is an artefact: it is a machine, not a living being (but built by one). However, like so many statements about living creatures, this one also has exceptions. The distinction between one and two bodies is poorly defined if the whole system is made of only a few molecules. This happens most clearly inside bacteria. Organisms such as Escherichia coli, the well-known bacterium found in the human gut, or bacteria from the Salmonella family, all swim using flagella. Flagella are thin filaments, similar to tiny hairs that stick out of the cell membrane. In the 1970s it was shown that each flagellum, made of one or a few long molecules with a diameter of a few tens of nanometres, does Motion Mountain – The Adventure of Physics Vol. V, page 282 in fact turn about its axis. Bacteria are able to rotate their flagella in both clockwise and anticlockwise directions, can achieve more than 1000 turns per second, and can turn all their flagella in perfect Ref. 76 synchronization. These wheels are so tiny that they do not need a mechanical connec- Ref. 77 tion; Figure 58 shows a number of motor models found in bacteria. The motion and the construction of these amazing structures are shown in more details in the films Figure 59 and Figure 60. In summary, wheels actually do exist in living beings, albeit only tiny ones. The growth and motion of these wheels are wonders of nature. Macroscopic wheels in living beings copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net are not possible, though rolling motion is. Curiosities and fun challenges ab ou t kinematics Challenge 167 s What is the biggest wheel ever made? ∗∗ A football is shot, by a goalkeeper, with around 30 m/s. Use a video to calculate the dis- tance it should fly and compare it with the distances realized in a soccer match. Where Challenge 168 s does the difference come from? ∗∗ A train starts to travel at a constant speed of 10 m/s between two cities A and B, 36 km * Rolling is also known for the Namibian wheel spiders of the Carparachne genus; films of their motion can be found on the internet. ** Despite the disadvantage of not being able to use rotating parts and of being restricted to one piece only, nature’s moving constructions, usually called animals, often outperform human-built machines. As an example, compare the size of the smallest flying systems built by evolution with those built by humans. (See, e.g., pixelito.reference.be.) There are two reasons for this discrepancy. First, nature’s systems have integrated repair and maintenance systems. Second, nature can build large structures inside containers with small openings. In fact, nature is very good at what people do when they build sailing ships inside glass Challenge 166 s bottles. The human body is full of such examples; can you name a few? 3 how to describe motion – kinematics 93 Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 58 Some types of ﬂagellar motors found in nature; the images are taken by cryotomography. All yellow scale bars are 10 nm long (© S. Chen & al., EMBO Journal, Wiley & Sons). 94 3 how to describe motion – kinematics F I G U R E 59 The rotational motion of a bacterial ﬂagellum, and its reversal Motion Mountain – The Adventure of Physics (QuickTime ﬁlm © Osaka University). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 60 The growth of a bacterial ﬂagellum, showing the molecular assembly (QuickTime ﬁlm © Osaka University). apart. The train will take one hour for the journey. At the same time as the train, a fast dove starts to fly from A to B, at 20 m/s. Being faster than the train, the dove arrives at B first. The dove then flies back towards A; when it meets the train, it turns back again, to city B. It goes on flying back and forward until the train reaches B. What distance did Challenge 169 e the dove cover? 3 how to describe motion – kinematics 95 Motion Mountain – The Adventure of Physics F I G U R E 61 Are comets, such as the beautiful comet McNaught seen in 2007, images or bodies? How can you show it? (And why is the tail curved?) (© Robert McNaught) F I G U R E 62 The parabola of safety around a cannon, shown in red. The highest points of all trajectories copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net form an ellipse, shown in blue. (© Theon) ∗∗ Figure 62 illustrates that around a cannon, there is a line outside which you cannot be hit. Already in the 17th century, Evangelista Torricelli showed, without algebra, that the Challenge 170 e line is a parabola, and called it the parabola of safety. Can you show this as well? Can you confirm that the highest points of all trajectories lie on an ellipse? The parabola of safety also appears in certain water fountains. ∗∗ Balance a pencil vertically (tip upwards!) on a piece of paper near the edge of a table. Challenge 171 e How can you pull out the paper without letting the pencil fall? 96 3 how to describe motion – kinematics F I G U R E 63 Observation of sonoluminescence with a simple set-up that focuses ultrasound in water (© Detlef Lohse). ∗∗ Motion Mountain – The Adventure of Physics Is a return flight by aeroplane – from a point A to B and back to A – faster if the wind Challenge 172 e blows or if it does not? ∗∗ The level of acceleration that a human can survive depends on the duration over which one is subjected to it. For a tenth of a second, 30 𝑔 = 300 m/s2 , as generated by an ejector seat in an aeroplane, is acceptable. (It seems that the record acceleration a human has survived is about 80 𝑔 = 800 m/s2 .) But as a rule of thumb it is said that accelerations of 15 𝑔 = 150 m/s2 or more are fatal. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ The highest microscopic accelerations are observed in particle collisions, where values up to 1035 m/s2 are achieved. The highest macroscopic accelerations are probably found in the collapsing interiors of supernovae, exploding stars which can be so bright as to be vis- ible in the sky even during the daytime. A candidate on Earth is the interior of collapsing bubbles in liquids, a process called cavitation. Cavitation often produces light, an effect Ref. 78 discovered by Frenzel and Schultes in 1934 and called sonoluminescence. (See Figure 63.) It appears most prominently when air bubbles in water are expanded and contracted by underwater loudspeakers at around 30 kHz and allows precise measurements of bubble motion. At a certain threshold intensity, the bubble radius changes at 1500 m/s in as little Ref. 79 as a few μm, giving an acceleration of several 1011 m/s2 . ∗∗ Legs are easy to build. Nature has even produced a millipede, Illacme plenipes, that has 750 legs. The animal is 3 to 4 cm long and about 0.5 mm wide. This seems to be the record so far. In contrast to its name, no millipede actually has a thousand legs. Summary of kinematics The description of everyday motion of mass points with three coordinates as (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) is simple, precise and complete. This description of paths is the basis 3 how to describe motion – kinematics 97 of kinematics. As a consequence, space is described as a three-dimensional Euclidean space and velocity and acceleration as Euclidean vectors. The description of motion with paths assumes that the motion of objects can be fol- lowed along their paths. Therefore, the description often does not work for an important case: the motion of images. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Chapter 4 F ROM OB J E C T S A N D I M AG E S TO C ON SE RVAT ION W alking through a forest we observe two rather different types of motion: e see the breeze move the leaves, and at the same time, on the ground, Ref. 80 e see their shadows move. Shadows are a simple type of image. Both objects and images are able to move; both change position over time. Running tigers, falling snow- flakes, and material ejected by volcanoes, but also the shadow following our body, the Motion Mountain – The Adventure of Physics beam of light circling the tower of a lighthouse on a misty night, and the rainbow that constantly keeps the same apparent distance from us are examples of motion. Both objects and images differ from their environment in that they have boundaries defining their size and shape. But everybody who has ever seen an animated cartoon knows that images can move in more surprising ways than objects. Images can change their size and shape, they can even change colour, a feat only a few objects are able to perform.** Images can appear and disappear without a trace, multiply, interpenetrate, go backwards in time and defy gravity or any other force. Images, even ordinary shadows, can move faster than light. Images can float in space and keep the same distance from approaching objects. Objects can do almost none of this. In general, the ‘laws of cartoon copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Ref. 82 physics’ are rather different from those in nature. In fact, the motion of images does not seem to follow any rules, in contrast to the motion of objects. We feel the need for precise criteria allowing the two cases to be distinguished. Making a clear distinction between images and objects is performed using the same method that children or animals use when they stand in front of a mirror for the first time: they try to touch what they see. Indeed, ⊳ If we are able to touch what we see – or more precisely, if we are able to move it with a collision – we call it an object, otherwise an image.*** ** Excluding very slow changes such as the change of colour of leaves in the Autumn, in nature only certain crystals, the octopus and other cephalopods, the chameleon and a few other animals achieve this. Of man- made objects, television, computer displays, heated objects and certain lasers can do it. Do you know more Challenge 173 s examples? An excellent source of information on the topic of colour is the book by K. Nassau, The Physics and Chemistry of Colour – the fifteen causes of colour, J. Wiley & Sons, 1983. In the popular science domain, the most beautiful book is the classic work by the Flemish astronomer Marcel G. J. Minnaert, Light and Colour in the Outdoors, Springer, 1993, an updated version based on his wonderful book series, De Ref. 81 natuurkunde van ‘t vrije veld, Thieme & Cie, 1937. Reading it is a must for all natural scientists. On the web, there is also the – simpler, but excellent – webexhibits.org/causesofcolour website. *** One could propose including the requirement that objects may be rotated; however, this requirement, surprisingly, gives difficulties in the case of atoms, as explained on page 85 in Volume IV. 4 from objects and images to conservation 99 push F I G U R E 64 In which direction does the bicycle turn? Vol. IV, page 139 Images cannot be touched, but objects can. Images cannot hit each other, but objects can. And as everybody knows, touching something means feeling that it resists movement. Certain bodies, such as butterflies, pose little resistance and are moved with ease, others, such as ships, resist more, and are moved with more difficulty. ⊳ The resistance to motion – more precisely, to change of motion – is called Motion Mountain – The Adventure of Physics inertia, and the difficulty with which a body can be moved is called its (in- ertial) mass. Images have neither inertia nor mass. Summing up, for the description of motion we must distinguish bodies, which can be touched and are impenetrable, from images, which cannot and are not. Everything Challenge 174 s visible is either an object or an image; there is no third possibility. (Do you agree?) If the object is so far away that it cannot be touched, such as a star or a comet, it can be difficult to decide whether one is dealing with an image or an object; we will encounter copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net this difficulty repeatedly. For example, how would you show that comets – such as the beautiful example of Figure 61 – are objects and not images, as Galileo (falsely) claimed? Challenge 175 s In the same way that objects are made of matter, images are made of radiation. Im- Ref. 83 ages are the domain of shadow theatre, cinema, television, computer graphics, belief sys- tems and drug experts. Photographs, motion pictures, ghosts, angels, dreams and many hallucinations are images (sometimes coupled with brain malfunction). To understand images, we need to study radiation (plus the eye and the brain). However, due to the importance of objects – after all, we are objects ourselves – we study the latter first. Motion and contact “ Democritus affirms that there is only one type ” of movement: That resulting from collision. Ref. 84 Aetius, Opinions. When a child rides a unicycle, she or he makes use of a general rule in our world: one body acting on another puts it in motion. Indeed, in about six hours, anybody can learn to ride and enjoy a unicycle. As in all of life’s pleasures, such as toys, animals, women, machines, children, men, the sea, wind, cinema, juggling, rambling and loving, some- thing pushes something else. Thus our first challenge is to describe the transfer of motion due to contact – and to collisions – in more precise terms. 100 4 from objects and images to conservation 𝑣1 𝑣2 𝑣1 + Δ𝑣1 𝑣2 + Δ𝑣2 Motion Mountain – The Adventure of Physics F I G U R E 65 Collisions deﬁne mass. F I G U R E 66 The standard kilogram (until 2019) (© BIPM). Contact is not the only way to put something into motion; a counter-example is an apple falling from a tree or one magnet pulling another. Non-contact influences are more fascinating: nothing is hidden, but nevertheless something mysterious happens. Contact copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net motion seems easier to grasp, and that is why one usually starts with it. However, despite this choice, non-contact interactions cannot be avoided. Our choice to start with contact will lead us to a similar experience to that of riding a bicycle. (See Figure 64.) If we ride a bicycle at a sustained speed and try to turn left by pushing the right-hand steering bar, we will turn right. By the way, this surprising effect, also known to motorbike riders, obviously works only above a certain minimum speed. Can you determine what this Challenge 176 s speed is? Be careful! Too strong a push will make you fall. Something similar will happen to us as well; despite our choice of contact motion, the rest of our walk will rapidly force us to study non-contact interactions. What is mass? “ Δός μοί (φησι) ποῦ στῶ καὶ κινῶ τὴν γῆν. ” Da ubi consistam, et terram movebo.* Archimedes When we push something we are unfamiliar with, such as when we kick an object on the street, we automatically pay attention to the same aspect that children explore when * ‘Give me a place to stand, and I’ll move the Earth.’ Archimedes (c. 283–212), Greek scientist and engineer. Ref. 85 This phrase is attributed to him by Pappus. Already Archimedes knew that the distinction used by lawyers between movable and immovable objects made no sense. 4 from objects and images to conservation 101 F I G U R E 67 Antoine Lavoisier (1743 –1794) and his wife. Motion Mountain – The Adventure of Physics they stand in front of a mirror for the first time, or when they see a red laser spot for the first time. They check whether the unknown entity can be pushed or caught, and they pay attention to how the unknown object moves under their influence. All these are col- lision experiments. The high-precision version of any collision experiment is illustrated in Figure 65. Repeating such experiments with various pairs of objects, we find: ⊳ A fixed quantity 𝑚𝑖 can be ascribed to every object 𝑖, determined by the relation 𝑚2 Δ𝑣 =− 1 (17) copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net 𝑚1 Δ𝑣2 where Δ𝑣 is the velocity change produced by the collision. The quantity 𝑚𝑖 is called the mass of the object 𝑖. The more difficult it is to move an object, the higher the mass value. In order to have mass values that are common to everybody, the mass value for one particular, selected object has to be fixed in advance. Until 2019, there really was one such special object in the world, shown in Figure 66; it was called the standard kilogram. It was kept with great care in a glass container in Sèvres near Paris. Until 2019, the standard kilogram determined the value of the mass of every other object in the world. The standard kilogram was touched only once every few years because otherwise dust, humidity, or scratches would change its mass. For example, the standard kilogram was not kept under vacuum, because this would lead to outgassing and thus to changes in its mass. All the care did not avoid the stability issues though, and in 2019, the kilogram unit has been redefined using the fundamental constants 𝐺 (indirectly, via the caesium transition frequency), 𝑐 and ℏ that are shown in Figure 1. Since that change, everybody can produce his or her own standard kilogram in the laboratory – provided that sufficient care is used. The mass thus measures the difficulty of getting something moving. High masses are harder to move than low masses. Obviously, only objects have mass; images don’t. (By Ref. 68 the way, the word ‘mass’ is derived, via Latin, from the Greek μαζα – bread – or the 102 4 from objects and images to conservation F I G U R E 68 Christiaan Huygens (1629 –1695). Hebrew ‘mazza’ – unleavened bread. That is quite a change in meaning.) Experiments with everyday life objects also show that throughout any collision, the sum of all masses is conserved: ∑ 𝑚𝑖 = const . (18) 𝑖 Motion Mountain – The Adventure of Physics The principle of conservation of mass was first stated by Antoine-Laurent Lavoisier.* Conservation of mass also implies that the mass of a composite system is the sum of the mass of the components. In short, mass is also a measure for the quantity of matter. In a famous experiment in the sixteenth century, for several weeks Santorio Santorio (Sanctorius) (1561–1636), a friend of Galileo, lived with all his food and drink supply, and also his toilet, on a large balance. He wanted to test mass conservation. How did the Challenge 177 s measured weight change with time? Various cult leaders pretended and still pretend that they can produce matter out of nothing. This would be an example of non-conservation of mass. How can you show that all such leaders are crooks? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 178 s Momentum and mass The definition of mass can also be given in another way. We can ascribe a number 𝑚𝑖 to every object 𝑖 such that for collisions free of outside interference the following sum is unchanged throughout the collision: ∑ 𝑚𝑖 𝑣𝑖 = const . (19) 𝑖 The product of the velocity 𝑣𝑖 and the mass 𝑚𝑖 is called the (linear) momentum of the body. The sum, or total momentum of the system, is the same before and after the colli- sion; momentum is a conserved quantity. * Antoine-Laurent Lavoisier (b. 1743 Paris , d. 1794 Paris), chemist and genius. Lavoisier was the first to un- derstand that combustion is a reaction with oxygen; he discovered the components of water and introduced mass measurements into chemistry. A famous story about his character: When he was (unjustly) sentenced to the guillotine during the French revolution, he decided to use the situation for a scientific experiment. He announced that he would try to blink his eyes as frequently as possible after his head was cut off, in order to show others how long it takes to lose consciousness. Lavoisier managed to blink eleven times. It is unclear whether the story is true or not. It is known, however, that it could be true. Indeed, after a decapitation Ref. 86 without pain or shock, a person can remain conscious for up to half a minute. 4 from objects and images to conservation 103 F I G U R E 69 Is this dangerous? ⊳ Momentum conservation defines mass. The two conservation principles (18) and (19) were first stated in this way by the import- ant physicist Christiaan Huygens:* Momentum and mass are conserved in the everyday Motion Mountain – The Adventure of Physics motion of objects. In particular, neither quantity can be defined for the motion of images. Some typical momentum values are given in Table 15. Momentum conservation implies that when a moving sphere hits a resting one of the same mass and without loss of energy, a simple rule determines the angle between the Challenge 179 s directions the two spheres take after the collision. Can you find this rule? It is particularly useful when playing billiards. We will find out later that the rule is not valid for speeds Vol. II, page 67 near that of light. Another consequence of momentum conservation is shown in Figure 69: a man is lying on a bed of nails with a large block of concrete on his stomach. Another man is copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net hitting the concrete with a heavy sledgehammer. As the impact is mostly absorbed by Challenge 180 s the concrete, there is no pain and no danger – unless the concrete is missed. Why? The above definition (17) of mass has been generalized by the physicist and philo- sopher Ernst Mach** in such a way that it is valid even if the two objects interact without contact, as long as they do so along the line connecting their positions. ⊳ The mass ratio between two bodies is defined as a negative inverse accelera- tion ratio, thus as 𝑚2 𝑎 =− 1 , (20) 𝑚1 𝑎2 * Christiaan Huygens (b. 1629 ’s Gravenhage, d. 1695 Hofwyck) was one of the main physicists and math- ematicians of his time. Huygens clarified the concepts of mechanics; he also was one of the first to show that light is a wave. He wrote influential books on probability theory, clock mechanisms, optics and astronomy. Among other achievements, Huygens showed that the Orion Nebula consists of stars, discovered Titan, the moon of Saturn, and showed that the rings of Saturn consist of rock. (This is in contrast to Saturn itself, whose density is lower than that of water.) ** Ernst Mach (1838 Chrlice–1916 Vaterstetten), Austrian physicist and philosopher. The mach unit for aero- plane speed as a multiple of the speed of sound in air (about 0.3 km/s) is named after him. He also studied the basis of mechanics. His thoughts about mass and inertia influenced the development of general relativ- ity and led to Mach’s principle, which will appear later on. He was also proud to be the last scientist denying – humorously, and against all evidence – the existence of atoms. 104 4 from objects and images to conservation TA B L E 15 Some measured momentum values. O b s e r va t i o n Momentum Images 0 Momentum of a green photon 1.2 ⋅ 10−27 Ns Average momentum of oxygen molecule in air 10−26 Ns X-ray photon momentum 10−23 Ns 𝛾 photon momentum 10−17 Ns Highest particle momentum in accelerators 1 fNs Highest possible momentum of a single elementary 6.5 Ns particle – the Planck momentum Fast billiard ball 3 Ns Flying rifle bullet 10 Ns Box punch 15 to 50 Ns Comfortably walking human 80 Ns Motion Mountain – The Adventure of Physics Lion paw strike c. 0.2 kNs Whale tail blow c. 3 kNs Car on highway 40 kNs Impact of meteorite with 2 km diameter 100 TNs Momentum of a galaxy in galaxy collision up to 1046 Ns where 𝑎 is the acceleration of each body during the interaction. This definition of mass has been explored in much detail in the physics community, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net mainly in the nineteenth century. A few points sum up the results: — The definition of mass implies the conservation of total momentum ∑ 𝑚𝑣. Mo- mentum conservation is not a separate principle. Conservation of momentum cannot be checked experimentally, because mass is defined in such a way that the momentum conservation holds. — The definition of mass implies the equality of the products 𝑚1 𝑎1 and −𝑚2 𝑎2 . Such products are called forces. The equality of acting and reacting forces is not a separate principle; mass is defined in such a way that the principle holds. — The definition of mass is independent of whether contact is involved or not, and whether the accelerations are due to electricity, gravitation, or other interactions.* Since the interaction does not enter the definition of mass, mass values defined with the help of the electric, nuclear or gravitational interaction all agree, as long as mo- mentum is conserved. All known interactions conserve momentum. For some un- fortunate historical reasons, the mass value measured with the electric or nuclear in- teractions is called the ‘inertial’ mass and the mass measured using gravity is called * As mentioned above, only central forces obey the relation (20) used to define mass. Central forces act Page 119 between the centre of mass of bodies. We give a precise definition later. However, since all fundamental forces are central, this is not a restriction. There seems to be one notable exception: magnetism. Is the Challenge 181 s definition of mass valid in this case? 4 from objects and images to conservation 105 TA B L E 16 Some measured mass values. O b s e r va t i o n Mass Probably lightest known object: neutrino c. 2 ⋅ 10−36 kg Mass increase due to absorption of one green photon 4.1 ⋅ 10−36 kg Lightest known charged object: electron 9.109 381 88(72) ⋅ 10−31 kg Atom of argon 39.962 383 123(3) u = 66.359 1(1) yg Lightest object ever weighed (a gold particle) 0.39 ag Human at early age (fertilized egg) 10−8 g Water adsorbed on to a kilogram metal weight 10−5 g Planck mass 2.2 ⋅ 10−5 g Fingerprint 10−4 g Typical ant 10−4 g Water droplet 1 mg Honey bee, Apis mellifera 0.1 g Motion Mountain – The Adventure of Physics Euro coin 7.5 g Blue whale, Balaenoptera musculus 180 Mg Heaviest living things, such as the fungus Armillaria 106 kg ostoyae or a large Sequoia Sequoiadendron giganteum Heaviest train ever 99.7 ⋅ 106 kg Largest ocean-going ship 400 ⋅ 106 kg Largest object moved by man (Troll gas rig) 687.5 ⋅ 106 kg Large antarctic iceberg 1015 kg Water on Earth 1021 kg copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Earth’s mass 5.98 ⋅ 1024 kg Solar mass 2.0 ⋅ 1030 kg Our galaxy’s visible mass 3 ⋅ 1041 kg Our galaxy’s estimated total mass 2 ⋅ 1042 kg virgo supercluster 2 ⋅ 1046 kg Total mass visible in the universe 1054 kg the ‘gravitational’ mass. As it turns out, this artificial distinction makes no sense; this becomes especially clear when we take an observation point that is far away from all the bodies concerned. — The definition of mass requires observers at rest or in inertial motion. By measuring the masses of bodies around us we can explore the science and art of ex- periments. An overview of mass measurement devices is given in Table 18 and Figure 71. Some measurement results are listed in Table 16. By measuring mass values around us we confirm the main properties of mass. First of all, mass is additive in everyday life, as the mass of two bodies combined is equal to the sum of the two separate masses. Furthermore, mass is continuous; it can seemingly take any positive value. Finally, mass is conserved in everyday life; the mass of a system, 106 4 from objects and images to conservation TA B L E 17 Properties of mass in everyday life. Masses Physical M at h e m at i c a l Defini- propert y name tion Can be distinguished distinguishability element of set Vol. III, page 285 Can be ordered sequence order Vol. IV, page 224 Can be compared measurability metricity Vol. IV, page 236 Can change gradually continuity completeness Vol. V, page 364 Can be added quantity of matter additivity Page 81 Beat any limit infinity unboundedness, openness Vol. III, page 286 Do not change conservation invariance 𝑚 = const Do not disappear impenetrability positivity 𝑚⩾0 defined as the sum of the mass of all constituents, does not change over time if the system Motion Mountain – The Adventure of Physics is kept isolated from the rest of the world. Mass is not only conserved in collisions but also during melting, evaporation, digestion and all other everyday processes. All the properties of everyday mass are summarized in Table 17. Later we will find that several of the properties are only approximate. High-precision experiments show devi- ations.* However, the definition of mass remains unchanged throughout our adventure. The definition of mass through momentum conservation implies that when an object falls, the Earth is accelerated upwards by a tiny amount. If we could measure this tiny amount, we could determine the mass of the Earth. Unfortunately, this measurement is Challenge 182 s impossible. Can you find a better way to determine the mass of the Earth? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net The definition of mass and momentum allows answering the question of Figure 70. A brick hangs from the ceiling; a second thread hangs down from the brick, and you Challenge 183 e can pull it. How can you tune your pulling method to make the upper thread break? The lower one? Summarizing Table 17, the mass of a body is thus most precisely described by a positive real number, often abbreviated 𝑚 or 𝑀. This is a direct consequence of the impenetrabil- ity of matter. Indeed, a negative (inertial) mass would mean that such a body would move in the opposite direction of any applied force or acceleration. Such a body could not be kept in a box; it would break through any wall trying to stop it. Strangely enough, neg- ative mass bodies would still fall downwards in the field of a large positive mass (though Challenge 184 e more slowly than an equivalent positive mass). Are you able to confirm this? However, a small positive mass object would float away from a large negative-mass body, as you can easily deduce by comparing the various accelerations involved. A positive and a negative mass of the same value would remain at a constant distance and spontaneously accelerate Challenge 185 e away along the line connecting the two masses. Note that both energy and momentum Vol. II, page 72 are conserved in all these situations.** Negative-mass bodies have never been observed. Vol. IV, page 192 Antimatter, which will be discussed later, also has positive mass. * For example, in order to define mass we must be able to distinguish bodies. This seems a trivial require- ment, but we discover that this is not always possible in nature. ** For more curiosities, see R. H. P rice, Negative mass can be positively amusing, American Journal of 4 from objects and images to conservation 107 ceiling thread brick thread hand Motion Mountain – The Adventure of Physics F I G U R E 70 Depending on the way you pull, either the upper of the lower thread snaps. What are the options? TA B L E 18 Some mass sensors. Measurement Sensor Range copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Precision scales balance, pendulum, or spring 1 pg to 103 kg Particle collision speed below 1 mg Sense of touch pressure sensitive cells 1 mg to 500 kg Doppler effect on light reflected off interferometer 1 mg to 100 g the object Cosmonaut body mass spring frequency around 70 kg measurement device Truck scales hydraulic balance 103 to 60 ⋅ 103 kg Ship weight water volume measurement up to 500 ⋅ 106 kg Physics 61, pp. 216–217, 1993. Negative mass particles in a box would heat up a box made of positive mass Page 110 while traversing its walls, and accelerating, i.e., losing energy, at the same time. They would allow one to build a perpetuum mobile of the second kind, i.e., a device circumventing the second principle of thermo- Challenge 186 e dynamics. Moreover, such a system would have no thermodynamic equilibrium, because its energy could decrease forever. The more one thinks about negative mass, the more one finds strange properties contra- Challenge 187 s dicting observations. By the way, what is the range of possible mass values for tachyons? 108 4 from objects and images to conservation Motion Mountain – The Adventure of Physics F I G U R E 71 Mass measurement devices: a vacuum balance used in 1890 by Dmitriy Ivanovich Mendeleyev, a modern laboratory balance, a device to measure the mass of a cosmonaut in space and copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net a truck scales (© Thinktank Trust, Mettler-Toledo, NASA, Anonymous). Is motion eternal? – C onservation of momentum “ Every body continues in the state of rest or of uniform motion in a straight line except in so ” far as it doesn’t. Arthur Eddington* The product 𝑝 = 𝑚𝑣 of mass and velocity is called the momentum of a particle; it de- scribes the tendency of an object to keep moving during collisions. The larger it is, the harder it is to stop the object. Like velocity, momentum has a direction and a magnitude: it is a vector. In French, momentum is called ‘quantity of motion’, a more appropriate term. In the old days, the term ‘motion’ was used instead of ‘momentum’, for example by Newton. The conservation of momentum, relation (19), therefore expresses the con- servation of motion during interactions. Momentum is an extensive quantity. That means that it can be said that it flows from one body to the other, and that it can be accumulated in bodies, in the same way that water flows and can be accumulated in containers. Imagining momentum as something * Arthur Eddington (1882–1944), British astrophysicist. 4 from objects and images to conservation 109 cork wine wine stone Challenge 188 s F I G U R E 72 What happens in these four situations? Ref. 87 that can be exchanged between bodies in collisions is always useful when thinking about the description of moving objects. Momentum is conserved. That explains the limitations you might experience when Motion Mountain – The Adventure of Physics being on a perfectly frictionless surface, such as ice or a polished, oil covered marble: you cannot propel yourself forward by patting your own back. (Have you ever tried to put a cat on such a marble surface? It is not even able to stand on its four legs. Neither Challenge 189 s are humans. Can you imagine why?) Momentum conservation also answers the puzzles of Figure 72. The conservation of momentum and mass also means that teleportation (‘beam me Challenge 190 s up’) is impossible in nature. Can you explain this to a non-physicist? Momentum conservation implies that momentum can be imagined to be like an invis- ible fluid. In an interaction, the invisible fluid is transferred from one object to another. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net In such transfers, the amount of fluid is always constant. Momentum conservation implies that motion never stops; it is only exchanged. On the other hand, motion often ‘disappears’ in our environment, as in the case of a stone dropped to the ground, or of a ball left rolling on grass. Moreover, in daily life we of- ten observe the creation of motion, such as every time we open a hand. How do these examples fit with the conservation of momentum? It turns out that apparent momentum disappearance is due to the microscopic aspects of the involved systems. A muscle only transforms one type of motion, namely that of the electrons in certain chemical compounds* into another, the motion of the fingers. The working of muscles is similar to that of a car engine transforming the motion of electrons in the fuel into motion of the wheels. Both systems need fuel and get warm in the process. We must also study the microscopic behaviour when a ball rolls on grass until it stops. The apparent disappearance of motion is called friction. Studying the situation carefully, we find that the grass and the ball heat up a little during this process. During friction, visible motion is transformed into heat. A striking observation of this effect for a bicycle Page 384 is shown below, in Figure 273. Later, when we discover the structure of matter, it will become clear that heat is the disorganized motion of the microscopic constituents of every material. When the microscopic constituents all move in the same direction, the object as a whole moves; when they oscillate randomly, the object is at rest, but is warm. Ref. 88 * The fuel of most processes in animals usually is adenosine triphosphate (ATP). 110 4 from objects and images to conservation Heat is a form of motion. Friction thus only seems to be disappearance of motion; in fact it is a transformation of ordered into unordered motion. Page 395 Despite momentum conservation, macroscopic perpetual motion does not exist, since friction cannot be completely eliminated.* Motion is eternal only at the microscopic scale. In other words, the disappearance and also the spontaneous appearance of mo- tion in everyday life is an illusion due to the limitations of our senses. For example, the motion proper of every living being exists before its birth, and stays after its death. The same happens with its energy. This result is probably the closest one can get to the idea of everlasting life from evidence collected by observation. It is perhaps less than a co- incidence that energy used to be called vis viva, or ‘living force’, by Leibniz and many others. Since motion is conserved, it has no origin. Therefore, at this stage of our walk we cannot answer the fundamental questions: Why does motion exist? What is its origin? The end of our adventure is nowhere near. More conservation – energy Motion Mountain – The Adventure of Physics When collisions are studied in detail, a second conserved quantity turns up. Experiments show that in the case of perfect, or elastic collisions – collisions without friction – the following quantity, called the kinetic energy 𝑇 of the system, is also conserved: 𝑇 = ∑ 21 𝑚𝑖 𝑣𝑖2 = const . (21) 𝑖 Kinetic energy is the ability that a body has to induce change in bodies it hits. Kinetic energy thus depends on the mass and on the square of the speed 𝑣 of a body. The full copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net name ‘kinetic energy’ was introduced by Gustave-Gaspard Coriolis.** Some measured energy values are given in Table 19. * Some funny examples of past attempts to built a perpetual motion machine are described in Stan- islav Michel, Perpetuum mobile, VDI Verlag, 1976. Interestingly, the idea of eternal motion came to Europe from India, via the Islamic world, around the year 1200, and became popular as it op- posed the then standard view that all motion on Earth disappears over time. See also the web.archive. org/web/20040812085618/http://www.geocities.com/mercutio78_99/pmm.html and the www.lhup.edu/ ~dsimanek/museum/unwork.htm websites. The conceptual mistake made by eccentrics and used by crooks is always the same: the hope of overcoming friction. (In fact, this applied only to the perpetual motion ma- chines of the second kind; those of the first kind – which are even more in contrast with observation – even try to generate energy from nothing.) If the machine is well constructed, i.e., with little friction, it can take the little energy it needs for the sustenance of its motion from very subtle environmental effects. For example, in the Victoria and Albert Ref. 89 Museum in London one can admire a beautiful clock powered by the variations of air pressure over time. Low friction means that motion takes a long time to stop. One immediately thinks of the motion of the planets. In fact, there is friction between the Earth and the Sun. (Can you guess one of the mechanisms?) Challenge 191 s But the value is so small that the Earth has already circled around the Sun for thousands of millions of years, and will do so for quite some time more. ** Gustave-Gaspard Coriolis (b. 1792 Paris, d. 1843 Paris) was engineer and mathematician. He introduced the modern concepts of ‘work’ and of ‘kinetic energy’, and explored the Coriolis effect discovered by Page 138 Laplace. Coriolis also introduced the factor 1/2 in the kinetic energy 𝑇, in order that the relation d𝑇/d𝑣 = 𝑝 Challenge 192 s would be obeyed. (Why?) 4 from objects and images to conservation 111 The experiments and ideas mentioned so far can be summarized in the following definition: ⊳ (Physical) energy is the measure of the ability to generate motion. A body has a lot of energy if it has the ability to move many other bodies. Energy is a number; energy, in contrast to momentum, has no direction. The total momentum of two equal masses moving with opposite velocities is zero; but their total energy is not, and it increases with velocity. Energy thus also measures motion, but in a different way than momentum. Energy measures motion in a more global way. An equivalent definition is the following: ⊳ Energy is the ability to perform work. Here, the physical concept of work is just the precise version of what is meant by work in everyday life. As usual, (physical) work is the product of force and distance in direction Motion Mountain – The Adventure of Physics of the force. In other words, work is the scalar product of force and distance. Physical work is a quantity that describes the effort of pushing of an object along a distance. As a result, in physics, work is a form of energy. Another, equivalent definition of energy will become clear shortly: ⊳ Energy is what can be transformed into heat. Energy is a word taken from ancient Greek; originally it was used to describe character, and meant ‘intellectual or moral vigour’. It was taken into physics by Thomas Young copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net (1773–1829) in 1807 because its literal meaning is ‘force within’. (The letters 𝐸, 𝑊, 𝐴 and several others are also used to denote energy.) Both energy and momentum measure how systems change. Momentum tells how sys- tems change over distance: momentum is action (or change) divided by distance. Mo- mentum is needed to compare motion here and there. Energy measures how systems change over time: energy is action (or change) divided by time. Energy is needed to compare motion now and later. Do not be surprised if you do not grasp the difference between momentum and energy straight away: physicists took about a century to figure it out! So you are allowed to take some time to get used to it. Indeed, for many decades, English physicists insisted on using the same term for both concepts; this was due to Newton’s insistence that – no joke – the existence of god implied that energy was the same as momentum. Leibniz, instead, knew that energy increases with the square of the speed and proved Newton wrong. In 1722, Willem Jacob ’s Gravesande even showed the difference between energy and momentum Ref. 90 experimentally. He let metal balls of different masses fall into mud from different heights. By comparing the size of the imprints he confirmed that Newton was wrong both with his physical statements and his theological ones. One way to explore the difference between energy and momentum is to think about the following challenges. Which running man is more difficult to stop? One of mass 𝑚 running at speed 𝑣, or one with mass 𝑚/2 and speed 2𝑣, or one with mass 𝑚/2 and speed Challenge 193 e √2 𝑣? You may want to ask a rugby-playing friend for confirmation. 112 4 from objects and images to conservation TA B L E 19 Some measured energy values. O b s e r va t i o n Energy Average kinetic energy of oxygen molecule in air 6 zJ Green photon energy 0.37 aJ X-ray photon energy 1 fJ 𝛾 photon energy 1 pJ Highest particle energy in accelerators 0.1 μJ Kinetic energy of a flying mosquito 0.2 μJ Comfortably walking human 20 J Flying arrow 50 J Right hook in boxing 50 J Energy in torch battery 1 kJ Energy in explosion of 1 g TNT 4.1 kJ Energy of 1 kcal 4.18 kJ Motion Mountain – The Adventure of Physics Flying rifle bullet 10 kJ One gram of fat 38 kJ One gram of gasoline 44 kJ Apple digestion 0.2 MJ Car on highway 0.3 to 1 MJ Highest laser pulse energy 1.8 MJ Lightning flash up to 1 GJ Planck energy 2.0 GJ Small nuclear bomb (20 ktonne) 84 TJ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Earthquake of magnitude 7 2 PJ Largest nuclear bomb (50 Mtonne) 210 PJ Impact of meteorite with 2 km diameter 1 EJ Yearly machine energy use 420 EJ Rotation energy of Earth 2 ⋅ 1029 J Supernova explosion 1044 J Gamma-ray burst up to 1047 J Energy content 𝐸 = 𝑐2 𝑚 of Sun’s mass 1.8 ⋅ 1047 J Energy content of Galaxy’s central black hole 4 ⋅ 1053 J Another distinction between energy and momentum is illustrated by athletics: the real long jump world record, almost 10 m, is still kept by an athlete who in the early twentieth century ran with two weights in his hands, and then threw the weights behind him at the Challenge 194 s moment he took off. Can you explain the feat? When a car travelling at 100 m/s runs head-on into a parked car of the same kind and Challenge 195 s make, which car receives the greatest damage? What changes if the parked car has its brakes on? To get a better feeling for energy, here is an additional aspect. The world consumption of energy by human machines (coming from solar, geothermal, biomass, wind, nuclear, 4 from objects and images to conservation 113 F I G U R E 73 Robert Mayer (1814–1878). hydro, gas, oil, coal, or animal sources) in the year 2000 was about 420 EJ,* for a world Ref. 91 population of about 6000 million people. To see what this energy consumption means, we translate it into a personal power consumption; we get about 2.2 kW. The watt W is the unit of power, and is simply defined as 1 W = 1 J/s, reflecting the definition of (physical) power as energy used per unit time. The precise wording is: power is energy flowing per time through a defined closed surface. See Table 20 for some power values Motion Mountain – The Adventure of Physics found in nature, and Table 21 for some measurement devices. As a working person can produce mechanical work of about 100 W, the average hu- man energy consumption corresponds to about 22 humans working 24 hours a day. In particular, if we look at the energy consumption in First World countries, the average inhabitant there has machines working for him or her that are equivalent to several hun- Challenge 196 s dred ‘servants’. Machines do a lot of good. Can you point out some of these machines? Kinetic energy is thus not conserved in everyday life. For example, in non-elastic colli- sions, such as that of a piece of chewing gum hitting a wall, kinetic energy is lost. Friction destroys kinetic energy. At the same time, friction produces heat. It was one of the im- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net portant conceptual discoveries of physics that total energy is conserved if one includes the discovery that heat is a form of energy. Friction is thus a process transforming kinetic energy, i.e., the energy connected with the motion of a body, into heat. On a microscopic scale, energy is always conserved. Any example for the non-conservation of energy is only apparent. ** Indeed, without energy conservation, the concept of time would not be definable! We will show this im- portant connection shortly. In summary, in addition to mass and momentum, everyday linear motion also con- serves energy. To discover the last conserved quantity, we explore another type of motion: rotation. Page 453 * For the explanation of the abbreviation E, see Appendix B. ** In fact, the conservation of energy was stated in its full generality in public only in 1842, by Julius Robert Mayer. He was a medical doctor by training, and the journal Annalen der Physik refused to publish his paper, as it supposedly contained ‘fundamental errors’. What the editors called errors were in fact mostly – but not only – contradictions of their prejudices. Later on, Helmholtz, Thomson-Kelvin, Joule and many others acknowledged Mayer’s genius. However, the first to have stated energy conservation in its modern form was the French physicist Sadi Carnot (1796–1832) in 1820. To him the issue was so clear that he did not publish the result. In fact he went on and discovered the second ‘law’ of thermodynamics. Today, energy conservation, also called the first ‘law’ of thermodynamics, is one of the pillars of physics, as it is valid in all its domains. 114 4 from objects and images to conservation TA B L E 20 Some measured power values. O b s e r va t i o n Power Radio signal from the Galileo space probe sending from Jupiter 10 zW Power of flagellar motor in bacterium 0.1 pW Power consumption of a typical cell 1 pW sound power at the ear at hearing threshold 2.5 pW CR-R laser, at 780 nm 40-80 mW Sound output from a piano playing fortissimo 0.4 W Dove (0.16 kg) basal metabolic rate 0.97 W Rat (0.26 kg) basal metabolic rate 1.45 W Pigeon (0.30 kg) basal metabolic rate 1.55 W Hen (2.0 kg) basal metabolic rate 4.8 W Incandescent light bulb light output 1 to 5 W Dog (16 kg) basal metabolic rate 20 W Motion Mountain – The Adventure of Physics Sheep (45 kg) basal metabolic rate 50 W Woman (60 kg) basal metabolic rate 68 W Man (70 kg) basal metabolic rate 87 W Incandescent light bulb electricity consumption 25 to 100 W A human, during one work shift of eight hours 100 W Cow (400 kg) basal metabolic rate 266 W One horse, for one shift of eight hours 300 W Steer (680 kg) basal metabolic rate 411 W Eddy Merckx, the great bicycle athlete, during one hour 500 W copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Metric horse power power unit (75 kg ⋅ 9.81 m/s2 ⋅ 1 m/s) 735.5 W British horse power power unit 745.7 W Large motorbike 100 kW Electrical power station output 0.1 to 6 GW World’s electrical power production in 2000 Ref. 91 450 GW Power used by the geodynamo 200 to 500 GW Limit on wind energy production Ref. 92 18 to 68 TW Input on Earth surface: Sun’s irradiation of Earth Ref. 93 0.17 EW Input on Earth surface: thermal energy from inside of the Earth 32 TW Input on Earth surface: power from tides (i.e., from Earth’s rotation) 3 TW Input on Earth surface: power generated by man from fossil fuels 8 to 11 TW Lost from Earth surface: power stored by plants’ photosynthesis 40 TW World’s record laser power 1 PW Output of Earth surface: sunlight reflected into space 0.06 EW Output of Earth surface: power radiated into space at 287 K 0.11 EW Peak power of the largest nuclear bomb 5 YW Sun’s output 384.6 YW Maximum power in nature, 𝑐5 /4𝐺 9.1 ⋅ 1051 W 4 from objects and images to conservation 115 TA B L E 21 Some power sensors. Measurement Sensor Range Heart beat as power meter deformation sensor and clock 75 to 2 000 W Fitness power meter piezoelectric sensor 75 to 2 000 W Electricity meter at home rotating aluminium disc 20 to 10 000 W Power meter for car engine electromagnetic brake up to 1 MW Laser power meter photoelectric effect in up to 10 GW semiconductor Calorimeter for chemical reactions temperature sensor up to 1 MW Calorimeter for particles light detector up to a few μJ/ns Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 74 Some power measurement devices: a bicycle power meter, a laser power meter, and an electrical power meter (© SRAM, Laser Components, Wikimedia). The cross product, or vector product The discussion of rotation is easiest if we introduce an additional way to multiply vectors. This new product between two vectors 𝑎 and 𝑏 is called the cross product or vector product 𝑎 × 𝑏. The result of the vector product is another vector; thus it differs from the scalar product, whose result is a scalar, i.e., a number. The result of the vector product is that vector — that is orthogonal to both vectors to be multiplied, — whose orientation is given by the right-hand rule, and — whose length is given by the surface area of the parallelogram spanned by the two vectors, i.e., by 𝑎𝑏 sin ∢(𝑎, 𝑏). The definition implies that the cross product vanishes if and only if the vectors are par- 116 4 from objects and images to conservation Challenge 197 e allel. From the definition you can also show that the vector product has the properties 𝑎 × 𝑏 = −𝑏 × 𝑎 , 𝑎 × (𝑏 + 𝑐) = 𝑎 × 𝑏 + 𝑎 × 𝑐 , 𝜆𝑎 × 𝑏 = 𝜆(𝑎 × 𝑏) = 𝑎 × 𝜆𝑏 , 𝑎 × 𝑎 = 0 , 𝑎(𝑏 × 𝑐) = 𝑏(𝑐 × 𝑎) = 𝑐(𝑎 × 𝑏) , 𝑎 × (𝑏 × 𝑐) = (𝑎𝑐)𝑏 − (𝑎𝑏)𝑐 , (𝑎 × 𝑏)(𝑐 × 𝑑) = 𝑎(𝑏 × (𝑐 × 𝑑)) = (𝑎𝑐)(𝑏𝑑) − (𝑏𝑐)(𝑎𝑑) , (𝑎 × 𝑏) × (𝑐 × 𝑑) = ((𝑎 × 𝑏)𝑑)𝑐 − ((𝑎 × 𝑏)𝑐)𝑑 , 𝑎 × (𝑏 × 𝑐) + 𝑏 × (𝑐 × 𝑎) + 𝑐 × (𝑎 × 𝑏) = 0 . (22) The vector product exists only in vector spaces with three dimensions. We will explore Vol. IV, page 234 more details on this connection later on. The vector product is useful to describe systems that rotate – and (thus) also systems with magnetic forces. The motion of an orbiting body is always perpendicular both to the axis and to the line that connects the body with the axis. In rotation, axis, radius and velocity form a right-handed set of mutually orthogonal vectors. This connection lies at Motion Mountain – The Adventure of Physics the origin of the vector product. Challenge 198 e Confirm that the best way to calculate the vector product 𝑎 × 𝑏 component by com- ponent is given by the symbolic determinant 𝑒 𝑎 𝑏 + − + 𝑥 𝑥 𝑥 𝑎 × 𝑏 = 𝑒𝑦 𝑎𝑦 𝑏𝑦 or, sloppily 𝑎 × 𝑏 = 𝑎𝑥 𝑎𝑦 𝑎𝑧 . (23) 𝑏 𝑏 𝑏 𝑒𝑧 𝑎𝑧 𝑏𝑧 𝑥 𝑦 𝑧 These symbolic determinants are easy to remember and easy to perform, both with letters copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net and with numerical values. (Here, 𝑒𝑥 is the unit basis vector in the 𝑥 direction.) Written out, the symbolic determinants are equivalent to the relation 𝑎 × 𝑏 = (𝑎𝑦 𝑏𝑧 − 𝑏𝑦 𝑎𝑧 , 𝑏𝑥 𝑎𝑧 − 𝑎𝑥 𝑏𝑧 , 𝑎𝑥 𝑏𝑦 − 𝑏𝑥 𝑎𝑦 ) (24) which is harder to remember, though. Challenge 199 e Show that the parallelepiped spanned by three arbitrary vectors 𝑎, 𝑏 and 𝑐 has the volume 𝑉 = 𝑐 (𝑎 × 𝑏). Show that the pyramid or tetrahedron formed by the same three Challenge 200 e vectors has one sixth of that volume. Rotation and angular momentum Rotation keeps us alive. Without the change of day and night, we would be either fried or frozen to death, depending on our location on our planet. But rotation appears in many other settings, as Table 22 shows. A short exploration of rotation is thus appropriate. All objects have the ability to rotate. We saw before that a body is described by its reluctance to move, which we called mass; similarly, a body also has a reluctance to turn. This quantity is called its moment of inertia and is often abbreviated Θ – pronounced ‘theta’. The speed or rate of rotation is described by angular velocity, usually abbreviated 𝜔 – pronounced ‘omega’. A few values found in nature are given in Table 22. The observables that describe rotation are similar to those describing linear motion, 4 from objects and images to conservation 117 TA B L E 22 Some measured rotation frequencies. O b s e r va t i o n Angular velocity 𝜔 = 2π/𝑇 Galactic rotation 2π ⋅ 0.14 ⋅ 10−15 / s 6 = 2π /(220 ⋅ 10 a) Average Sun rotation around its axis 2π ⋅3.8 ⋅ 10−7 / s = 2π / 30 d Typical lighthouse 2π ⋅ 0.08/ s Pirouetting ballet dancer 2π ⋅ 3/ s Ship’s diesel engine 2π ⋅ 5/ s Helicopter rotor 2π ⋅ 5.3/ s Washing machine up to 2π ⋅ 20/ s Bacterial flagella 2π ⋅ 100/ s Fast CD recorder up to 2π ⋅ 458/ s Racing car engine up to 2π ⋅ 600/ s Fastest turbine built 2π ⋅ 103 / s Motion Mountain – The Adventure of Physics Fastest pulsars (rotating stars) up to at least 2π ⋅ 716/ s Ultracentrifuge > 2π ⋅ 3 ⋅ 103 / s Dental drill up to 2π ⋅ 13 ⋅ 103 / s Technical record 2π ⋅ 333 ⋅ 103 / s Proton rotation 2π ⋅ 1020 / s Highest possible, Planck angular velocity 2π⋅ 1035 / s copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net as shown in Table 24. Like mass, the moment of inertia is defined in such a way that the sum of angular momenta 𝐿 – the product of moment of inertia and angular velocity – is conserved in systems that do not interact with the outside world: ∑ Θ𝑖 𝜔𝑖 = ∑ 𝐿 𝑖 = const . (25) 𝑖 𝑖 In the same way that the conservation of linear momentum defines mass, the conser- vation of angular momentum defines the moment of inertia. Angular momentum is a concept introduced in the 1730s and 1740s by Leonhard Euler and Daniel Bernoulli. The moment of inertia can be related to the mass and shape of a body. If the body is imagined to consist of small parts or mass elements, the resulting expression is Θ = ∑ 𝑚𝑛 𝑟𝑛2 , (26) 𝑛 where 𝑟𝑛 is the distance from the mass element 𝑚𝑛 to the axis of rotation. Can you con- Challenge 201 e firm the expression? Therefore, the moment of inertia of a body depends on the chosen Challenge 202 s axis of rotation. Can you confirm that this is so for a brick? In contrast to the case of mass, there is no conservation of the moment of inertia. In fact, the value of the moment of inertia depends both on the direction and on the 118 4 from objects and images to conservation TA B L E 23 Some measured angular momentum values. O b s e r va t i o n Angular momentum Smallest observed value in nature, ℏ/2, in elementary 0.53 ⋅ 10−34 Js matter particles (fermions) Spinning top 5 ⋅ 10−6 Js CD (compact disc) playing c. 0.029 Js Walking man (around body axis) c. 4 Js Dancer in a pirouette 5 Js Typical car wheel at 30 m/s 10 Js Typical wind generator at 12 m/s (6 Beaufort) 104 Js Earth’s atmosphere 1 to 2 ⋅ 1026 Js Earth’s oceans 5 ⋅ 1024 Js Earth around its axis 7.1 ⋅ 1033 Js Moon around Earth 2.9 ⋅ 1034 Js Motion Mountain – The Adventure of Physics Earth around Sun 2.7 ⋅ 1040 Js Sun around its axis 1.1 ⋅ 1042 Js Jupiter around Sun 1.9 ⋅ 1043 Js Solar System around Sun 3.2 ⋅ 1043 Js Milky Way 1068 Js All masses in the universe 0 (within measurement error) TA B L E 24 Correspondence between linear and rotational motion. Q ua nt i t y Linear motion R o tat i o na l copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net motion State time 𝑡 time 𝑡 position 𝑥 angle 𝜑 momentum 𝑝 = 𝑚𝑣 angular momentum 𝐿 = Θ𝜔 energy 𝑚𝑣2 /2 energy Θ𝜔2 /2 Motion velocity 𝑣 angular velocity 𝜔 acceleration 𝑎 angular acceleration 𝛼 Reluctance to move mass 𝑚 moment of inertia Θ Motion change force 𝑚𝑎 torque Θ𝛼 location of the axis used for its definition. For each axis direction, one distinguishes an intrinsic moment of inertia, when the axis passes through the centre of mass of the body, from an extrinsic moment of inertia, when it does not.* In the same way, we distinguish * Extrinsic and intrinsic moment of inertia are related by Θext = Θint + 𝑚𝑑2 , (27) where 𝑑 is the distance between the centre of mass and the axis of extrinsic rotation. This relation is called Challenge 203 s Steiner’s parallel axis theorem. Are you able to deduce it? 4 from objects and images to conservation 119 middle finger: "r x p" 2 𝐿 = 𝑟 × 𝑝 = Θ𝜔 = 𝑚𝑟 𝜔 fingers in rotation sense; thumb 𝑟 shows index: "p" 𝐴 angular thumb: "r" momentum 𝑝 = 𝑚𝑣 = 𝑚𝜔 × 𝑟 F I G U R E 75 Angular momentum and other quantities for a point particle in circular motion, and the two versions of the right-hand rule. Motion Mountain – The Adventure of Physics frictionless axis copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 76 Can the ape reach the F I G U R E 77 How a snake turns itself around its axis. banana? intrinsic and extrinsic angular momenta. (By the way, the centre of mass of a body is that imaginary point which moves straight during vertical fall, even if the body is rotating. Challenge 204 s Can you find a way to determine its location for a specific body?) We now define the rotational energy as 𝐿2 𝐸rot = 1 2 Θ 𝜔2 = . (28) 2Θ The expression is similar to the expression for the kinetic energy of a particle. For rotating objects with a fixed shape, rotational energy is conserved. Can you guess how much larger the rotational energy of the Earth is compared with Challenge 205 s the yearly electricity usage of humanity? In fact, if you could find a way to harness the Earth’s rotational energy, you would become famous. Every object that has an orientation also has an intrinsic angular momentum. (What Challenge 206 s about a sphere?) Therefore, point particles do not have intrinsic angular momenta – at 120 4 from objects and images to conservation least in classical physics. (This statement will change in quantum theory.) The extrinsic angular momentum 𝐿 of a point particle is defined as 𝐿=𝑟×𝑝 (29) where 𝑝 is the momentum of the particle and 𝑟 the position vector. The angular mo- mentum thus points along the rotation axis, following the right-hand rule, as shown in Figure 75. A few values observed in nature are given in Table 23. The definition implies Challenge 207 e that the angular momentum can also be determined using the expression 2𝐴(𝑡)𝑚 𝐿= , (30) 𝑡 where 𝐴(𝑡) is the area swept by the position vector 𝑟 of the particle during time 𝑡. For example, by determining the swept area with the help of his telescope, Johannes Kepler discovered in the year 1609 that each planet orbiting the Sun has an angular momentum Motion Mountain – The Adventure of Physics value that is constant over time. A physical body can rotate simultaneously about several axes. The film of Figure 108 Page 148 shows an example: The top rotates around its body axis and around the vertical at the same time. A detailed exploration shows that the exact rotation of the top is given by Challenge 208 e the vector sum of these two rotations. To find out, ‘freeze’ the changing rotation axis at Page 161 a specific time. Rotations thus are a type of vectors. As in the case of linear motion, rotational energy and angular momentum are not always conserved in the macroscopic world: rotational energy can change due to fric- tion, and angular momentum can change due to external forces (torques). But for closed (undisturbed) systems, both angular momentum and rotational energy are always con- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net served. In particular, on a microscopic scale, most objects are undisturbed, so that con- servation of rotational energy and angular momentum usually holds on microscopic scales. Angular momentum is conserved. This statement is valid for any axis of a physical system, provided that external forces (torques) play no role. To make the point, Jean-Marc Ref. 2 Lévy-Leblond poses the problem of Figure 76. Can the ape reach the banana without leaving the plate, assuming that the plate on which the ape rests can turn around the axis Challenge 209 s without any friction? We note that many effects of rotation are the same as for acceleration: both accelera- tion and rotation of a car pushed us in our seats. Therefore, many sensors for rotation are Page 86 the same as the acceleration sensors we explored above. But a few sensors for rotation Page 141 are fundamentally new. In particular, we will meet the gyroscope shortly. On a frictionless surface, as approximated by smooth ice or by a marble floor covered by a layer of oil, it is impossible to move forward. In order to move, we need to push against something. Is this also the case for rotation? Surprisingly, it is possible to turn even without pushing against something. You can check this on a well-oiled rotating office chair: simply rotate an arm above the head. After each turn of the hand, the orientation of the chair has changed by a small amount. In- deed, conservation of angular momentum and of rotational energy do not prevent bodies from changing their orientation. Cats learn this in their youth. After they have learned 4 from objects and images to conservation 121 ωr ω d = vp P ωR r ω R = vaxis d R C F I G U R E 78 The velocities and unit vectors F I G U R E 79 A simulated photograph of a for a rolling wheel. rolling wheel with spokes. Motion Mountain – The Adventure of Physics the trick, if cats are dropped legs up, they can turn themselves in such a way that they Ref. 94 always land feet first. Snakes also know how to rotate themselves, as Figure 77 shows. Also humans have the ability: during the Olympic Games you can watch board divers and gymnasts perform similar tricks. Rotation thus differs from translation in this im- Challenge 210 d portant aspect. (Why?) Rolling wheels Rotation is an interesting phenomenon in many ways. A rolling wheel does not turn copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net around its axis, but around its point of contact. Let us show this. A wheel of radius 𝑅 is rolling if the speed of the axis 𝑣axis is related to the angular velocity 𝜔 by 𝑣 𝜔 = axis . (31) 𝑅 For any point P on the wheel, with distance 𝑟 from the axis, the velocity 𝑣P is the sum of the motion of the axis and the motion around the axis. Figure 78 shows that 𝑣P is orthogonal to 𝑑, the distance between the point P and the contact point of the wheel. Challenge 211 e The figure also shows that the length ratio between 𝑣P and 𝑑 is the same as between 𝑣axis and 𝑅. As a result, we can write 𝑣P = 𝜔 × 𝑑 , (32) which shows that a rolling wheel does indeed rotate about its point of contact with the ground. Surprisingly, when a wheel rolls, some points on it move towards the wheel’s axis, some stay at a fixed distance and others move away from it. Can you determine where Challenge 212 s these various points are located? Together, they lead to an interesting pattern when a Ref. 95 rolling wheel with spokes, such as a bicycle wheel, is photographed, as show in Figure 79. Ref. 96 With these results you can tackle the following beautiful challenge. When a turning bicycle wheel is deposed on a slippery surface, it will slip for a while, then slip and roll, 122 4 from objects and images to conservation F I G U R E 80 The measured motion of a walking human (© Ray McCoy). Motion Mountain – The Adventure of Physics and finally roll only. How does the final speed depend on the initial speed and on the Challenge 213 d friction? How d o we walk and run? “ ” Golf is a good walk spoiled. The Allens Why do we move our arms when walking or running? To save energy or to be graceful? In fact, whenever a body movement is performed with as little energy as possible, it is both copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net natural and graceful. This correspondence can indeed be taken as the actual definition of grace. The connection is common knowledge in the world of dance; it is also a central Ref. 20 aspect of the methods used by actors to learn how to move their bodies as beautifully as possible. To convince yourself about the energy savings, try walking or running with your arms fixed or moving in the opposite direction to usual: the effort required is considerably higher. In fact, when a leg is moved, it produces a torque around the body axis which has to be counterbalanced. The method using the least energy is the swinging of arms, as depicted in Figure 80. Since the arms are lighter than the legs, they must move further from the axis of the body, to compensate for the momentum; evolution has therefore moved the attachment of the arms, the shoulders, farther apart than those of the legs, the hips. Animals on two legs but without arms, such as penguins or pigeons, have more difficulty walking; they have to move their whole torso with every step. Ref. 97 Measurements show that all walking animals follow 𝑣max walking = (2.2 ± 0.2 m/s) √𝑙/m . (33) Indeed, walking, the moving of one leg after the other, can be described as a concaten- ation of (inverted) pendulum swings. The pendulum length is given by the leg length 𝑙. The typical time scale of a pendulum is 𝑡 ∼ √𝑙/𝑔 . The maximum speed of walking then 4 from objects and images to conservation 123 becomes 𝑣 ∼ 𝑙/𝑡 ∼ √𝑔𝑙 , which is, up to a constant factor, the measured result. Which muscles do most of the work when walking, the motion that experts call gait? Ref. 98 In 1980, Serge Gracovetsky found that in human gait a large fraction of the power comes from the muscles along the spine, not from those of the legs. (Indeed, people without legs are also able to walk. However, a number of muscles in the legs must work in order to walk normally.) When you take a step, the lumbar muscles straighten the spine; this automatically makes it turn a bit to one side, so that the knee of the leg on that side automatically comes forward. When the foot is moved, the lumbar muscles can relax, and then straighten again for the next step. In fact, one can experience the increase in Challenge 214 e tension in the back muscles when walking without moving the arms, thus confirming where the human engine, the so-called spinal engine is located. Human legs differ from those of apes in a fundamental aspect: humans are able to run. In fact the whole human body has been optimized for running, an ability that no other primate has. The human body has shed most of its hair to achieve better cooling, has evolved the ability to run while keeping the head stable, has evolved the right length of arms for proper balance when running, and even has a special ligament in the back that Motion Mountain – The Adventure of Physics works as a shock absorber while running. In other words, running is the most human of all forms of motion. Curiosities and fun challenges ab ou t mass, conservation and rotation “ It is a mathematical fact that the casting of this pebble from my hand alters the centre of gravity ” of the universe. Thomas Carlyle,* Sartor Resartus III. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net A cup with water is placed on a weighing scale, as shown in Figure 81. How does the Challenge 216 e mass result change if you let a piece of metal attached to a string hang into the water? ∗∗ Take ten coins of the same denomination. Put nine of them on a table and form a closed loop with them of any shape you like, as shown in Figure 82. (The nine coins thus look like a section of pearl necklace where the pearls touch each other.) Now take the tenth coin and let it roll around the loop, thus without ever sliding it. How many turns does Challenge 217 e this last coin make during one round? ∗∗ Conservation of momentum is best studied by playing and exploring billiards, snooker or pool. The best introduction is provided by the trickshot films found across the internet. Are you able to use momentum conservation to deduce ways for improving your billiards Challenge 218 e game? Another way to explore momentum conservation is to explore the ball-chain, or ball collision pendulum, that was invented by Edme Mariotte. Decades later, Newton claimed it as his, as he often did with other people’s results. Playing with the toy is fun – and explaining its behaviour even more. Indeed, if you lift and let go three balls on one side, Challenge 215 s * Thomas Carlyle (1797–1881), Scottish essayist. Do you agree with the quotation? 124 4 from objects and images to conservation Motion Mountain – The Adventure of Physics F I G U R E 81 How does the displayed weight value change when an object hangs into the water? How many rotations? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 82 How many rotations does the tenth coin perform in one round? you will see three balls departing on the other side; for the explanation of this behaviour the conservation of momentum and energy conservation are not sufficient, as you should Challenge 219 d be able to find out. Are you able to build a high-precision ball-chain? ∗∗ Challenge 220 s There is a well-known way to experience 81 sunrises in just 80 days. How? ∗∗ 4 from objects and images to conservation 125 F I G U R E 83 The ball-chain or cradle invented by Mariotte allows exploring momentum conservation, energy conservation, and the difﬁculties of precision manufacturing (© www. questacon.edu.au). Motion Mountain – The Adventure of Physics Walking is a source of many physics problems. When climbing a mountain, the most Ref. 99 energy-effective way is not always to follow the steepest ascent; indeed, for steep slopes, zig-zagging is more energy efficient. Why? And can you estimate the slope angle at which Challenge 221 s this will happen? ∗∗ Asterix and his friends from the homonymous comic strip, fear only one thing: that the Challenge 222 e sky might fall down. Is the sky an object? An image? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Death is a physical process and thus can be explored. In general, animals have a lifespan 𝑇 that scales with fourth root of their mass 𝑀. In other terms, 𝑇 = 𝑀1/4 . This is valid from bacteria to insects to blue whales. Animals also have a power consumption per mass, or metabolic rate per mass, that scales with the inverse fourth root. We conclude that death occurs for all animals when a certain fixed energy consumption per mass has been achieved. This is indeed the case; death occurs for most animals when they have Ref. 100 consumed around 1 GJ/kg. (But quite a bit later for humans.) This surprisingly simple result is valid, on average, for all known animals. Note that the argument is only valid when different species are compared. The de- pendence on mass is not valid when specimen of the same species are compared. (You cannot live longer by eating less.) In short, animals die after they metabolized 1 GJ/kg. In other words, once we ate all the calories we were designed for, we die. ∗∗ A car at a certain speed uses 7 litres of gasoline per 100 km. What is the combined air Challenge 223 s and rolling resistance? (Assume that the engine has an efficiency of 25 %.) ∗∗ A cork is attached to a thin string a metre long. The string is passed over a long rod 126 4 from objects and images to conservation F I G U R E 84 Is it safe to let the cork go? atmosphere mountain height h ocean plain ocean ocean Motion Mountain – The Adventure of Physics ocean crust solid solidcontinental continentalcrust crust ocean crust depth d liquid magma of the mantle F I G U R E 85 A simple model for continents and mountains. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net held horizontally, and a wine glass is attached at the other end. If you let go the cork in Challenge 224 s Figure 84, nothing breaks. Why not? And what happens exactly? ∗∗ In 1907, Duncan MacDougalls, a medical doctor, measured the weight of dying people, Ref. 101 in the hope to see whether death leads to a mass change. He found a sudden decrease between 10 and 20 g at the moment of death. He attributed it to the soul exiting the body. Challenge 225 s Can you find a more satisfying explanation? ∗∗ It is well known that the weight of a one-year-old child depends on whether it wants to be carried or whether it wants to reach the floor. Does this contradict mass conservation? Challenge 226 e ∗∗ The Earth’s crust is less dense (2.7 kg/l) than the Earth’s mantle (3.1 kg/l) and floats on it. As a result, the lighter crust below a mountain ridge must be much deeper than below a plain. If a mountain rises 1 km above the plain, how much deeper must the crust be Challenge 227 s below it? The simple block model shown in Figure 85 works fairly well; first, it explains why, near mountains, measurements of the deviation of free fall from the vertical line 4 from objects and images to conservation 127 lead to so much lower values than those expected without a deep crust. Later, sound measurements confirmed directly that the continental crust is indeed thicker beneath mountains. ∗∗ All homogeneous cylinders roll down an inclined plane in the same way. True or false? Challenge 228 e And what about spheres? Can you show that spheres roll faster than cylinders? ∗∗ Challenge 229 s Which one rolls faster: a soda can filled with liquid or a soda can filled with ice? (And how do you make a can filled with ice?) ∗∗ Take two cans of the same size and weight, one full of ravioli and one full of peas. Which Challenge 230 e one rolls faster on an inclined plane? Motion Mountain – The Adventure of Physics ∗∗ Another difference between matter and images: matter smells. In fact, the nose is a matter sensor. The same can be said of the tongue and its sense of taste. ∗∗ Take a pile of coins. You can push out the coins, starting with the one at the bottom, by shooting another coin over the table surface. The method also helps to visualize two- Challenge 231 e dimensional momentum conservation. ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net In early 2004, two men and a woman earned £ 1.2 million in a single evening in a Lon- don casino. They did so by applying the formulae of Galilean mechanics. They used the method pioneered by various physicists in the 1950s who built various small computers that could predict the outcome of a roulette ball from the initial velocity imparted by the Ref. 102 croupier. In the case of Britain, the group added a laser scanner to a smartphone that measured the path of a roulette ball and predicted the numbers where it would arrive. In this way, they increased the odds from 1 in 37 to about 1 in 6. After six months of investigations, Scotland Yard ruled that they could keep the money they won. In fact around the same time, a few people earned around 400 000 euro over a few weeks by using the same method in Germany, but with no computer at all. In certain casinos, machines were throwing the roulette ball. By measuring the position of the zero to the incoming ball with the naked eye, these gamblers were able to increase the odds of the bets they placed during the last allowed seconds and thus win a considerable sum purely through fast reactions. ∗∗ Challenge 232 s Does the universe rotate? ∗∗ The toy of Figure 86 shows interesting behaviour: when a number of spheres are lifted 128 4 from objects and images to conservation before the hit observed after the hit F I G U R E 86 A well-known toy. before the hit observed after the hit V=0 v V‘ v’ 0 F I G U R E 87 An elastic collision 2L,2M L, M that seems not to Motion Mountain – The Adventure of Physics obey energy conservation. and dropped to hit the resting ones, the same number of spheres detach on the other side, whereas the previously dropped spheres remain motionless. At first sight, all this seems to follow from energy and momentum conservation. However, energy and momentum conservation provide only two equations, which are insufficient to explain or determine the behaviour of five spheres. Why then do the spheres behave in this way? And why do copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 233 d they all swing in phase when a longer time has passed? ∗∗ A surprising effect is used in home tools such as hammer drills. We remember that when a small ball elastically hits a large one at rest, both balls move after the hit, and the small Ref. 103 one obviously moves faster than the large one. Despite this result, when a short cylinder hits a long one of the same diameter and material, but with a length that is some integer multiple of that of the short one, something strange happens. After the hit, the small cylinder remains almost at rest, whereas the large one moves, as shown in Figure 87. Even though the collision is elastic, conservation of energy seems not to hold in this case. (In fact this is the reason that demonstrations of elastic collisions in schools are Challenge 234 d always performed with spheres.) What happens to the energy? ∗∗ Is the structure shown in Figure 88 possible? ∗∗ Does a wall get a stronger jolt when it is hit by a ball rebounding from it or when it is hit Challenge 235 s by a ball that remains stuck to it? ∗∗ 4 from objects and images to conservation 129 wall ladder F I G U R E 88 Is this possible? F I G U R E 89 How does the ladder fall? Motion Mountain – The Adventure of Physics Housewives know how to extract a cork of a wine bottle using a cloth or a shoe. Can you Challenge 236 s imagine how? They also know how to extract the cork with the cloth if the cork has fallen inside the bottle. How? ∗∗ The sliding ladder problem, shown schematically in Figure 89, asks for the detailed mo- tion of the ladder over time. The problem is more difficult than it looks, even if friction is not taken into account. Can you say whether the lower end always touches the floor, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 237 s or if is lifted into the air for a short time interval? ∗∗ A homogeneous ladder of length 5 m and mass 30 kg leans on a wall. The angle is 30°; the static friction coefficient on the wall is negligible, and on the floor it is 0.3. A person of mass 60 kg climbs the ladder. What is the maximum height the person can climb before the ladder starts sliding? This and many puzzles about ladders can be found on www. mathematische-basteleien.de/leiter.htm. ∗∗ Ref. 104 A common fly on the stern of a 30 000 ton ship of 100 m length tilts it by less than the diameter of an atom. Today, distances that small are easily measured. Can you think of Challenge 238 s at least two methods, one of which should not cost more than 2000 euro? ∗∗ Is the image of three stacked spinning tops shown in Figure 90 a true photograph, show- ing a real observation, or is it the result of digital composition, showing an impossible Challenge 239 ny situation? ∗∗ Challenge 240 s How does the kinetic energy of a rifle bullet compare to that of a running man? 130 4 from objects and images to conservation F I G U R E 90 Is this a possible situation or is it a fake photograph? (© Wikimedia) Motion Mountain – The Adventure of Physics ∗∗ Challenge 241 s What happens to the size of an egg when one places it in a jar of vinegar for a few days? ∗∗ What is the amplitude of a pendulum oscillating in such a way that the absolute value of Challenge 242 s its acceleration at the lowest point and at the return point are equal? ∗∗ Can you confirm that the value of the acceleration of a drop of water falling through mist copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 243 d is 𝑔/7? ∗∗ You have two hollow spheres: they have the same weight, the same size and are painted in the same colour. One is made of copper, the other of aluminium. Obviously, they fall with Challenge 244 s the same speed and acceleration. What happens if they both roll down a tilted plane? ∗∗ Challenge 245 s What is the shape of a rope when rope jumping? ∗∗ Challenge 246 s How can you determine the speed of a rifle bullet with only a scale and a metre stick? ∗∗ Why does a gun make a hole in a door but cannot push it open, in exact contrast to what Challenge 247 e a finger can do? ∗∗ Challenge 248 s What is the curve described by the midpoint of a ladder sliding down a wall? ∗∗ 4 from objects and images to conservation 131 Motion Mountain – The Adventure of Physics F I G U R E 91 A commercial clock that needs no special energy source, because it takes its energy from the environment (© Jaeger-LeCoultre). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net A high-tech company, see www.enocean.com, sells electric switches for room lights that have no cables and no power cell (battery). You can glue such a switch to the centre of a Challenge 249 s window pane. How is this possible? ∗∗ For over 50 years now, a famous Swiss clockmaker is selling table clocks with a rotating pendulum that need no battery and no manual rewinding, as they take up energy from the environment. A specimen is shown in Figure 91. Can you imagine how this clock Challenge 250 s works? ∗∗ Ship lifts, such as the one shown in Figure 92, are impressive machines. How does the Challenge 251 s weight of the lift change when the ship enters? ∗∗ Challenge 252 e How do you measure the mass of a ship? ∗∗ All masses are measured by comparing them, directly or indirectly, to the standard kilo- gram in Sèvres near Paris. For a number of years, there was serious doubt that the stand- 132 4 from objects and images to conservation Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 92 The spectacular ship lift at Strépy-Thieux in Belgium. What engine power is needed to lift a ship, if the right and left lifts were connected by ropes or by a hydraulic system? (© Jean-Marie Hoornaert) 4 from objects and images to conservation 133 F I G U R E 93 The famous Celtic wobble stone – above and right – and a version made by bending a spoon – bottom left (© Ed Keath). Motion Mountain – The Adventure of Physics ard kilogram was losing weight, possibly through outgassing, with an estimated rate of around 0.5 μg/a. This was an awkward situation, and there has been a vast, worldwide effort to find a better definition of the kilogram. Such an improved definition had to be simple, precise, and make trips to Sèvres unnecessary. Finally, an alternative was found and defined in 2019. ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Which engine is more efficient: a moped or a human on a bicycle? ∗∗ Both mass and moment of inertia can be defined and measured both with and without Challenge 253 e contact. Can you do so? ∗∗ Figure 93 shows the so-called Celtic wobble stone, also called anagyre or rattleback, a stone Ref. 103 that starts rotating on a plane surface when it is put into up-and-down oscillation. The size can vary between a few centimetres and a few metres. By simply bending a spoon one can realize a primitive form of this strange device, if the bend is not completely symmetrical. The rotation is always in the same direction. If the stone is put into rotation in the wrong direction, after a while it stops and starts rotating in the other sense! Can Challenge 254 d you explain the effect that seems to contradict the conservation of angular momentum? ∗∗ A beautiful effect, the chain fountain, was discovered in 2013 by Steve Mould. Certain chains, when flowing out of a container, first shoot up in the air. See the video at www. youtube.com/embed/_dQJBBklpQQ and the story of the discovery at stevemould.com. Challenge 255 ny Can you explain the effect to your grandmother? 134 4 from objects and images to conservation Summary on conservation in motion “ The gods are not as rich as one might think: what they give to one, they take away from the ” other. Antiquity We have encountered four conservation principles that are valid for the motion of all closed systems in everyday life: — conservation of total linear momentum, — conservation of total angular momentum, — conservation of total energy, — conservation of total mass. None of these conservation principles applies to the motion of images. These principles thus allow us to distinguish objects from images. The conservation principles are among the great results in science. They limit the sur- prises that nature can offer: conservation means that linear momentum, angular mo- Motion Mountain – The Adventure of Physics mentum, and mass–energy can neither be created from nothing, nor can they disappear into nothing. Conservation limits creation. The quote below the section title expresses this idea. Page 280 Later on we will find out that these results could have been deduced from three simple observations: closed systems behave the same independently of where they are, in what direction they are oriented and of the time at which they are set up. In more abstract terms, physicists like to say that all conservation principles are consequences of the in- variances, or symmetries, of nature. Later on, the theory of special relativity will show that energy and mass are conserved only when taken together. Many adventures still await us. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Chapter 5 F ROM T H E ROTAT ION OF T H E E A RT H TO T H E R E L AT I V I T Y OF MOT ION “ ” Eppur si muove! Anonymous** I s the Earth rotating? The search for definite answers to this question gives an nteresting cross-section of the history of classical physics. In the fourth century, Motion Mountain – The Adventure of Physics n ancient Greece, Hicetas and Philolaus, already stated that the Earth rotates. Then, in the year 265 b ce, Aristarchus of Samos was the first to explore the issue in detail. He had measured the parallax of the Moon (today known to be up to 0.95°) and of the Sun (today known to be 8.8 ).*** The parallax is an interesting effect; it is the angle describing the difference between the directions of a body in the sky when seen by an observer on the surface of the Earth and when seen by a hypothetical observer at the Earth’s centre. (See Figure 94.) Aristarchus noticed that the Moon and the Sun wobble across the sky, and this wobble has a period of 24 hours. He concluded that the Earth rotates. It seems that Aristarchus received death threats for his conclusion. Aristarchus’ observation yields an even more powerful argument than the trails of the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 256 e stars shown in Figure 95. Can you explain why? (And how do the trails look at the most Challenge 257 s populated places on Earth?) Experiencing Figure 95 might be one reason that people dreamt and still dream about reaching the poles. Because the rotation and the motion of the Earth make the poles ex- tremely cold places, the adventure of reaching them is not easy. Many tried unsuccess- Ref. 136 fully. A famous crook, Robert Peary, claimed to have reached the North Pole in 1909. (In fact, Roald Amundsen reached both the South and the North Pole first.) Among oth- ers, Peary claimed to have taken a picture there, but that picture, which went round the Challenge 258 s world, turned out to be one of the proofs that he had not been there. Can you imagine how? If the Earth rotates instead of being at rest, said the unconvinced, the speed at the equator has the substantial value of 0.46 km/s. How did Galileo explain why we do not Challenge 259 e feel or notice this speed? Vol. II, page 17 Measurements of the aberration of light also show the rotation of the Earth; it can be detected with a telescope while looking at the stars. The aberration is a change of the ** ‘And yet she moves’ is the sentence about the Earth attributed, most probably incorrectly, to Galileo since the 1640s. It is true, however, that at his trial he was forced to publicly retract the statement of a moving Earth to save his life. For more details of this famous story, see the section on page 335. *** For the definition of the concept of angle, see page 67, and for the definition of the measurement units for angle see Appendix B. 136 5 from the rotation of the earth rotating Moon sky Earth or and Sun stars N F I G U R E 94 The parallax – not drawn to scale. expected light direction, which we will discuss shortly. At the Equator, Earth rotation adds an angular deviation of 0.32 , changing sign every 12 hours, to the aberration due to Motion Mountain – The Adventure of Physics the motion of the Earth around the Sun, about 20.5 . In modern times, astronomers have found a number of additional proofs for the rotation of the Earth, but none is accessible to the man on the street. Also the measurements showing that the Earth is not a sphere, but is flattened at the poles, confirmed the rotation of the Earth. Figure 96 illustrates the situation. Again, how- ever, this eighteenth-century measurement by Maupertuis* is not accessible to everyday observation. Then, in the years 1790 to 1792 in Bologna, Giovanni Battista Guglielmini (1763–1817) finally succeeded in measuring what Galileo and Newton had predicted to be the simplest copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net proof for the Earth’s rotation. On the rotating Earth, objects do not fall vertically, but are slightly deviated to the east. This deviation appears because an object keeps the larger ho- rizontal velocity it had at the height from which it started falling, as shown in Figure 97. Guglielmini’s result was the first non-astronomical proof of the Earth’s rotation. The experiments were repeated in 1802 by Johann Friedrich Benzenberg (1777–1846). Using metal balls which he dropped from the Michaelis tower in Hamburg – a height of 76 m – Benzenberg found that the deviation to the east was 9.6 mm. Can you confirm that the value measured by Benzenberg almost agrees with the assumption that the Earth turns Challenge 260 d once every 24 hours? There is also a much smaller deviation towards the Equator, not measured by Guglielmini, Benzenberg or anybody after them up to this day; however, it completes the list of effects on free fall by the rotation of the Earth. Both deviations from vertical fall are easily understood if we use the result (described Page 192 below) that falling objects describe an ellipse around the centre of the rotating Earth. The elliptical shape shows that the path of a thrown stone does not lie on a plane for an * Pierre Louis Moreau de Maupertuis (1698–1759), physicist and mathematician, was one of the key figures in the quest for the principle of least action, which he named in this way. He was also the founding president of the Berlin Academy of Sciences. Maupertuis thought that the principle reflected the maximization of goodness in the universe. This idea was thoroughly ridiculed by Voltaire in his Histoire du Docteur Akakia et du natif de Saint-Malo, 1753. (Read it at gallica.bnf.fr/ark:/12148/bpt6k6548988f.texteImage.) Maupertuis performed his measurement of the Earth to distinguish between the theory of gravitation of Newton and that of Descartes, who had predicted that the Earth is elongated at the poles, instead of flattened. to the relativity of motion 137 Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 95 The motion of the stars during the night, observed on 1 May 2012 from the South Pole, together with the green light of an aurora australis (© Robert Schwartz). sphere Earth 5 km 5 km Equator F I G U R E 96 Earth’s deviation from spherical shape due to its rotation (exaggerated). observer standing on Earth; for such an observer, the exact path of a stone thus cannot 138 5 from the rotation of the earth 𝑣ℎ = 𝜔(𝑅 + ℎ) ℎ North ℎ 𝑣 = 𝜔𝑅 𝜑 North Equator East F I G U R E 97 The deviations of free fall towards the east and South towards the Equator due to the rotation of the Earth. Motion Mountain – The Adventure of Physics c. 0.2 Hz copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net c. 3 m F I G U R E 98 A typical carousel allows observing the Coriolis effect in its most striking appearance: if a person lets a ball roll with the proper speed and direction, the ball is deﬂected so strongly that it comes back to her. be drawn on a flat piece of paper! In 1798, Pierre Simon Laplace* explained how bodies move on the rotating Earth and Ref. 106 showed that they feel an apparent force. In 1835, Gustave-Gaspard Coriolis then reformu- lated and simplified the description. Imagine a ball that rolls over a table. For a person on the floor, the ball rolls in a straight line. Now imagine that the table rotates. For the per- son on the floor, the ball still rolls in a straight line. But for a person on the rotating table, the ball traces a curved path. In short, any object that travels in a rotating background is subject to a transversal acceleration. The acceleration, discovered by Laplace, is nowadays called Coriolis acceleration or Coriolis effect. On a rotating background, travelling objects deviate from the straight line. The best way to understand the Coriolis effect is to exper- * Pierre Simon Laplace (b. 1749 Beaumont-en-Auge, d. 1827 Paris), important mathematician. His famous treatise Traité de mécanique céleste appeared in five volumes between 1798 and 1825. He was the first to propose that the Solar System was formed from a rotating gas cloud, and one of the first people to imagine and explore black holes. to the relativity of motion 139 Motion Mountain – The Adventure of Physics F I G U R E 99 Cyclones, with their low pressure centre, differ in rotation sense between the southern hemisphere, here cyclone Larry in 2006, and the northern hemisphere, here hurricane Katrina in 2005. (Courtesy NOAA) ience it yourself; this can be done on a carousel, as shown in Figure 98. Watching films copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Ref. 107 on the internet on the topic is also helpful. You will notice that on a rotating carousel it is not easy to hit a target by throwing or rolling a ball. Also the Earth is a rotating background. On the northern hemisphere, the rotation is anticlockwise. As the result, any moving object is slightly deviated to the right (while the magnitude of its velocity stays constant). On Earth, like on all rotating backgrounds, the Coriolis acceleration 𝑎C results from the change of distance to the rotation axis. Can you Challenge 261 s deduce the analytical expression for the Coriolis effect, namely 𝑎C = −2𝜔 × 𝑣? On Earth, the Coriolis acceleration generally has a small value. Therefore it is best ob- served either in large-scale or high-speed phenomena. Indeed, the Coriolis acceleration determines the handedness of many large-scale phenomena with a spiral shape, such as the directions of cyclones and anticyclones in meteorology – as shown in Figure 99 – the general wind patterns on Earth and the deflection of ocean currents and tides. These phe- nomena have opposite handedness on the northern and the southern hemisphere. Most beautifully, the Coriolis acceleration explains why icebergs do not follow the direction Ref. 108 of the wind as they drift away from the polar caps. The Coriolis acceleration also plays a role in the flight of cannon balls (that was the original interest of Coriolis), in satellite Ref. 109 launches, in the motion of sunspots and even in the motion of electrons in molecules. All these Coriolis accelerations are of opposite sign on the northern and southern hemi- spheres and thus prove the rotation of the Earth. For example, in the First World War, many naval guns missed their targets in the southern hemisphere because the engineers had compensated them for the Coriolis effect in the northern hemisphere. 140 5 from the rotation of the earth 𝜓1 N Earth's centre 𝜑 Eq u ato r 𝜓0 𝜓1 F I G U R E 100 The turning motion of a pendulum showing the rotation of the Earth. Only in 1962, after several earlier attempts by other researchers, Asher Shapiro was the Motion Mountain – The Adventure of Physics Ref. 110 first to verify that the Coriolis effect has a tiny influence on the direction of the vortex formed by the water flowing out of a bath-tub. Instead of a normal bath-tub, he had to use a carefully designed experimental set-up because, contrary to an often-heard assertion, no such effect can be seen in a real bath-tub. He succeeded only by carefully eliminat- ing all disturbances from the system; for example, he waited 24 hours after the filling of the reservoir (and never actually stepped in or out of it!) in order to avoid any left-over motion of water that would disturb the effect, and built a carefully designed, completely rotationally-symmetric opening mechanism. Others have repeated the experiment in the Ref. 110 southern hemisphere, finding opposite rotation direction and thus confirming the result. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net In other words, the handedness of usual bath-tub vortices is not caused by the rotation of the Earth, but results from the way the water starts to flow out. (A number of crooks in Quito, a city located on the Equator, show gullible tourists that the vortex in a sink changes when crossing the Equator line drawn on the road.) But let us go on with the story about the Earth’s rotation. In 1851, the physician-turned-physicist Jean Bernard Léon Foucault (b. 1819 Paris, d. 1868 Paris) performed an experiment that removed all doubts and rendered him world-famous practically overnight. He suspended a 67 m long pendulum* in the Panthéon in Paris and showed the astonished public that the direction of its swing changed over time, rotating slowly. To anybody with a few minutes of patience to watch the change of direction, the experiment proved that the Earth rotates. If the Earth did not rotate, the swing of the pendulum would always continue in the same direction. On a rotating Earth, in Paris, the direction changes to the right, in clockwise sense, as shown in Figure 100. The swing direction does not change if the pendulum is located at the Equator, and it changes to the left in the southern hemisphere.** A modern version of the Challenge 262 d * Why was such a long pendulum necessary? Understanding the reasons allows one to repeat the experiment Ref. 111 at home, using a pendulum as short as 70 cm, with the help of a few tricks. To observe Foucault’s effect with a simple set-up, attach a pendulum to your office chair and rotate the chair slowly. Several pendulum an- imations, with exaggerated deviation, can be found at commons.wikimedia.org/wiki/Foucault_pendulum. ** The discovery also shows how precision and genius go together. In fact, the first person to observe the to the relativity of motion 141 pendulum can be observed via the web cam at pendelcam.kip.uni-heidelberg.de; high speed films of the pendulum’s motion during day and night can also be downloaded at www.kip.uni-heidelberg.de/oeffwiss/pendel/zeitraffer/. The time over which the orientation of the pendulum’s swing performs a full turn – the precession time – can be calculated. Study a pendulum starting to swing in the North– Challenge 263 d South direction and you will find that the precession time 𝑇Foucault is given by 23 h 56 min 𝑇Foucault = (34) sin 𝜑 where 𝜑 is the latitude of the location of the pendulum, e.g. 0° at the Equator and 90° at the North Pole. This formula is one of the most beautiful results of Galilean kinematics.* Foucault was also the inventor and namer of the gyroscope. He built the device, shown in Figure 101 and Figure 102, in 1852, one year after his pendulum. With it, he again demonstrated the rotation of the Earth. Once a gyroscope rotates, the axis stays fixed in space – but only when seen from distant stars or galaxies. (By the way, this is not the same as talking about absolute space. Why?) For an observer on Earth, the axis direction Motion Mountain – The Adventure of Physics Challenge 264 s changes regularly with a period of 24 hours. Gyroscopes are now routinely used in ships and in aeroplanes to give the direction of north, because they are more precise and more reliable than magnetic compasses. The most modern versions use laser light running in circles instead of rotating masses.** In 1909, Roland von Eötvös measured a small but surprising effect: due to the rotation of the Earth, the weight of an object depends on the direction in which it moves. As a result, a balance in rotation around the vertical axis does not stay perfectly horizontal: Challenge 266 s the balance starts to oscillate slightly. Can you explain the origin of the effect? Ref. 112 In 1910, John Hagen published the results of an even simpler experiment, proposed copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net by Louis Poinsot in 1851. Two masses are put on a horizontal bar that can turn around a vertical axis, a so-called isotomeograph. Its total mass was 260 kg. If the two masses are slowly moved towards the support, as shown in Figure 103, and if the friction is kept low enough, the bar rotates. Obviously, this would not happen if the Earth were not rotating. Challenge 267 s Can you explain the observation? This little-known effect is also useful for winning bets between physicists. In 1913, Arthur Compton showed that a closed tube filled with water and some small Ref. 113 floating particles (or bubbles) can be used to show the rotation of the Earth. The device is called a Compton tube or Compton wheel. Compton showed that when a horizontal tube filled with water is rotated by 180°, something happens that allows one to prove that the Earth rotates. The experiment, shown in Figure 104, even allows measuring the latitude Challenge 268 d of the point where the experiment is made. Can you guess what happens? Another method to detect the rotation of the Earth using light was first realized in 1913 Georges Sagnac:*** he used an interferometer to produce bright and dark fringes effect was Vincenzo Viviani, a student of Galileo, as early as 1661! Indeed, Foucault had read about Viviani’s work in the publications of the Academia dei Lincei. But it took Foucault’s genius to connect the effect to the rotation of the Earth; nobody had done so before him. * The calculation of the period of Foucault’s pendulum assumes that the precession rate is constant during a rotation. This is only an approximation (though usually a good one). Challenge 265 s ** Can you guess how rotation is detected in this case? *** Georges Sagnac (b. 1869 Périgeux, d. 1928 Meudon-Bellevue) was a physicist in Lille and Paris, friend 142 5 from the rotation of the earth Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 101 The gyroscope: the original system by Foucault with its freely movable spinning top, the mechanical device to bring it to speed, the optical device to detect its motion, the general construction principle, and a modern (triangular) ring laser gyroscope, based on colour change of rotating laser light instead of angular changes of a rotating mass (© CNAM, JAXA). Vol. IV, page 56 of light with two light beams, one circulating in clockwise direction, and the second circulating in anticlockwise direction. The interference fringes are shifted when the whole Ref. 114 system rotates; the faster it rotates, the larger the shift. A modern, high-precision version of the experiment, which uses lasers instead of lamps, is shown in Figure 105. (More Vol. III, page 104 details on interference and fringes are found in volume III.) Sagnac also determined the relationship between the fringe shift and the details of the experiment. The rotation of a complete ring interferometer with angular frequency (vector) Ω produces a fringe shift Challenge 269 s of angular phase Δ𝜑 given by 8π Ω 𝑎 Δ𝜑 = (35) 𝑐𝜆 where 𝑎 is the area (vector) enclosed by the two interfering light rays, 𝜆 their wavelength and 𝑐 the speed of light. The effect is now called the Sagnac effect after its discoverer. It had of the Curies, Langevin, Perrin, and Borel. Sagnac also deduced from his experiment that the speed of light was independent of the speed of its source, and thus confirmed a prediction of special relativity. to the relativity of motion 143 Motion Mountain – The Adventure of Physics F I G U R E 102 A three-dimensional model of Foucault’s original gyroscope: in the pdf verion of this text, the model can be rotated and zoomed by moving copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the cursor over it (© Zach Joseph Espiritu). 144 5 from the rotation of the earth North water- 𝑟 𝑚 𝑚 filled East tube West South Motion Mountain – The Adventure of Physics F I G U R E 103 Showing the rotation of the Earth F I G U R E 104 Demonstrating the rotation of the through the rotation of an axis. Earth with water. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 105 A modern precision ring laser interferometer (© Bundesamt für Kartographie und Geodäsie, Carl Zeiss). Ref. 115 already been predicted 20 years earlier by Oliver Lodge.* Today, Sagnac interferometers are the central part of laser gyroscopes – shown in Figure 101 – and are found in every passenger aeroplane, missile and submarine, in order to measure the changes of their motion and thus to determine their actual position. A part of the fringe shift is due to the rotation of the Earth. Modern high-precision Sagnac interferometers use ring lasers with areas of a few square metres, as shown in * Oliver Lodge (b. 1851, Stoke, d. on-Trent-1940, Wiltshire) was a physicist and spiritualist who studied electromagnetic waves and tried to communicate with the dead. A strange but influential figure, his ideas are often cited when fun needs to be made of physicists; for example, he was one of those (rare) physicists who believed that at the end of the nineteenth-century physics was complete. to the relativity of motion 145 mirror massive metal rod typically 1.5 m F I G U R E 106 Observing the rotation of the Earth in two seconds. Motion Mountain – The Adventure of Physics Figure 105. Such a ring interferometer is able to measure variations in the rotation rates of the Earth of less than one part per million. Indeed, over the course of a year, the length of a day varies irregularly by a few milliseconds, mostly due to influences from the Sun or Ref. 116 the Moon, due to weather changes and due to hot magma flows deep inside the Earth.* But also earthquakes, the El Niño effect in the climate and the filling of large water dams have effects on the rotation of the Earth. All these effects can be studied with such high- precision interferometers; they can also be used for research into the motion of the soil due to lunar tides or earthquakes, and for checks on the theory of special relativity. Finally, in 1948, Hans Bucka developed the simplest experiment so far to show the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Ref. 117 rotation of the Earth. A metal rod allows anybody to detect the rotation of the Earth after only a few seconds of observation, using the set-up of Figure 106. The experiment Challenge 270 s can be easily be performed in class. Can you guess how it works? In summary: all observations show that the Earth surface rotates at 464 m/s at the Equator, a larger value than that of the speed of sound in air, which is about 340 m/s at usual conditions. The rotation of the Earth also implies an acceleration, at the Equator, of 0.034 m/s2 . We are in fact whirling through the universe. How d oes the E arth rotate? Is the rotation of the Earth, the length of the day, constant over geological time scales? That is a hard question. If you find a method leading to an answer, publish it! (The same Ref. 118 is true for the question of whether the length of the year is constant.) Only a few methods are known, as we will find out shortly. The rotation of the Earth is not even constant during a human lifespan. It varies by a few parts in 108 . In particular, on a ‘secular’ time scale, the length of the day increases by about 1 to 2 ms per century, mainly because of the friction by the Moon and the melting of the polar ice caps. This was deduced by studying historical astronomical observations * The growth of leaves on trees and the consequent change in the Earth’s moment of inertia, already thought of in 1916 by Harold Jeffreys, is way too small to be seen, as it is hidden by larger effects. 146 5 from the rotation of the earth Ref. 119 of the ancient Babylonian and Arab astronomers. Additional ‘decadic’ changes have an amplitude of 4 or 5 ms and are due to the motion of the liquid part of the Earth’s core. (The centre of the Earth’s core is solid; this was discovered in 1936 by the Danish seis- mologist Inge Lehmann (1888–1993); her discovery was confirmed most impressively by two British seismologists in 2008, who detected shear waves of the inner core, thus confirming Lehmann’s conclusion. There is a liquid core around the solid core.) The seasonal and biannual changes of the length of the day – with an amplitude of 0.4 ms over six months, another 0.5 ms over the year, and 0.08 ms over 24 to 26 months – are mainly due to the effects of the atmosphere. In the 1950s the availability of precision measurements showed that there is even a 14 and a 28 day period with an amplitude of 0.2 ms, due to the Moon. In the 1970s, when wind oscillations with a length scale of about 50 days were discovered, they were also found to alter the length of the day, with an amplitude of about 0.25 ms. However, these last variations are quite irregular. Also the oceans influence the rotation of the Earth, due to the tides, ocean currents, Ref. 120 wind forcing, and atmospheric pressure forcing. Further effects are due to the ice sheet variations and due to water evaporation and rainfalls. Last but not least, flows in the Motion Mountain – The Adventure of Physics interior of the Earth, both in the mantle and in the core, change the rotation. For example, earthquakes, plate motion, post-glacial rebound and volcanic eruptions all influence the rotation. But why does the Earth rotate at all? The rotation originated in the rotating gas cloud at the origin of the Solar System. This connection explains that the Sun and all planets, except two, turn around their axes in the same direction, and that they also all orbit the Ref. 121 Sun in that same direction. But the complete story is outside the scope of this text. The rotation around its axis is not the only motion of the Earth; it performs other motions as well. This was already known long ago. In 128 b ce, the Greek astronomer copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Hipparchos discovered what is today called the (equinoctial) precession. He compared a measurement he made himself with another made 169 years before. Hipparchos found that the Earth’s axis points to different stars at different times. He concluded that the sky was moving. Today we prefer to say that the axis of the Earth is moving. (Why?) Challenge 271 e During a period of 25 800 years the axis draws a cone with an opening angle of 23.5°. This motion, shown in Figure 107, is generated by the tidal forces of the Moon and the Sun on the equatorial bulge of the Earth that result from its flattening. The Sun and the Moon try to align the axis of the Earth at right angles to the Earth’s path; this torque leads to the precession of the Earth’s axis. Precession is a motion common to all rotating systems: it appears in planets, spinning tops and atoms. (Precession is also at the basis of the surprise related to the suspended wheel shown on page 244.) Precession is most easily seen in spinning tops, be they sus- pended or not. An example is shown in Figure 108; for atomic nuclei or planets, just imagine that the suspending wire is missing and the rotating body less flattened. On the Earth, precession leads to the upwelling of deep water in the equatorial Atlantic Ocean and regularly changes the ecology of algae. In addition, the axis of the Earth is not even fixed relative to the Earth’s surface. In 1884, by measuring the exact angle above the horizon of the celestial North Pole, Friedrich Küstner (1856–1936) found that the axis of the Earth moves with respect to the Earth’s crust, as Bessel had suggested 40 years earlier. As a consequence of Küstner’s discovery, the International Latitude Service was created. The polar motion Küstner dis- to the relativity of motion 147 nutation period year 15000: is 18.6 years year 2000: North pole is North pole is Vega in Polaris in Lyra Ursa minor precession N Moon’s path Moon equatorial Motion Mountain – The Adventure of Physics bulge Eq equatorial uat bulge or S Earth’s path F I G U R E 107 The precession and the nutation of the Earth’s axis. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net covered turned out to consist of three components: a small linear drift – not yet under- stood – a yearly elliptical motion due to seasonal changes of the air and water masses, and a circular motion* with a period of about 1.2 years due to fluctuations in the pres- sure at the bottom of the oceans. In practice, the North Pole moves with an amplitude Ref. 122 of about 15 m around an average central position, as shown in Figure 109. Short-term variations of the North Pole position, due to local variations in atmospheric pressure, to Ref. 123 weather changes and to the tides, have also been measured. The high precision of the GPS system is possible only with the help of the exact position of the Earth’s axis; and only with this knowledge can artificial satellites be guided to Mars or other planets. The details of the motion of the Earth have been studied in great detail. Table 25 gives an overview of the knowledge and precision that is available today. In 1912, the meteorologist and geophysicist Alfred Wegener (1880–1930) discovered an even larger effect. After studying the shapes of the continental shelves and the geolo- gical layers on both sides of the Atlantic, he conjectured that the continents move, and that they are all fragments of a single continent that broke up 200 million years ago.** * The circular motion, a wobble, was predicted by the great Swiss mathematician Leonhard Euler (1707– 1783). In a disgusting story, using Euler’s and Bessel’s predictions and Küstner’s data, in 1891 Seth Chandler claimed to be the discoverer of the circular component. ** In this old continent, called Gondwanaland, there was a huge river that flowed westwards from Chad 148 5 from the rotation of the earth F I G U R E 108 Precession of a suspended spinning top Motion Mountain – The Adventure of Physics (mpg ﬁlm © Lucas Barbosa) copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 109 The motion of the North Pole – roughly speaking, the Earth’s effective polhode – from 2003 to 2007, including the prediction until 2008 (left) and the average position since 1900 (right) – with 0.1 arcsecond being around 3.1 m on the surface of the Earth – not showing the diurnal and semidiurnal variations of a fraction of a millisecond of arc due to the tides (from hpiers.obspm.fr/ eop-pc). Even though at first derided across the world, Wegener’s discoveries were correct. Modern satellite measurements, shown in Figure 110, confirm this model. For example, the American continent moves away from the European continent by about 23 mm every to Guayaquil in Ecuador. After the continent split up, this river still flowed to the west. When the Andes appeared, the water was blocked, and many millions of years later, it reversed its flow. Today, the river still flows eastwards: it is called the Amazon River. to the relativity of motion 149 F I G U R E 110 The continental plates are the objects of tectonic motion (HoloGlobe Motion Mountain – The Adventure of Physics project, NASA). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 111 The tectonic plates of the Earth, with the relative speeds at the boundaries. (© NASA) year, as shown in Figure 111. There are also speculations that this velocity may have been much higher at certain periods in the past. The way to check this is to look at the mag- netization of sedimental rocks. At present, this is still a hot topic of research. Following the modern version of the model, called plate tectonics, the continents (with a density of 150 5 from the rotation of the earth F I G U R E 112 The angular size of the Sun changes due to the elliptical motion of the Earth (© Anthony Ayiomamitis). Motion Mountain – The Adventure of Physics F I G U R E 113 Friedrich Wilhelm Bessel (1784–1846). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Page 126 2.7 ⋅ 103 kg/m3 ) float on the fluid mantle of the Earth (with a density of 3.1 ⋅ 103 kg/m3 ) Vol. III, page 223 like pieces of cork on water, and the convection inside the mantle provides the driving Ref. 124 mechanism for the motion. Does the E arth move? Also the centre of the Earth is not at rest in the universe. In the third century b ce Aristarchus of Samos maintained that the Earth turns around the Sun. Experiments such as that of Figure 112 confirm that the orbit is an ellipse. However, a fundamental difficulty of the heliocentric system is that the stars look the same all year long. How can this be, if the Earth travels around the Sun? The distance between the Earth and the Sun has been known since the seventeenth century, but it was only in 1837 that Friedrich Wil- helm Bessel* became the first person to observe the parallax of a star. This was a result of extremely careful measurements and complex calculations: he discovered the Bessel functions in order to realize it. He was able to find a star, 61 Cygni, whose apparent pos- ition changed with the month of the year. Seen over the whole year, the star describes a small ellipse in the sky, with an opening of 0.588 (this is the modern value). After care- * Friedrich Wilhelm Bessel (1784–1846), Westphalian astronomer who left a successful business career to dedicate his life to the stars, and became the foremost astronomer of his time. to the relativity of motion 151 TA B L E 25 Modern measurement data about the motion of the Earth (from hpiers.obspm.fr/eop-pc). O b s e r va b l e Symbol Va l u e Mean angular velocity of Earth Ω 72.921 150(1) μrad/s Nominal angular velocity of Earth (epoch ΩN 72.921 151 467 064 μrad/s 1820) Conventional mean solar day (epoch 1820) d 86 400 s Conventional sidereal day dsi 86 164.090 530 832 88 s Ratio conv. mean solar day to conv. sidereal day 𝑘 = d/dsi 1.002 737 909 350 795 Conventional duration of the stellar day dst 86 164.098 903 691 s Ratio conv. mean solar day to conv. stellar day 𝑘 = d/dst 1.002 737 811 911 354 48 General precession in longitude 𝑝 5.028 792(2) /a Obliquity of the ecliptic (epoch 2000) 𝜀0 23° 26 21.4119 Küstner-Chandler period in terrestrial frame 𝑇KC 433.1(1.7) d Quality factor of the Küstner-Chandler peak 𝑄KC 170 Motion Mountain – The Adventure of Physics Free core nutation period in celestial frame 𝑇F 430.2(3) d Quality factor of the free core nutation 𝑄F 2 ⋅ 104 Astronomical unit AU 149 597 870.691(6) km Sidereal year (epoch 2000) 𝑎si 365.256 363 004 d = 365 d 6 h 9 min 9.76 s Tropical year 𝑎tr 365.242 190 402 d = 365 d 5 h 48 min 45.25 s Mean Moon period 𝑇M 27.321 661 55(1) d Earth’s equatorial radius 𝑎 6 378 136.6(1) m 8.0101(2) ⋅ 1037 kg m2 copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net First equatorial moment of inertia 𝐴 Longitude of principal inertia axis 𝐴 𝜆𝐴 −14.9291(10)° Second equatorial moment of inertia 𝐵 8.0103(2) ⋅ 1037 kg m2 Axial moment of inertia 𝐶 8.0365(2) ⋅ 1037 kg m2 Equatorial moment of inertia of mantle 𝐴m 7.0165 ⋅ 1037 kg m2 Axial moment of inertia of mantle 𝐶m 7.0400 ⋅ 1037 kg m2 Earth’s flattening 𝑓 1/298.25642(1) Astronomical Earth’s dynamical flattening ℎ = (𝐶 − 𝐴)/𝐶 0.003 273 794 9(1) Geophysical Earth’s dynamical flattening 𝑒 = (𝐶 − 𝐴)/𝐴 0.003 284 547 9(1) Earth’s core dynamical flattening 𝑒f 0.002 646(2) Second degree term in Earth’s gravity potential 𝐽2 = −(𝐴 + 𝐵 − 1.082 635 9(1) ⋅ 10−3 2𝐶)/(2𝑀𝑅2 ) Secular rate of 𝐽2 d𝐽2 /d𝑡 −2.6(3) ⋅ 10−11 /a Love number (measures shape distortion by 𝑘2 0.3 tides) Secular Love number 𝑘s 0.9383 Mean equatorial gravity 𝑔eq 9.780 3278(10) m/s2 Geocentric constant of gravitation 𝐺𝑀 3.986 004 418(8) ⋅ 1014 m3 /s2 Heliocentric constant of gravitation 𝐺𝑀⊙ 1.327 124 420 76(50) ⋅ 1020 m3 /s2 Moon-to-Earth mass ratio 𝜇 0.012 300 038 3(5) 152 5 from the rotation of the earth precession ellipticity change rotation axis Earth Sun Sun rotation tilt change axis Earth perihelion shift Sun Motion Mountain – The Adventure of Physics Earth orbital inclination change P Sun P Sun F I G U R E 114 Changes in the Earth’s motion around the Sun, as seen from different observers outside the orbital plane. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net fully eliminating all other possible explanations, he deduced that the change of position was due to the motion of the Earth around the Sun, when seen from distant stars. From the size of the ellipse he determined the distance to the star to be 105 Pm, or 11.1 light Challenge 272 s years. Bessel had thus managed, for the first time, to measure the distance of a star. By doing so he also proved that the Earth is not fixed with respect to the stars in the sky. The motion of the Earth was not a surprise. It confirmed the result of the mentioned aberration of Vol. II, page 17 light, discovered in 1728 by James Bradley* and to be discussed below. When seen from the sky, the Earth indeed revolves around the Sun. With the improvement of telescopes, other motions of the Earth were discovered. In 1748, James Bradley announced that there is a small regular change of the precession, which he called nutation, with a period of 18.6 years and an angular amplitude of 19.2 . Nutation occurs because the plane of the Moon’s orbit around the Earth is not exactly * James Bradley (b. 1693 Sherborne , d. 1762 Chalford), was an important astronomer. He was one of the first astronomers to understand the value of precise measurement, and thoroughly modernized the Greenwich observatory. He discovered, independently of Eustachio Manfredi, the aberration of light, and showed with it that the Earth moves. In particular, the discovery allowed him to measure the speed of light and confirm the value of 0.3 Gm/s. He later discovered the nutation of the Earth’s axis. to the relativity of motion 153 the same as the plane of the Earth’s orbit around the Sun. Are you able to confirm that Challenge 273 e this situation produces nutation? Astronomers also discovered that the 23.5° tilt – or obliquity – of the Earth’s axis, the angle between its intrinsic and its orbital angular momentum, actually changes from 22.1° to 24.5° with a period of 41 000 years. This motion is due to the attraction of the Sun and the deviations of the Earth from a spherical shape. In 1941, during the Second World War, the Serbian astronomer Milutin Milankovitch (1879–1958) retreated into solitude and explored the consequences. In his studies he realized that this 41 000 year period of the obliquity, together with an average period of 22 000 years due to precession,* gives rise to more than 20 ice ages in the last 2 million years. This happens through stronger or weaker irradiation of the poles by the Sun. The changing amounts of melted ice then lead to changes in average temperature. The last ice age had its peak about 20 000 years ago and ended around 11 800 years ago; the next is still far away. A spectacular confirm- ation of the relation between ice age cycles and astronomy came through measurements of oxygen isotope ratios in ice cores and sea sediments, which allow the average temper- Ref. 125 ature over the past million years to be tracked. Figure 115 shows how closely the temper- Motion Mountain – The Adventure of Physics ature follows the changes in irradiation due to changes in obliquity and precession. The Earth’s orbit also changes its eccentricity with time, from completely circular to slightly oval and back. However, this happens in very complex ways, not with periodic regularity, and is due to the influence of the large planets of the solar system on the Earth’s orbit. The typical time scale is 100 000 to 125 000 years. In addition, the Earth’s orbit changes in inclination with respect to the orbits of the other planets; this seems to happen regularly every 100 000 years. In this period the in- clination changes from +2.5° to −2.5° and back. Even the direction in which the ellipse points changes with time. This so-called peri- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net helion shift is due in large part to the influence of the other planets; a small remaining part will be important in the chapter on general relativity. The perihelion shift of Mercury was the first piece of data confirming Einstein’s theory. Obviously, the length of the year also changes with time. The measured variations are of the order of a few parts in 1011 or about 1 ms per year. However, knowledge of these changes and of their origins is much less detailed than for the changes in the Earth’s rotation. The next step is to ask whether the Sun itself moves. Indeed it does. Locally, it moves with a speed of 19.4 km/s towards the constellation of Hercules. This was shown by Wil- liam Herschel in 1783. But globally, the motion is even more interesting. The diameter of the galaxy is at least 100 000 light years, and we are located 26 000 light years from the centre. (This has been known since 1918; the centre of the galaxy is located in the direc- tion of Sagittarius.) At our position, the galaxy is 1 300 light years thick; presently, we are Ref. 126 68 light years ‘above’ the centre plane. The Sun, and with it the Solar System, takes about 225 million years to turn once around the galactic centre, its orbital velocity being around 220 km/s. It seems that the Sun will continue moving away from the galaxy plane until it is about 250 light years above the plane, and then move back, as shown in Figure 116. The oscillation period is estimated to be around 62 million years, and has been suggested as * In fact, the 25 800 year precession leads to three insolation periods, of 23 700, 22 400 and 19 000 years, due to the interaction between precession and perihelion shift. 154 5 from the rotation of the earth P re c e s s ion P a ra me te r ( 1 0 −3 ) A −40 −20 0 20 40 4 B 0 ΔT S ( ° C ) −4 −8 C 0 R F (W m ) Motion Mountain – The Adventure of Physics −2 −1 −2 D 2 24. 0 0. 4 (W m ) −2 O bliquity ( ° ) 1 (° C ) 23. 5 obl 0 0. 0 23. 0 obl ΔT S copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net −1 RF 22.5 −0. 4 −2 0 100 200 300 400 500 600 700 800 Age (1000 years before present) F I G U R E 115 Modern measurements showing how Earth’s precession parameter (black curve A) and obliquity (black curve D) inﬂuence the average temperature (coloured curve B) and the irradiation of the Earth (blue curve C) over the past 800 000 years: the obliquity deduced by Fourier analysis from the irradiation data RF (blue curve D) and the obliquity deduced by Fourier analysis from the temperature (red curve D) match the obliquity known from astronomical data (black curve D); sharp cooling events took place whenever the obliquity rose while the precession parameter was falling (marked red below the temperature curve) (© Jean Jouzel/Science from Ref. 125). the mechanism for the mass extinctions of animal life on Earth, possibly because some gas cloud or some cosmic radiation source may be periodically encountered on the way. The issue is still a hot topic of research. The motion of the Sun around the centre of the Milky Way implies that the planets of the Solar System can be seen as forming helices around the Sun. Figure 117 illustrates their helical path. We turn around the galaxy centre because the formation of galaxies, like that of plan- etary systems, always happens in a whirl. By the way, can you confirm from your own to the relativity of motion 155 120 000 al = 1.2 Zm our galaxy orbit of our local star system 500 al = 5 Em Sun's path F I G U R E 116 The 50 000 al = 500 Em motion of the Sun around the galaxy. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 117 The helical motion of the ﬁrst four planets around the path traced by the Sun during its travel around the centre of the Milky Way. Brown: Mercury, white: Venus, blue: Earth, red: Mars. (QuickTime ﬁlm © Rhys Taylor at www.rhysy.net). Challenge 274 s observation that our galaxy itself rotates? Finally, we can ask whether the galaxy itself moves. Its motion can indeed be observed because it is possible to give a value for the motion of the Sun through the universe, defining it as the motion against the background radiation. This value has been measured Ref. 127 to be 370 km/s. (The velocity of the Earth through the background radiation of course depends on the season.) This value is a combination of the motion of the Sun around the galaxy centre and of the motion of the galaxy itself. This latter motion is due to the 156 5 from the rotation of the earth Motion Mountain – The Adventure of Physics F I G U R E 118 Driving through snowﬂakes shows the effects of relative motion in three dimensions. Page 206 Similar effects are seen when the Earth speeds through the universe. (© Neil Provo at neilprovo.com). gravitational attraction of the other, nearby galaxies in our local group of galaxies.* In summary, the Earth really moves, and it does so in rather complex ways. As Henri copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Poincaré would say, if we are in a given spot today, say the Panthéon in Paris, and come back to the same spot tomorrow at the same time, we are in fact 31 million kilometres away. This state of affairs would make time travel extremely difficult even if it were pos- sible (which it is not); whenever you went back to the past, you would have to get to the old spot exactly! Is velo cit y absolu te? – The theory of everyday relativity Why don’t we feel all the motions of the Earth? The two parts of the answer were already given in 1632. First of all, as Galileo explained, we do not feel the accelerations of the Earth because the effects they produce are too small to be detected by our senses. Indeed, many of the mentioned accelerations do induce measurable effects only in high-precision Vol. II, page 155 experiments, e.g. in atomic clocks. But the second point made by Galileo is equally important: it is impossible to feel the high speed at which we are moving. We do not feel translational, unaccelerated motions because this is impossible in principle. Galileo discussed the issue by comparing the ob- servations of two observers: one on the ground and another on the most modern means of unaccelerated transportation of the time, a ship. Galileo asked whether a man on the * This is roughly the end of the ladder. Note that the expansion of the universe, to be studied later, produces no motion. to the relativity of motion 157 ground and a man in a ship moving at constant speed experience (or ‘feel’) anything different. Einstein used observers in trains. Later it became fashionable to use travellers Challenge 275 e in rockets. (What will come next?) Galileo explained that only relative velocities between bodies produce effects, not the absolute values of the velocities. For the senses and for all measurements we find: ⊳ There is no difference between constant, undisturbed motion, however rapid it may be, and rest. This is called Galileo’s principle of relativity. Indeed, in everyday life we feel motion only if the means of transportation trembles – thus if it accelerates – or if we move against the air. Therefore Galileo concludes that two observers in straight and undisturbed motion against each other cannot say who is ‘really’ moving. Whatever their relative speed, neither of them ‘feels’ in motion.* Rest is relative. Or more clearly: rest is an observer-dependent concept. This result of Galilean physics is so important that Poincaré introduced the expression ‘theory of relativity’ and Einstein repeated the principle explicitly when he published his famous Motion Mountain – The Adventure of Physics theory of special relativity. However, these names are awkward. Galilean physics is also a theory of relativity! The relativity of rest is common to all of physics; it is an essential aspect of motion. In summary, undisturbed or uniform motion has no observable effect; only change of motion does. Velocity cannot be felt; acceleration can. As a result, every physicist can deduce something simple about the following statement by Wittgenstein: * In 1632, in his Dialogo, Galileo writes: ‘Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal: jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that, you will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship’s motion is common to all the things contained in it, and to the air also. That is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable differences would be seen in some of the effects noted.’ (Translation by Stillman Drake) 158 5 from the rotation of the earth Daß die Sonne morgen aufgehen wird, ist eine Hypothese; und das heißt: wir wissen nicht, ob sie aufgehen wird.* The statement is wrong. Can you explain why Wittgenstein erred here, despite his strong Challenge 276 s desire not to? Is rotation relative? When we turn rapidly, our arms lift. Why does this happen? How can our body detect whether we are rotating or not? There are two possible answers. The first approach, pro- moted by Newton, is to say that there is an absolute space; whenever we rotate against this space, the system reacts. The other answer is to note that whenever the arms lift, the stars also rotate, and in exactly the same manner. In other words, our body detects rotation because we move against the average mass distribution in space. The most cited discussion of this question is due to Newton. Instead of arms, he ex- plored the water in a rotating bucket. In a rotating bucket, the water surface forms a Motion Mountain – The Adventure of Physics concave shape, whereas the surface is flat for a non-rotating bucket. Newton asked why this is the case. As usual for philosophical issues, Newton’s answer was guided by the mysticism triggered by his father’s early death. Newton saw absolute space as a mystical and religious concept and was not even able to conceive an alternative. Newton thus saw rotation as an absolute type of motion. Most modern scientists have fewer personal problems and more common sense than Newton; as a result, today’s consensus is that rotation effects are due to the mass distribution in the universe: ⊳ Rotation is relative. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net A number of high-precision experiments confirm this conclusion; thus it is also part of Einstein’s theory of relativity. Curiosities and fun challenges ab ou t rotation and relativit y When travelling in the train, you can test Galileo’s statement about the everyday relativ- ity of motion. Close your eyes and ask somebody to turn you around several times: are Challenge 277 e you able to say in which direction the train is running? ∗∗ A good bathroom scales, used to determine the weight of objects, does not show a con- Challenge 278 s stant weight when you step on it and stay motionless. Why not? ∗∗ Challenge 279 s If a gun located at the Equator shoots a bullet vertically, where does the bullet fall? ∗∗ Challenge 280 s Why are most rocket launch sites as near as possible to the Equator? * ‘That the Sun will rise to-morrow, is an hypothesis; and that means that we do not know whether it will rise.’ This well-known statement is found in Ludwig Wittgenstein, Tractatus, 6.36311. to the relativity of motion 159 ∗∗ At the Equator, the speed of rotation of the Earth is 464 m/s, or about Mach 1.4; the latter number means that it is 1.4 times the speed of sound. This supersonic motion has two intriguing consequences. First of all, the rotation speed determines the size of typical weather phenomena. This size, the so-called Rossby radius, is given by the speed of sound (or some other typical speed) divided by twice the local rotation speed, multiplied by the radius of the Earth. At moderate latitudes, the Rossby radius is about 2000 km. This is a sizeable fraction of the Earth’s radius, so that only a few large weather systems are present on Earth at any specific time. If the Earth rotated more slowly, the weather would be determined by short-lived, local flows and have no general regularities. If the Earth rotated more rapidly, the weather would be much more violent – as on Jupiter – but the small Rossby radius implies that large weather structures have a huge lifetime, such as the red spot on Jupiter, which lasted for several centuries. In a sense, the rotation of the Earth has the speed that provides the most interesting weather. The other consequence of the value of the Earth’s rotation speed concerns the thick- Motion Mountain – The Adventure of Physics ness of the atmosphere. Mach 1 is also, roughly speaking, the thermal speed of air mo- lecules. This speed is sufficient for an air molecule to reach the characteristic height of the atmosphere, about 6 km. On the other hand, the speed of rotation Ω of the Earth de- Page 137 termines its departure ℎ from sphericity: the Earth is flattened, as we saw above. Roughly speaking, we have 𝑔ℎ = Ω2 𝑅2 /2, or about 12 km. (This is correct to within 50 %, the ac- tual value is 21 km.) We thus find that the speed of rotation of the Earth implies that its flattening is comparable to the thickness of the atmosphere. ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net The Coriolis effect influences rivers and their shores. This surprising connection was made in 1860 by Karl Ernst von Baer who found that in Russia, many rivers flowing north in lowlands had right shores that are steep and high, and left shores that are low and flat. Challenge 281 e (Can you explain the details?) He also found that rivers in the southern hemisphere show the opposite effect. ∗∗ The Coriolis effect saves lives and helps people. Indeed, it has an important application for navigation systems; the typical uses are shown in Figure 119. Insects use vibrating masses to stabilize their orientation, to determine their direction of travel and to find their way. Most two-winged insects, or diptera, use vibrating halteres for navigation: in particular, bees, house-flies, hover-flies and crane flies use them. Other insects, such as moths, use vibrating antennae for navigation. Cars, satellites, mobile phones, remote- controlled helicopter models, and computer games also use tiny vibrating masses as ori- entation and navigation sensors, in exactly the same way as insects do. In all these navigation applications, one or a few tiny masses are made to vibrate; if the system to which they are attached turns, the change of orientation leads to a Coriolis effect. The effect is measured by detecting the ensuing change in geometry; the change, and thus the signal strength, depends on the angular velocity and its direction. Such orientation sensors are therefore called vibrating Coriolis gyroscopes. Their development 160 5 from the rotation of the earth Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 119 The use of the Coriolis effect in insects – here a crane ﬂy and a hovering ﬂy – and in micro-electromechanic systems (size about a few mm); all provide navigation signals to the systems to which they are attached (© Pinzo, Sean McCann, ST Microelectronics). to the relativity of motion 161 and production is a sizeable part of high-tech business – and of biological evolution. ∗∗ A wealthy and quirky customer asked his architect to plan and build a house whose four Challenge 282 e walls all faced south. How did the architect realize the request? ∗∗ Would travelling through interplanetary space be healthy? People often fantasize about long trips through the cosmos. Experiments have shown that on trips of long duration, cosmic radiation, bone weakening, muscle degeneration and psychological problems are the biggest dangers. Many medical experts question the viability of space travel lasting longer than a couple of years. Other dangers are rapid sunburn, at least near the Sun, and exposure to the vacuum. So far only one man has experienced vacuum without Ref. 128 protection. He lost consciousness after 14 seconds, but survived unharmed. ∗∗ Motion Mountain – The Adventure of Physics Challenge 283 s In which direction does a flame lean if it burns inside a jar on a rotating turntable? ∗∗ Galileo’s principle of everyday relativity states that it is impossible to determine an abso- lute velocity. It is equally impossible to determine an absolute position, an absolute time Challenge 284 s and an absolute direction. Is this correct? ∗∗ Does centrifugal acceleration exist? Most university students go through the shock of copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net meeting a teacher who says that it doesn’t because it is a ‘fictitious’ quantity, in the face of what one experiences every day in a car when driving around a bend. Simply ask the teacher who denies it to define ‘existence’. (The definition physicists usually use is given Vol. III, page 324 later on.) Then check whether the definition applies to the term and make up your own Challenge 285 s mind. Whether you like the term ‘centrifugal acceleration’ or avoid it by using its negative, the so-called centripetal acceleration, you should know how it is calculated. We use a simple trick. For an object in circular motion of radius 𝑟, the magnitude 𝑣 of the velocity 𝑣 = d𝑥/d𝑡 is 𝑣 = 2π𝑟/𝑇. The vector 𝑣 behaves over time exactly like the position of the object: it rotates continuously. Therefore, the magnitude 𝑎 of the centrifugal/centripetal acceleration 𝑎 = d𝑣/d𝑡 is given by the corresponding expression, namely 𝑎 = 2π𝑣/𝑇. Eliminating 𝑇, we find that the centrifugal/centripetal acceleration 𝑎 of a body rotating at speed 𝑣 at radius 𝑟 is given by 𝑣2 𝑎= = 𝜔2 𝑟 . (36) 𝑟 This is the acceleration we feel when sitting in a car that goes around a bend. ∗∗ Rotational motion holds a little surprise for anybody who studies it carefully. Angular momentum is a quantity with a magnitude and a direction. However, it is not a vector, 162 5 from the rotation of the earth water ping-pong ball string stone F I G U R E 120 How does the ball move when the jar is accelerated in direction of the arrow? as any mirror shows. The angular momentum of a body circling in a plane parallel to a mirror behaves in a different way from a usual arrow: its mirror image is not reflected Challenge 286 e if it points towards the mirror! You can easily check this for yourself. For this reason, Challenge 287 e angular momentum is called a pseudo-vector. (Are rotations pseudo-vectors?) The fact Page 284 has no important consequences in classical physics; but we have to keep it in mind for Vol. V, page 245 later, when we explore nuclear physics. Motion Mountain – The Adventure of Physics ∗∗ What is the best way to transport a number of full coffee or tea cups while at the same Challenge 288 s time avoiding spilling any precious liquid? ∗∗ A ping-pong ball is attached by a string to a stone, and the whole is put under water in a jar. The set-up is shown in Figure 120. Now the jar is accelerated horizontally, for Challenge 289 s example in a car. In which direction does the ball move? What do you deduce for a jar copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net at rest? ∗∗ The Moon recedes from the Earth by 3.8 cm a year, due to friction. Can you find the Challenge 290 s mechanism responsible for the effect? ∗∗ What are earthquakes? Earthquakes are large examples of the same process that make a door squeak. The continental plates correspond to the metal surfaces in the joints of the door. Earthquakes can be described as energy sources. The Richter scale is a direct measure of this energy. The Richter magnitude 𝑀s of an earthquake, a pure number, is defined from its energy 𝐸 in joule via log(𝐸/1 J) − 4.8 𝑀s = . (37) 1.5 The strange numbers in the expression have been chosen to put the earthquake values as near as possible to the older, qualitative Mercalli scale (now called EMS98) that classifies the intensity of earthquakes. However, this is not fully possible; the most sensitive instru- ments today detect earthquakes with magnitudes of −3. The highest value ever measured to the relativity of motion 163 F I G U R E 121 What happens when the ape climbs? Motion Mountain – The Adventure of Physics was a Richter magnitude of 10, in Chile in 1960. Magnitudes above 12 are probably im- Challenge 291 s possible. Can you show why? ∗∗ What is the motion of the point on the surface of the Earth that has Sun in its zenith – i.e., vertically above it – when seen on a map of the Earth during one day? And day after Challenge 292 ny day? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Can it happen that a satellite dish for geostationary TV satellites focuses the sunshine Challenge 293 s onto the receiver? ∗∗ Why is it difficult to fire a rocket from an aeroplane in the direction opposite to the Challenge 294 s motion of the plane? ∗∗ An ape hangs on a rope. The rope hangs over a wheel and is attached to a mass of equal weight hanging down on the other side, as shown in Figure 121. The rope and the wheel Challenge 295 s are massless and frictionless. What happens when the ape climbs the rope? ∗∗ Challenge 296 s Can a water skier move with a higher speed than the boat pulling him? ∗∗ You might know the ‘Dynabee’, a hand-held gyroscopic device that can be accelerated Challenge 297 d to high speed by proper movements of the hand. How does it work? ∗∗ 164 5 from the rotation of the earth Motion Mountain – The Adventure of Physics F I G U R E 122 The motion of the angular velocity of a tumbling brick. The tip of the angular velocity vector moves along the yellow curve, the polhode. It moves together with the tumbling object, as does the elliptical mesh representing the energy ellipsoid; the herpolhode is the white curve and lies in the plane bright blue cross pattern that represents the invariable plane (see text). The animation behind the screenshot illustrates the well-known nerd statement: the polhode rolls without slipping on the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net herpolhode lying in the invariable plane. The full animation is available online at www.ialms.net/sim/ 3d-rigid-body-simulation/. Any rotating, irregular, free rigid body follows such a motion. For the Earth, which is rotating and irregular, but neither fully free nor fully rigid, this description is only an approximation. (© Svetoslav Zabunov) It is possible to make a spinning top with a metal paper clip. It is even possible to make Challenge 298 s one of those tops that turn onto their head when spinning. Can you find out how? ∗∗ The moment of inertia of a body depends on the shape of the body; usually, the angular momentum and the angular velocity do not point in the same direction. Can you confirm Challenge 299 s this with an example? ∗∗ Challenge 300 s What is the moment of inertia of a homogeneous sphere? ∗∗ The complete moment of inertia of a rigid body is determined by the values along its three principal axes. These values are all equal for a sphere and for a cube. Does it mean Challenge 301 s that it is impossible to distinguish a sphere from a cube by their inertial behaviour? to the relativity of motion 165 ∗∗ Here is some mathematical fun about the rotation of a free rigid body. Even for a free ro- tating rigid body, such as a brick rotating in free space, the angular velocity is, in general, not constant: the brick tumbles while rotating. In this motion, both energy and angular momentum are constant, but the angular velocity is not. In particular, not even the dir- ection of angular velocity is constant; in other words, the north pole changes with time. In fact, the north pole changes with time both for an observer on the body and for an observer in free space. How does the north pole, the end point of the angular velocity vector, move? Page 276 The moment of inertia is a tensor and can thus represented by an ellipsoid. In addition, the motion of a free rigid body is described, in the angular velocity space, by the kinetic energy ellipsoid – because its rotation energy is constant. When a free rigid body moves, the energy ellipsoid – not the moment of inertia ellipsoid – rolls on the invariable plane Challenge 302 e that is perpendicular to the initial (and constant) angular momentum of the body. This is the mathematical description of the tumbling motion. The curve traced by the angular velocity, or the extended north pole of the body, on the invariable plane is called the Motion Mountain – The Adventure of Physics herpolhode. It is an involved curve that results from two superposed conical motions. For an observer on the rotating body, another curve is more interesting. The pole of the rotating body traces a curve on the energy ellipsoid – which is itself attached to the body. This curve is called the polhode. The polhode is a closed curve i three dimensions, the herpolhode is an open curve in two dimensions. The curves were named by Louis Poinsot and are especially useful for the description of the motion of rotating irregularly shaped bodies. For an complete visualization, see the excellent website www.ialms.net/ sim/3d-rigid-body-simulation/. The polhode is a circle only if the rigid body has rotational symmetry; the pole motion copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Page 147 is then called precession; we encountered it above. As shown in Figure 109, the measured polhode of the Earth is not of the expected shape; this irregularity is due to several effects, among them the non-rigidity of our planet. Even though the angular momentum of a freely tumbling brick is fixed in space, it is Challenge 303 e not fixed in the body frame. Can you confirm this? In the body frame, the end of the angular momentum of a tumbling brick moves along still another curve, given by the intersection of a sphere and an ellipsoid, as Jacques Binet pointed out. ∗∗ Is it true that the Moon in the first quarter in the northern hemisphere looks like the Challenge 304 s Moon in the last quarter in the southern hemisphere? ∗∗ An impressive confirmation that the Earth is a sphere can be seen at sunset, if we turn, against our usual habit, our back to the Sun. On the eastern sky we can then see the impressive rise of the Earth’s shadow. We can admire the vast shadow rising over the whole horizon, clearly having the shape of a segment of a huge circle. Figure 124 shows an example. In fact, more precise investigations show that it is not the shadow of the Earth alone, but the shadow of its ionosphere. ∗∗ 166 5 from the rotation of the earth Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 123 Long exposures of the stars at night – one when facing north, above the Gemini telescope in Hawaii, and one above the Alps that includes the celestial equator, with the geostationary satellites on it (© Gemini Observatory/AURA, Michael Kunze). to the relativity of motion 167 F I G U R E 124 The shadow of the Earth – here a panoramic photograph taken at the South Pole – shows Motion Mountain – The Adventure of Physics that the Earth is round (© Ian R. Rees). Challenge 305 s How would Figure 123 look if taken at the Equator? ∗∗ Precision measurements show that not all planets move in exactly the same plane. Mer- cury shows the largest deviation. In fact, no planet moves exactly in an ellipse, nor even in a plane around the Sun. Almost all of these effects are too small and too complex to explain here. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ Since the Earth is round, there are many ways to drive from one point on the Earth to another along a circle segment. This freedom of choice has interesting consequences for volleyballs and for men watching women. Take a volleyball and look at its air inlet. If you want to move the inlet to a different position with a simple rotation, you can choose Challenge 306 e the rotation axis in many different ways. Can you confirm this? In other words, when we look in a given direction and then want to look in another, the eye can accomplish this change in different ways. The option chosen by the human eye had already been studied by medical scientists in the eighteenth century. It is called Listing’s ‘law’.* It states that all Challenge 307 s axes that nature chooses lie in one plane. Can you imagine its position in space? Many men have a real interest that this mechanism is strictly followed; if not, looking at women on the beach could cause the muscles moving their eyes to get knotted up. ∗∗ Imagine to cut open a soft mattress, glue a steel ball into it, and glue the mattress together again. Now imagine that we use a magnetic field to rotate the steel ball glued inside. Intu- * If you are interested in learning in more detail how nature and the eye cope with the complexities of three dimensions, see the schorlab.berkeley.edu/vilis/whatisLL.htm and www.physpharm.fmd.uwo.ca/ undergrad/llconsequencesweb/ListingsLaw/perceptual1.htm websites. 168 5 from the rotation of the earth Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 125 A steel ball glued in a mattress can rotate forever. (QuickTime ﬁlm © Jason Hise). itively, we think that the ball can only be rotated by a finite angle, whose value would be limited by the elasticity of the mattress. But in reality, the steel ball can be rotated infin- itely often! This surprising possibility is a consequence of the tethered rotation shown in Page 90 Figure 54 and Figure 55. Such a continuous rotation in a mattress is shown in Figure 125. Challenge 308 r And despite its fascination, nobody has yet realized the feat. Can you? Legs or wheels? – Again The acceleration and deceleration of standard wheel-driven cars is never much greater than about 1 𝑔 = 9.8 m/s2 , the acceleration due to gravity on our planet. Higher accel- erations are achieved by motorbikes and racing cars through the use of suspensions that divert weight to the axes and by the use of spoilers, so that the car is pushed downwards to the relativity of motion 169 F I G U R E 126 A basilisk lizard (Basiliscus basiliscus) running on water, with a total length of about 25 cm, showing how the propulsing leg pushes into the water (© TERRA). with more than its own weight. Modern spoilers are so efficient in pushing a car towards the track that racing cars could race on the roof of a tunnel without falling down. Motion Mountain – The Adventure of Physics Through the use of special tyres the downward forces produced by aerodynamic ef- fects are transformed into grip; modern racing tyres allow forward, backward and side- ways accelerations (necessary for speed increase, for braking and for turning corners) of about 1.1 to 1.3 times the load. Engineers once believed that a factor 1 was the theoretical limit and this limit is still sometimes found in textbooks; but advances in tyre technology, mostly by making clever use of interlocking between the tyre and the road surface as in a gear mechanism, have allowed engineers to achieve these higher values. The highest ac- celerations, around 4 𝑔, are achieved when part of the tyre melts and glues to the surface. Special tyres designed to make this happen are used for dragsters, but high-performance radio-controlled model cars also achieve such values. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net How do wheels compare to using legs? High jump athletes can achieve peak ac- celerations of about 2 to 4 𝑔, cheetahs over 3 𝑔, bushbabies up to 13 𝑔, locusts about Ref. 129 18 𝑔, and fleas have been measured to accelerate about 135 𝑔. The maximum accelera- tion known for animals is that of click beetles, a small insect able to accelerate at over 2000 m/s2 = 200 𝑔, about the same as an airgun pellet when fired. Legs are thus definit- ively more efficient accelerating devices than wheels – a cheetah can easily beat any car or motorbike – and evolution developed legs, instead of wheels, to improve the chances of an animal in danger getting to safety. In short, legs outperform wheels. But there are other reasons for using legs instead of Challenge 309 s wheels. (Can you name some?) For example, legs, unlike wheels, allow walking on water. Most famous for this ability is the basilisk, * a lizard living in Central America and shown in Figure 126. This reptile is up to 70 cm long and has a mass of up to 500 g. It looks like a miniature Tyrannosaurus rex and is able to run over water surfaces on its hind legs. The motion has been studied in detail with high-speed cameras and by measurements using aluminium models of the animal’s feet. The experiments show that the feet slapping on Ref. 130 the water provides only 25 % of the force necessary to run above water; the other 75 % is provided by a pocket of compressed air that the basilisks create between their feet and the water once the feet are inside the water. In fact, basilisks mainly walk on air. (Both * In the Middle Ages, the term ‘basilisk’ referred to a mythical monster supposed to appear shortly before the end of the world. Today, it is a small reptile in the Americas. 170 5 from the rotation of the earth F I G U R E 127 A water strider, total size F I G U R E 128 A water walking robot, total about 10 mm (© Charles Lewallen). size about 20 mm (© AIP). Ref. 131 effects used by basilisks are also found in fast canoeing.) It was calculated that humans are also able to walk on water, provided their feet hit the water with a speed of 100 km/h Motion Mountain – The Adventure of Physics using the simultaneous physical power of 15 sprinters. Quite a feat for all those who ever did so. There is a second method of walking and running on water; this second method even allows its users to remain immobile on top of the water surface. This is what water striders, insects of the family Gerridae with an overall length of up to 15 mm, are able to do (together with several species of spiders), as shown in Figure 127. Like all insects, the water strider has six legs (spiders have eight). The water strider uses the back and front legs to hover over the surface, helped by thousands of tiny hairs attached to its body. The hairs, together with the surface tension of water, prevent the strider from getting wet. If copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net you put shampoo into the water, the water strider sinks and can no longer move. The water strider uses its large middle legs as oars to advance over the surface, reach- ing speeds of up to 1 m/s doing so. In short, water striders actually row over water. The same mechanism is used by the small robots that can move over water and were de- Ref. 132 veloped by Metin Sitti and his group, as shown in Figure 128. Robot design is still in its infancy. No robot can walk or even run as fast as the animal system it tries to copy. For two-legged robots, the most difficult kind, the speed record is around 3.5 leg lengths per second. In fact, there is a race going on in robotics depart- ments: each department tries to build the first robot that is faster, either in metres per second or in leg lengths per second, than the original four-legged animal or two-legged human. The difficulties of realizing this development goal show how complicated walk- ing motion is and how well nature has optimized living systems. Legs pose many interesting problems. Engineers know that a staircase is comfortable to walk only if for each step the depth 𝑙 plus twice the height ℎ is a constant: 𝑙 + 2ℎ = Challenge 310 s 0.63 ± 0.02 m. This is the so-called staircase formula. Why does it hold? Challenge 311 s Most animals have an even number of legs. Do you know an exception? Why not? In fact, one can argue that no animal has less than four legs. Why is this the case? On the other hand, all animals with two legs have legs side by side, whereas most systems with two wheels have them one behind the other. Why is this not the other way Challenge 312 e round? Legs are very efficient actuators. As Figure 129 shows, most small animals can run to the relativity of motion 171 100 Relative running speed (body length/s) 10 Motion Mountain – The Adventure of Physics 1 0.01 0.1 1 10 100 1000 10000 Body mass (kg) F I G U R E 129 The graph shows how the relative running speed changes with the mass of terrestrial mammal species, for 142 different species. The graph also illustrates how the running performance changes above 30 kg. Filled squares show Rodentia; open squares show Primata; ﬁlled diamonds Proboscidae; open diamonds Marsupialia; ﬁlled triangles Carnivora; open triangles Artiodactyla; ﬁlled circles Perissodactyla; open circles Lagomorpha (© José Iriarte-Díaz/JEB). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Ref. 133 with about 25 body lengths per second. For comparison, almost no car achieves such a speed. Only animals that weigh more than about 30 kg, including humans, are slower. Legs also provide simple distance rulers: just count your steps. In 2006, it was dis- covered that this method is used by certain ant species, such as Cataglyphis fortis. They Ref. 134 can count to at least 25 000, as shown by Matthias Wittlinger and his team. These ants use the ability to find the shortest way back to their home even in structureless desert terrain. Ref. 135 Why do 100 m sprinters run faster than ordinary people? A thorough investigation shows that the speed 𝑣 of a sprinter is given by 𝐹c 𝑣 = 𝑓 𝐿 stride = 𝑓 𝐿 c , (38) 𝑊 where 𝑓 is the frequency of the legs, 𝐿 stride is the stride length, 𝐿 c is the contact length – the length that the sprinter advances during the time the foot is in contact with the floor – 𝑊 the weight of the sprinter, and 𝐹c the average force the sprinter exerts on the floor during contact. It turns out that the frequency 𝑓 is almost the same for all sprinters; the only way to be faster than the competition is to increase the stride length 𝐿 stride . Also the contact length 𝐿 c varies little between athletes. Increasing the stride length thus requires that the athlete hits the ground with strong strokes. This is what athletic training for 172 5 from the rotation of the earth sprinters has to achieve. Summary on Galilean relativit y The Earth rotates. The acceleration is so small that we do not feel it. The speed of rotation is large, but we do not feel it, because there is no way to do so. Undisturbed or inertial motion cannot be felt or measured. It is thus impossible to distinguish motion from rest. The distinction between rest and motion depends on the observer: Motion of bodies is relative. That is why the soil below our feet seems so stable to us, even though it moves with high speed across the universe. Since motion is relative, speed values depend on the observer. Later on, we will dis- cover that one example of motion in nature has a speed value that is not relative: the mo- tion of light. But we continue first with the study of motion transmitted over distance, without the use of any contact at all. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Chapter 6 MOT ION DU E TO G R AV I TAT ION “ ” Caddi come corpo morto cade. Dante, Inferno, c. V, v. 142.** T he first and main method to generate motion without any contact hat we discover in our environment is height. Waterfalls, snow, rain, Motion Mountain – The Adventure of Physics he ball of your favourite game and falling apples all rely on it. It was one of the fundamental discoveries of physics that height has this property because there is an interaction between every body and the Earth. Gravitation produces an acceleration along the line connecting the centres of gravity of the body and the Earth. Note that in order to make this statement, it is necessary to realize that the Earth is a body in the same way as a stone or the Moon, that this body is finite and that therefore it has a centre and a mass. Today, these statements are common knowledge, but they are by no means evident from everyday personal experience. In several myths about the creation or the organization of the world, such as the bib- lical one or the Indian one, the Earth is not an object, but an imprecisely defined entity, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net such as an island floating or surrounded by water with unclear boundaries and unclear Challenge 313 s method of suspension. Are you able to convince a friend that the Earth is round and not flat? Can you find another argument apart from the roundness of the Earth’s shadow when it is visible on the Moon, shown in Figure 134? Gravitation as a limit to uniform motion A productive way to define gravitation, or gravity for short, appears when we note that no object around us moves along a straight line. In nature, there is a limit to steady, or constant motion: ⊳ Gravity prevents uniform motion, i.e., it prevents constant and straight mo- tion. In nature, we never observe bodies moving at constant speed along a straight line. Speak- ing with the vocabulary of kinematics: Gravity introduces an acceleration for every phys- ical body. The gravitation of the objects in the environment curves the path of a body, changes its speed, or both. This limit has two aspects. First, gravity prevents unlimited uniform motion: ** ‘I fell like dead bodies fall.’ Dante Alighieri (1265, Firenze–1321, Ravenna), the powerful Italian poet. 174 6 motion due to gravitation culmination zenith of a star or n idia planet: celestial mer me 90° – ϕ + δ North rid ian Pole star or planet nia rid celestial t me e Equator plan r or sta of merid 90° – ϕ ϕ–δ δ ation lin ian c de latitude ϕ 90° – ϕ Motion Mountain – The Adventure of Physics North observer South F I G U R E 130 Some important concepts when observing the stars and at night. ⊳ Motion cannot be straight for ever. Motion is not boundless. We will learn later what this means for the universe as a whole. Secondly, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ⊳ Motion cannot be constant and straight even during short time intervals. In other words, if we measure with sufficient precision, we will always find deviations from uniform motion. (Physicist also say that motion is never inertial.) These limits apply no matter how powerful the motion is and how free the moving body is from external influence. In nature, gravitation prevents the steady, uniform motion of atoms, billiard balls, planets, stars and even galaxies. Gravitation is the first limitation to motion that we discover in nature. We will dis- cover two additional limits to motion later on in our walk. These three fundamental Page 8 limits are illustrated in Figure 1. To achieve a precise description of motion, we need to take each limit of motion into account. This is our main aim in the rest of our adventure. Gravity affects all bodies, even if they are distant from each other. How exactly does gravitation affect two bodies that are far apart? We ask the experts for measuring distant objects: the astronomers. Gravitation in the sky The gravitation of the Earth forces the Moon in an orbit around it. The gravitation of the Sun forces the Earth in an orbit around it and sets the length of the year. Similarly, the gravitation of the Sun determines the motion of all the other planets across the sky. We 6 motion due to gravitation 175 Motion Mountain – The Adventure of Physics F I G U R E 131 ‘Planet’ means ‘wanderer’. This composed image shows the retrograde motion of planet Mars across the sky – the Pleiades star cluster is at the top left – when the planet is on the other side of the Sun. The pictures were taken about a week apart and superimposed. The motion is one of the many examples that are fully explained by universal gravitation (© Tunc Tezel). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net usually imagine to be located at the centre of the Sun and then say that the planets ‘orbit the Sun’. The Sun thus prevents the planets from moving in straight lines and forces them into orbits. How can we check this? First of all, looking at the sky at night, we can check that the planets always stay within Page 222 the zodiac, a narrow stripe across the sky. The centre line of the zodiac gives the path of the Sun and is called the ecliptic, since the Moon must be located on it to produce an Page 223 eclipse. This shows that planets move (approximately) in a single, common plane.* To learn more about the motion in the sky, astronomers have performed numerous measurements of the movements of the Moon and the planets. The most industrious of all was Tycho Brahe,** who organized an industrial-scale search for astronomical facts sponsored by his king. His measurements were the basis for the research of his young assistant, the Swabian astronomer Johannes Kepler*** who found the first precise * The apparent height of the ecliptic changes with the time of the year and is the reason for the changing seasons. Therefore seasons are a gravitational effect as well. ** Tycho Brahe (b. 1546 Scania, d. 1601 Prague), famous astronomer, builder of Uraniaborg, the astronom- ical castle. He consumed almost 10 % of the Danish gross national product for his research, which produced the first star catalogue and the first precise position measurements of planets. *** Johannes Kepler (1571 Weil der Stadt–1630 Regensburg) studied Protestant theology and became a teacher of mathematics, astronomy and rhetoric. He helped his mother to defend herself successfully in a trial where she was accused of witchcraft. His first book on astronomy made him famous, and he became assistant to Tycho Brahe and then, at his teacher’s death, the Imperial Mathematician. He was the first to 176 6 motion due to gravitation d d Sun planet F I G U R E 132 The motion of a planet around the Sun, showing its semimajor axis 𝑑, which is also the spatial average of its distance from the Sun. Vol. III, page 324 description of planetary motion. This is not an easy task, as the observation of Figure 131 shows. In his painstaking research on the movements of the planets in the zodiac, Kepler Motion Mountain – The Adventure of Physics discovered several ‘laws’, i.e., patterns or rules. The motion of all the planets follow the same rules, confirming that the Sun determines their orbits. The three main ones are as follows: 1. Planets move on ellipses with the Sun located at one focus (1609). 2. Planets sweep out equal areas in equal times (1609). 3. All planets have the same ratio 𝑇2 /𝑑3 between the orbit duration 𝑇 and the semimajor axis 𝑑 (1619). Kepler’s results are illustrated in Figure 132. The sheer work required to deduce the three copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ‘laws’ was enormous. Kepler had no calculating machine available. The calculation tech- nology he used was the recently discovered logarithms. Anyone who has used tables of logarithms to perform calculations can get a feeling for the amount of work behind these three discoveries. Finally, in 1684, all observations by Kepler about planets and stones were condensed into an astonishingly simple result by the English physicist Robert Hooke and a few oth- ers:* ⊳ Every body of mass 𝑀 attracts any other body towards its centre with an acceleration whose magnitude 𝑎 is given by 𝑀 𝑎=𝐺 (39) 𝑟2 where 𝑟 is the centre-to-centre distance of the two bodies. use mathematics in the description of astronomical observations, and introduced the concept and field of ‘celestial physics’. * Robert Hooke (1635–1703), important English physicist and secretary of the Royal Society. Apart from discovering the inverse square relation and many others, such as Hooke’s ‘law’, he also wrote the Micro- graphia, a beautifully illustrated exploration of the world of the very small. 6 motion due to gravitation 177 Orbit of planet Sun (origin) S F Reflection of Circle of largest point C possible planet along tangent: distance from Sun fixed in space at energy E < 0 Tangent P to planet planet Motion Mountain – The Adventure of Physics R = – k/E motion position = constant C Projection of planet position P onto circle of largest planet distance F I G U R E 133 The proof that a planet moves in an ellipse (magenta) around the Sun, given an inverse square distance relation for gravitation. The proof – detailed in the text – is based on the relation copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net SP+PF=R. Since R is constant, the orbit is an ellipse This is called universal gravitation, or the universal ‘law’ of gravitation, because it is valid both in the sky and on Earth, as we will see shortly. The proportionality constant 𝐺 is called the gravitational constant; it is one of the fundamental constants of nature, like the Page 182 speed of light 𝑐 or the quantum of action ℏ. More about 𝐺 will be said shortly. The effect of gravity thus decreases with increasing distance; the effect depends on the inverse distance squared of the bodies under consideration. If bodies are small compared with the distance 𝑟, or if they are spherical, expression (39) is correct as it stands; for non-spherical shapes the acceleration has to be calculated separately for each part of the bodies and then added together. Ref. 147 Why is the usual planetary orbit an ellipse? The simplest argument is given in Fig- ure 133. We know that the acceleration due to gravity varies as 𝑎 = 𝐺𝑀/𝑟2 . We also know that an orbiting body of mass 𝑚 has a constant energy 𝐸 < 0. We then can draw, around the Sun, the circle with radius 𝑅 = −𝐺𝑀𝑚/𝐸, which gives the largest distance that a body with energy 𝐸 can be from the Sun. We now project the planet position 𝑃 onto this circle, thus constructing a position 𝐶. We then reflect 𝐶 along the tangent to get a position 𝐹. This last position 𝐹 is fixed in space and time, as a simple argument shows. Challenge 314 s (Can you find it?) As a result of the construction, the distance sum SP+PF is constant in time, and given by the radius 𝑅 = −𝐺𝑀𝑚/𝐸. Since the distance sum is constant, the 178 6 motion due to gravitation orbit is an ellipse, because an ellipse is precisely the curve that appears when this sum is constant. (Remember that an ellipse can be drawn with a piece of rope in this way.) Point 𝐹, like the Sun, is a focus of the ellipse. The construction thus shows that the motion of a planet defines two foci and follows an elliptical orbit defined by these two foci. In short, we have deduced the first of Kepler’s ‘laws’ from the expression of universal gravitation. The second of Kepler’s ‘laws’, about equal swept areas, implies that planets move faster when they are near the Sun. It is a simple way to state the conservation of angular Challenge 315 e momentum. What does the third ‘law’ state? Can you confirm that also the second and third of Kepler’s ‘laws’ follow from Hooke’s Challenge 316 s expression of universal gravity? Publishing this result – which was obvious to Hooke – was one of the achievements of Newton. Try to repeat this achievement; it will show you not only the difficulties, but also the possibilities of physics, and the joy that puzzles give. Newton solved these puzzles with geometric drawings – though in quite a complex manner. It is well known that Newton was not able to write down, let alone handle, dif- Ref. 28 ferential equations at the time he published his results on gravitation. In fact, Newton’s notation and calculation methods were poor. (Much poorer than yours!) The English Motion Mountain – The Adventure of Physics mathematician Godfrey Hardy* used to say that the insistence on using Newton’s in- tegral and differential notation, rather than the earlier and better method, still common today, due to his rival Leibniz – threw back English mathematics by 100 years. To sum up, Kepler, Hooke and Newton became famous because they brought order to the description of planetary motion. They showed that all motion due to gravity follows from the same description, the inverse square distance. For this reason, the inverse square distance relation 𝑎 = 𝐺𝑀/𝑟2 is called the universal law of gravity. Achieving this unifica- tion of motion description, though of small practical significance, was widely publicized. The main reasons were the age-old prejudices and fantasies linked with astrology. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net In fact, the inverse square distance relation explains many additional phenomena. It explains the motion and shape of the Milky Way and of the other galaxies, the motion of many weather phenomena, and explains why the Earth has an atmosphere but the Moon Challenge 317 s does not. (Can you explain this?) Gravitation on E arth This inverse square dependence of gravitational acceleration is often, but incorrectly, called Newton’s ‘law’ of gravitation. Indeed, the occultist and physicist Isaac Newton proved more elegantly than Hooke that the expression agreed with all astronomical and terrestrial observations. Above all, however, he organized a better public relations cam- Ref. 137 paign, in which he falsely claimed to be the originator of the idea. Newton published a simple proof showing that the description of astronomical grav- itation also gives the correct description for stones thrown through the air, down here on ‘father Earth’. To achieve this, he compared the acceleration 𝑎m of the Moon with that of stones 𝑔. For the ratio between these two accelerations, the inverse square rela- tion predicts a value 𝑔/𝑎m = 𝑑2m/𝑅2 , where 𝑑m the distance of the Moon and 𝑅 is the radius of the Earth. The Moon’s distance can be measured by triangulation, comparing * Godfrey Harold Hardy (1877–1947) was an important number theorist, and the author of the well-known A Mathematician’s Apology. He also ‘discovered’ the famous Indian mathematician Srinivasa Ramanujan, and brought him to Britain. 6 motion due to gravitation 179 F I G U R E 134 How to compare the radius of the Earth with that of the Moon during a partial lunar eclipse (© Anthony Ayiomamitis). Motion Mountain – The Adventure of Physics the position of the Moon against the starry background from two different points on Earth.* The result is 𝑑m /𝑅 = 60 ± 3, depending on the orbital position of the Moon, so that an average ratio 𝑔/𝑎m = 3.6 ⋅ 103 is predicted from universal gravity. But both accelerations can also be measured directly. At the surface of the Earth, stones are sub- ject to an acceleration due to gravitation with magnitude 𝑔 = 9.8 m/s2 , as determined by measuring the time that stones need to fall a given distance. For the Moon, the definition of acceleration, 𝑎 = d𝑣/d𝑡, in the case of circular motion – roughly correct here – gives 𝑎m = 𝑑m(2π/𝑇)2 , where 𝑇 = 2.4 Ms is the time the Moon takes for one orbit around copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the Earth.** The measurement of the radius of the Earth*** yields 𝑅 = 6.4 Mm, so that * The first precise – but not the first – measurement was achieved in 1752 by the French astronomers Lalande and La Caille, who simultaneously measured the position of the Moon seen from Berlin and from Le Cap. ** This expression for the centripetal acceleration is deduced easily by noting that for an object in circular motion, the magnitude 𝑣 of the velocity 𝑣 = d𝑥/d𝑡 is given as 𝑣 = 2π𝑟/𝑇. The drawing of the vector 𝑣 over Challenge 318 s time, the so-called hodograph, shows that it behaves exactly like the position of the object. Therefore the magnitude 𝑎 of the acceleration 𝑎 = d𝑣/d𝑡 is given by the corresponding expression, namely 𝑎 = 2π𝑣/𝑇. *** This is the hardest quantity to measure oneself. The most surprising way to determine the Earth’s size is the following: watch a sunset in the garden of a house, with a stopwatch in hand, such as the one in your Ref. 138 mobile phone. When the last ray of the Sun disappears, start the stopwatch and run upstairs. There, the Sun is still visible; stop the stopwatch when the Sun disappears again and note the time 𝑡. Measure the height difference ℎ between the two eye positions where the Sun was observed. The Earth’s radius 𝑅 is then given Challenge 319 s by 𝑅 = 𝑘 ℎ/𝑡2 , with 𝑘 = 378 ⋅ 106 s2 . Ref. 139 There is also a simple way to measure the distance to the Moon, once the size of the Earth is known. Take a photograph of the Moon when it is high in the sky, and call 𝜃 its zenith angle, i.e., its angle from the vertical above you. Make another photograph of the Moon a few hours later, when it is just above the Page 69 horizon. On this picture, unlike the common optical illusion, the Moon is smaller, because it is further away. With a sketch the reason becomes immediately clear. If 𝑞 is the ratio of the two angular diameters, the Earth–Moon distance 𝑑m is given by the relation 𝑑2m = 𝑅2 + (2𝑅𝑞 cos 𝜃/(1 − 𝑞2 ))2 . Enjoy finding its Challenge 320 s derivation from the sketch. Another possibility is to determine the size of the Moon by comparing it with the size of the Earth in a lunar exclipe, as shown in Figure 134. The distance to the Moon is then computed from its angular size, about 0.5°. 180 6 motion due to gravitation Moon Earth figure to be inserted F I G U R E 135 A physicist’s and an artist’s view of the fall of the Moon: a diagram by Christiaan Huygens (not to scale) and a marble statue by Auguste Rodin. Motion Mountain – The Adventure of Physics the average Earth–Moon distance is 𝑑m = 0.38 Gm. One thus has 𝑔/𝑎m = 3.6 ⋅ 103 , in agreement with the above prediction. With this famous ‘Moon calculation’ we have thus shown that the inverse square property of gravitation indeed describes both the motion of the Moon and that of stones. You might want to deduce the value of the product 𝐺𝑀 Challenge 321 s for Earth. Universal gravitation thus describes all motion due to gravity – both on Earth and in the sky. This was an important step towards the unification of physics. Before this discov- ery, from the observation that on the Earth all motion eventually comes to rest, whereas copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net in the sky all motion is eternal, Aristotle and many others had concluded that motion in the sublunar world has different properties from motion in the translunar world. Sev- eral thinkers had criticized this distinction, notably the philosopher and rector of the Ref. 140 University of Paris, Jean Buridan.* The Moon calculation was the most important result showing this distinction to be wrong. This is the reason for calling Hooke’s expression (39) the universal gravitation. Universal gravitation allows us to answer another old question. Why does the Moon not fall from the sky? Well, the preceding discussion showed that fall is motion due to gravitation. Therefore the Moon actually is falling, with the peculiarity that instead of falling towards the Earth, it is continuously falling around it. Figure 135 illustrates the idea. The Moon is continuously missing the Earth.** The Moon is not the only object that falls around the Earth. Figure 137 shows another. * Jean Buridan (c. 1295 to c. 1366) was also one of the first modern thinkers to discuss the rotation of the Earth about an axis. ** Another way to put it is to use the answer of the Dutch physicist Christiaan Huygens (1629–1695): the Moon does not fall from the sky because of the centrifugal acceleration. As explained on page 161, this explanation is often out of favour at universities. Ref. 141 There is a beautiful problem connected to the left side of the figure: Which points on the surface of the Challenge 322 d Earth can be hit by shooting from a mountain? And which points can be hit by shooting horizontally? 6 motion due to gravitation 181 F I G U R E 136 A precision second pendulum, thus about 1 m in length; almost Motion Mountain – The Adventure of Physics at the upper end, the vacuum chamber that compensates for changes in atmospheric pressure; towards the lower end, the wide construction that compensates for temperature variations of pendulum length; at the very bottom, the screw that compensates for local variations of the gravitational acceleration, giving a ﬁnal precision of about 1 s per month (© Erwin Sattler OHG). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 137 The man in orbit feels no weight, the blue atmosphere, which is not, does (NASA). 182 6 motion due to gravitation TA B L E 26 Some measured values of the acceleration due to gravity. Place Va l u e Poles 9.83 m/s2 Trondheim 9.8215243 m/s2 Hamburg 9.8139443 m/s2 Munich 9.8072914 m/s2 Rome 9.8034755 m/s2 Equator 9.78 m/s2 Moon 1.6 m/s2 Sun 273 m/s2 Motion Mountain – The Adventure of Physics Properties of gravitation: 𝐺 and 𝑔 Gravitation implies that the path of a stone is not a parabola, as stated earlier, but actually copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Page 77 an ellipse around the centre of the Earth. This happens for exactly the same reason that Page 177 the planets move in ellipses around the Sun. Are you able to confirm this statement? Universal gravitation allows us to understand the puzzling acceleration value 𝑔 = 9.8 m/s2 we encountered in equation (6). The value is due to the relation 𝑔 = 𝐺𝑀Earth /𝑅2Earth . (40) The expression can be deduced from equation (39), universal gravity, by taking the Earth to be spherical. The everyday acceleration of gravity 𝑔 thus results from the size of the Earth, its mass, and the universal constant of gravitation 𝐺. Obviously, the value for 𝑔 is almost constant on the surface of the Earth, as shown in Table 26, because the Earth is almost a sphere. Expression (40) also explains why 𝑔 gets smaller as one rises above the Earth, and the deviations of the shape of the Earth from sphericity explain why 𝑔 is different at the poles and higher on a plateau. (What would 𝑔 be on the Moon? On Mars? Challenge 323 s On Jupiter?) By the way, it is possible to devise a simple machine, other than a yo-yo, that slows down the effective acceleration of gravity by a known amount, so that one can measure Challenge 324 s its value more easily. Can you imagine this machine? Note that 9.8 is roughly π2 . This is not a coincidence: the metre has been chosen in such a way to make this (roughly) correct. The period 𝑇 of a swinging pendulum, i.e., a 6 motion due to gravitation 183 Challenge 325 s back and forward swing, is given by* 𝑙 𝑇 = 2π√ , (41) 𝑔 where 𝑙 is the length of the pendulum, and 𝑔 = 9.8 m/s2 is the gravitational acceleration. (The pendulum is assumed to consist of a compact mass attached to a string of negligible mass.) The oscillation time of a pendulum depends only on the length of the string and on 𝑔, thus on the planet it is located on. If the metre had been defined such that 𝑇/2 = 1 s, the value of the normal acceleration Challenge 327 e 𝑔 would have been exactly π2 m/s2 = 9.869 604 401 09 m/s2 . Indeed, this was the first proposal for the definition of the metre; it was made in 1673 by Huygens and repeated in 1790 by Talleyrand, but was rejected by the conference that defined the metre because variations in the value of 𝑔 with geographical position, temperature-induced variations of the length of a pendulum and even air pressure variations induce errors that are too Motion Mountain – The Adventure of Physics large to yield a definition of useful precision. (Indeed, all these effects must be corrected in pendulum clocks, as shown in Figure 136.) Finally, the proposal was made to define the metre as 1/40 000 000 of the circum- ference of the Earth through the poles, a so-called meridian. This proposal was almost identical to – but much more precise than – the pendulum proposal. The meridian defin- ition of the metre was then adopted by the French national assembly on 26 March 1791, with the statement that ‘a meridian passes under the feet of every human being, and all meridians are equal’. (Nevertheless, the distance from the Equator to the poles is not Ref. 142 exactly 10 Mm; that is a strange story. One of the two geographers who determined the size of the first metre stick was dishonest. The data he gave for his measurements – the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net general method of which is shown in Figure 138 – was fabricated. Thus the first official metre stick in Paris was shorter than it should be.) Continuing our exploration of the gravitational acceleration 𝑔, we can still ask: Why does the Earth have the mass and size it has? And why does 𝐺 have the value it has? The first question asks for a history of the Solar System; it is still unanswered and is topic of research. The second question is addressed in Appendix B. If gravitation is indeed universal, and if all objects really attract each other, attraction should also occur between any two objects of everyday life. Gravity must also work side- ways. This is indeed the case, even though the effects are extremely small. Indeed, the effects are so small that they were measured only long after universal gravity had pre- dicted them. On the other hand, measuring this effect is the only way to determine the gravitational constant 𝐺. Let us see how to do it. * Formula (41) is noteworthy mainly for all that is missing. The period of a pendulum does not depend on the mass of the swinging body. In addition, the period of a pendulum does not depend on the amplitude. (This is true as long as the oscillation angle is smaller than about 15°.) Galileo discovered this as a student, when observing a chandelier hanging on a long rope in the dome of Pisa. Using his heartbeat as a clock he found that even though the amplitude of the swing got smaller and smaller, the time for the swing stayed the same. A leg also moves like a pendulum, when one walks normally. Why then do taller people tend to walk Challenge 326 s faster? Is the relation also true for animals of different size? 184 6 motion due to gravitation Motion Mountain – The Adventure of Physics F I G U R E 138 The measurements that lead to the deﬁnition of the metre (© Ken Alder). We note that measuring the gravitational constant 𝐺 is also the only way to determ- ine the mass of the Earth. The first to do so, in 1798, was the English physicist Henry Cavendish; he used the machine, ideas and method of John Michell who died when attempting the experiment. Michell and Cavendish* called the aim and result of their copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net experiments ‘weighing the Earth’. The idea of Michell was to suspended a horizontal handle, with two masses at the end, at the end of a long metal wire. He then approached two additional large masses at the two ends of the handle, avoiding any air currents, and measured how much the handle rotated. Figure 139 shows how to repeat this experiment in your basement, and Figure 140 how to perform it when you have a larger budget. The value the gravitational constant 𝐺 found in more elaborate versions of the Michell–Cavendish experiments is 𝐺 = 6.7 ⋅ 10−11 Nm2 /kg2 = 6.7 ⋅ 10−11 m3 /kg s2 . (42) Cavendish’s experiment was thus the first to confirm that gravity also works sideways. The experiment also allows deducing the mass 𝑀 of the Earth from its radius 𝑅 and the Challenge 328 e relation 𝑔 = 𝐺𝑀/𝑅2 . Therefore, the experiment also allows to deduce the average density of the Earth. Finally, as we will see later on, this experiment proves, if we keep in mind Vol. II, page 141 that the speed of light is finite and invariant, that space is curved. All this is achieved with this simple set-up! * Henry Cavendish (b. 1731 Nice, d. 1810 London) was one of the great geniuses of physics; rich, autistic, misogynist, unmarried and solitary, he found many rules of nature, but never published them. Had he done so, his name would be much more well-known. John Michell (1724–1793) was church minister, geologist and amateur astronomer. 6 motion due to gravitation 185 Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 139 Home experiments that allow determining the gravitational constant 𝐺, weighing the Earth, proving that gravity also works sideways and thus showing that gravity curves space. Top left and right: a torsion balance made of foam and lead, with pétanque (boules) masses as ﬁxed masses; centre right: a torsion balance made of wood and lead, with stones as ﬁxed masses; bottom: a time sequence showing how the stones do attract the lead (© John Walker). Ref. 143 Cavendish found a mass density of the Earth of 5.5 times that of water. This was a surprising result because rock only has 2.8 times the density of water. What is the origin Challenge 329 e of the large density value? We note that 𝐺 has a small value. Above, we mentioned that gravity limits motion. In fact, we can write the expression for universal gravitation in the following way: 𝑎𝑟2 =𝐺>0 (43) 𝑀 Gravity prevents uniform motion. In fact, we can say more: Gravitation is the smallest 186 6 motion due to gravitation Motion Mountain – The Adventure of Physics F I G U R E 140 A modern precision torsion balance experiment to measure the gravitational constant, performed at the University of Washington (© Eöt-Wash Group). 𝜑(𝑥, 𝑦) copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net 𝑦 𝑥 grad 𝜑 F I G U R E 141 The potential and the gradient, visualized for two spatial dimensions. possible effect of the environment on a moving body. All other effects come on top of grav- Challenge 330 d ity. However, it is not easy to put this statement in a simple formula. Gravitation between everyday objects is weak. For example, two average people 1 m apart feel an acceleration towards each other that is less than that exerted by a common Challenge 331 s fly when landing on the skin. Therefore we usually do not notice the attraction to other people. When we notice it, it is much stronger than that. The measurement of 𝐺 thus proves that gravitation cannot be the true cause of people falling in love, and also that erotic attraction is not of gravitational origin, but of a different source. The physical basis Vol. III, page 15 for love will be studied later in our walk: it is called electromagnetism. The gravitational potential Gravity has an important property: all effects of gravitation can also be described by an- other observable, namely the (gravitational) potential 𝜑. We then have the simple relation 6 motion due to gravitation 187 that the acceleration is given by the gradient of the potential 𝑎 = −∇𝜑 or 𝑎 = −grad 𝜑 . (44) The gradient is just a learned term for ‘slope along the steepest direction’. The gradient is defined for any point on a slope, is large for a steep one and small for a shallow one. The gradient points in the direction of steepest ascent, as shown in Figure 141. The gradient is abbreviated ∇, pronounced ‘nabla’, and is mathematically defined through the relation ∇𝜑 = (∂𝜑/∂𝑥, ∂𝜑/∂𝑦, ∂𝜑/∂𝑧) = grad 𝜑.* The minus sign in (44) is introduced by con- vention, in order to have higher potential values at larger heights. In everyday life, when the spherical shape of the Earth can be neglected, the gravitational potential is given by 𝜑 = 𝑔ℎ . (45) The potential 𝜑 is an interesting quantity; with a single number at every position in space we can describe the vector aspects of gravitational acceleration. It automatically gives that Motion Mountain – The Adventure of Physics gravity in New Zealand acts in the opposite direction to gravity in Paris. In addition, the potential suggests the introduction of the so-called potential energy 𝑈 by setting 𝑈 = 𝑚𝜑 (46) and thus allowing us to determine the change of kinetic energy 𝑇 of a body falling from a point 1 to a point 2 via 𝑇1 − 𝑇2 = 𝑈2 − 𝑈1 or 1 𝑚𝑣2 2 1 1 − 12 𝑚2 𝑣2 2 = 𝑚𝜑2 − 𝑚𝜑1 . (47) copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net In other words, the total energy, defined as the sum of kinetic and potential energy, is con- served in motion due to gravity. This is a characteristic property of gravitation. Gravity conserves energy and momentum. Page 110 Not all accelerations can be derived from a potential; systems with this property are called conservative. Observation shows that accelerations due to friction are not conser- vative, but accelerations due to electromagnetism are. In short, we can either say that gravity can be described by a potential, or say that it conserves energy and momentum. Both mean the same. When the non-spherical shape of the Earth can be neglected, the potential energy of an object at height ℎ is given by 𝑈 = 𝑚𝑔ℎ . (48) To get a feeling of how much energy this is, answer the following question. A car with mass 1 Mg falls down a cliff of 100 m. How much water can be heated from freezing point Challenge 332 s to boiling point with the energy of the car? * In two or more dimensions slopes are written ∂𝜑/∂𝑧 – where ∂ is still pronounced ‘d’ – because in those cases the expression 𝑑𝜑/𝑑𝑧 has a slightly different meaning. The details lie outside the scope of this walk. 188 6 motion due to gravitation The shape of the E arth Universal gravity also explains why the Earth and most planets are (almost) spherical. Since gravity increases with decreasing distance, a liquid body in space will always try to form a spherical shape. Seen on a large scale, the Earth is indeed liquid. We also know that the Earth is cooling down – that is how the crust and the continents formed. The sphericity of smaller solid objects encountered in space, such as the Moon, thus means that they used to be liquid in older times. The Earth is thus not flat, but roughly spherical. Therefore, the top of two tall buildings Challenge 333 s is further apart than their base. Can this effect be measured? Sphericity considerably simplifies the description of motion. For a spherical or a Challenge 334 e point-like body of mass 𝑀, the potential 𝜑 is 𝑀 𝜑 = −𝐺 . (49) 𝑟 A potential considerably simplifies the description of motion, since a potential is addit- Motion Mountain – The Adventure of Physics ive: given the potential of a point particle, we can calculate the potential and then the motion around any other irregularly shaped object.* Interestingly, the number 𝑑 of di- mensions of space is coded into the potential 𝜑 of a spherical mass: the dependence of 𝜑 Challenge 336 s on the radius 𝑟 is in fact 1/𝑟𝑑−2 . The exponent 𝑑 − 2 has been checked experimentally to Ref. 144 extremely high precision; no deviation of 𝑑 from 3 has ever been found. The concept of potential helps in understanding the shape of the Earth in more detail. Ref. 145 Since most of the Earth is still liquid when seen on a large scale, its surface is always hori- zontal with respect to the direction determined by the combination of the accelerations of gravity and rotation. In short, the Earth is not a sphere. It is not an ellipsoid either. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Ref. 146 The mathematical shape defined by the equilibrium requirement is called a geoid. The geoid shape, illustrated in Figure 142, differs from a suitably chosen ellipsoid by at most Challenge 337 ny 50 m. Can you describe the geoid mathematically? The geoid is an excellent approxima- tion to the actual shape of the Earth; sea level differs from it by less than 20 metres. The differences can be measured with satellite radar and are of great interest to geologists and geographers. For example, it turns out that the South Pole is nearer to the equatorial * Alternatively, for a general, extended body, the potential is found by requiring that the divergence of its gradient is given by the mass (or charge) density times some proportionality constant. More precisely, we have Δ𝜑 = 4π𝐺𝜌 (50) where 𝜌 = 𝜌(𝑥, 𝑡) is the mass volume density of the body and the so-called Laplace operator Δ, pronounced ‘delta’, is defined as Δ𝑓 = ∇∇𝑓 = ∂2 𝑓/∂𝑥2 +∂2 𝑓/∂𝑦2 +∂2 𝑓/∂𝑧2 . Equation (50) is called the Poisson equation for the potential 𝜑. It is named after Siméon-Denis Poisson (1781–1840), eminent French mathematician and physicist. The positions at which 𝜌 is not zero are called the sources of the potential. The so-called source term Δ𝜑 of a function is a measure for how much the function 𝜑(𝑥) at a point 𝑥 differs from the Challenge 335 e average value in a region around that point. (Can you show this, by showing that Δ𝜑 ∼ 𝜑̄ − 𝜑(𝑥)?) In other words, the Poisson equation (50) implies that the actual value of the potential at a point is the same as the average value around that point minus the mass density multiplied by 4π𝐺. In particular, in the case of empty space the potential at a point is equal to the average of the potential around that point. Often the concept of gravitational field is introduced, defined as 𝑔 = −∇𝜑. We avoid this in our walk because we will discover that, following the theory of relativity, gravity is not due to a field at all; in fact, even the concept of gravitational potential turns out to be only an approximation. 6 motion due to gravitation 189 F I G U R E 142 The shape of the Earth, with exaggerated height scale (© GeoForschungsZentrum Potsdam). Motion Mountain – The Adventure of Physics plane than the North Pole by about 30 m. This is probably due to the large land masses in the northern hemisphere. Page 137 Above we saw how the inertia of matter, through the so-called ‘centrifugal force’, increases the radius of the Earth at the Equator. In other words, the Earth is flattened at the poles. The Equator has a radius 𝑎 of 6.38 Mm, whereas the distance 𝑏 from the poles to the centre of the Earth is 6.36 Mm. The precise flattening (𝑎 − 𝑏)/𝑎 has the value Appendix B 1/298.3 = 0.0034. As a result, the top of Mount Chimborazo in Ecuador, even though its height is only 6267 m above sea level, is about 20 km farther away from the centre of copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the Earth than the top of Mount Sagarmatha* in Nepal, whose height above sea level is 8850 m. The top of Mount Chimborazo is in fact the point on the surface most distant from the centre of the Earth. The shape of the Earth has another important consequence. If the Earth stopped ro- tating (but kept its shape), the water of the oceans would flow from the Equator to the poles; all of Europe would be under water, except for the few mountains of the Alps that are higher than about 4 km. The northern parts of Europe would be covered by between 6 km and 10 km of water. Mount Sagarmatha would be over 11 km above sea level. We would also walk inclined. If we take into account the resulting change of shape of the Earth, the numbers come out somewhat smaller. In addition, the change in shape would produce extremely strong earthquakes and storms. As long as there are none of these Page 157 effects, we can be sure that the Sun will indeed rise tomorrow, despite what some philo- sophers pretended. Dynamics – how d o things move in various dimensions? The concept of potential is a powerful tool. If a body can move only along a – straight or curved – line, the concepts of kinetic and potential energy are sufficient to determine completely the way the body moves. * Mount Sagarmatha is sometimes also called Mount Everest. 190 6 motion due to gravitation F I G U R E 143 The change of the moon during the month, showing its libration (QuickTime Motion Mountain – The Adventure of Physics ﬁlm © Martin Elsässer) In fact, motion in one dimension follows directly from energy conservation. For a body moving along a given curve, the speed at every instant is given by energy conservation. If a body can move in two dimensions – i.e., on a flat or curved surface – and if the forces involved are internal (which is always the case in theory, but not in practice), the conservation of angular momentum can be used. The full motion in two dimensions thus copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net follows from energy and angular momentum conservation. For example, all properties of free fall follow from energy and angular momentum conservation. (Are you able to Challenge 338 s show this?) Again, the potential is essential. In the case of motion in three dimensions, a more general rule for determining motion is necessary. If more than two spatial dimensions are involved conservation is insufficient to determine how a body moves. It turns out that general motion follows from a simple principle: the time average of the difference between kinetic and potential energy must be as small as possible. This is called the least action principle. We will explain the details Page 248 of this calculation method later. But again, the potential is the main ingredient in the calculation of change, and thus in the description of any example of motion. For simple gravitational motions, motion is two-dimensional, in a plane. Most three- dimensional problems are outside the scope of this text; in fact, some of these problems are so hard that they still are subjects of research. In this adventure, we will explore three- dimensional motion only for selected cases that provide important insights. The Mo on How long is a day on the Moon? The answer is roughly 29 Earth-days. That is the time that it takes for an observer on the Moon to see the Sun again in the same position in the sky. One often hears that the Moon always shows the same side to the Earth. But this is 6 motion due to gravitation 191 F I G U R E 144 High resolution maps (not photographs) of the near side (left) and far side (right) of the Motion Mountain – The Adventure of Physics moon, showing how often the latter saved the Earth from meteorite impacts (courtesy USGS). wrong. As one can check with the naked eye, a given feature in the centre of the face of the Moon at full Moon is not at the centre one week later. The various motions leading to this change are called librations; they are shown in the film in Figure 143. The motions appear mainly because the Moon does not describe a circular, but an elliptical orbit around the Earth and because the axis of the Moon is slightly inclined, when compared with that of its rotation around the Earth. As a result, only around 45 % of the Moon’s surface is permanently hidden from Earth. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net The first photographs of the hidden area of the Moon were taken in the 1960s by a Soviet artificial satellite; modern satellites provided exact maps, as shown in Figure 144. Challenge 339 e (Just zoom into the figure for fun.) The hidden surface is much more irregular than the visible one, as the hidden side is the one that intercepts most asteroids attracted by the Earth. Thus the gravitation of the Moon helps to deflect asteroids from the Earth. The number of animal life extinctions is thus reduced to a small, but not negligible number. In other words, the gravitational attraction of the Moon has saved the human race from extinction many times over.* The trips to the Moon in the 1970s also showed that the Moon originated from the Earth itself: long ago, an object hit the Earth almost tangentially and threw a sizeable fraction of material up into the sky. This is the only mechanism able to explain the large Ref. 148 size of the Moon, its low iron content, as well as its general material composition. Ref. 149 The Moon is receding from the Earth at 3.8 cm a year. This result confirms the old de- duction that the tides slow down the Earth’s rotation. Can you imagine how this meas- Challenge 340 s urement was performed? Since the Moon slows down the Earth, the Earth also changes shape due to this effect. (Remember that the shape of the Earth depends on its speed of * The web pages www.minorplanetcenter.net/iau/lists/Closest.html and InnerPlot.html give an impression of the number of objects that almost hit the Earth every year. Without the Moon, we would have many additional catastrophes. 192 6 motion due to gravitation rotation.) These changes in shape influence the tectonic activity of the Earth, and maybe also the drift of the continents. The Moon has many effects on animal life. A famous example is the midge Clunio, Ref. 150 which lives on coasts with pronounced tides. Clunio spends between six and twelve weeks as a larva, sure then hatches and lives for only one or two hours as an adult flying insect, during which time it reproduces. The midges will only reproduce if they hatch during the low tide phase of a spring tide. Spring tides are the especially strong tides during the full and new moons, when the solar and lunar effects combine, and occur only every 14.8 days. In 1995, Dietrich Neumann showed that the larvae have two built- in clocks, a circadian and a circalunar one, which together control the hatching to pre- cisely those few hours when the insect can reproduce. He also showed that the circalunar clock is synchronized by the brightness of the Moon at night. In other words, the larvae monitor the Moon at night and then decide when to hatch: they are the smallest known astronomers. If insects can have circalunar cycles, it should come as no surprise that women also have such a cycle; however, in this case the precise origin of the cycle length is still un- Motion Mountain – The Adventure of Physics Ref. 151 known and a topic of research. The Moon also helps to stabilize the tilt of the Earth’s axis, keeping it more or less fixed relative to the plane of motion around the Sun. Without the Moon, the axis would change its direction irregularly, we would not have a regular day and night rhythm, we would have extremely large climate changes, and the evolution of life would have been Ref. 152 impossible. Without the Moon, the Earth would also rotate much faster and we would Ref. 153 have much less clement weather. The Moon’s main remaining effect on the Earth, the Page 146 precession of its axis, is responsible for the ice ages. The orbit of the Moon is still a topic of research. It is still not clear why the Moon orbit copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net is at a 5° to the ecliptic and how the orbit changed since the Moon formed. Possibly, the collision that led to the formation of the Moon tilted the rotation axis of the Earth and the Ref. 154 original Moon; then over thousands of millions of years, both axes moved in complicated ways towards the ecliptic, one more than the other. During this evolution, the distance to the Moon is estimated to have increased by a factor of 15. Orbits – conic sections and more The path of a body continuously orbiting another under the influence of gravity is an ellipse with the central body at one focus. A circular orbit is also possible, a circle being a special case of an ellipse. Single encounters of two objects can also be parabolas or hyper- bolas, as shown in Figure 145. Circles, ellipses, parabolas and hyperbolas are collectively known as conic sections. Indeed each of these curves can be produced by cutting a cone Challenge 341 e with a knife. Are you able to confirm this? If orbits are mostly ellipses, it follows that comets return. The English astronomer Ed- mund Halley (1656–1742) was the first to draw this conclusion and to predict the return of a comet. It arrived at the predicted date in 1756, after his death, and is now named after him. The period of Halley’s comet is between 74 and 80 years; the first recorded sighting was 22 centuries ago, and it has been seen at every one of its 30 passages since, the last time in 1986. Depending on the initial energy and the initial angular momentum of the body with 6 motion due to gravitation 193 hyperbola parabola mass circle ellipse Motion Mountain – The Adventure of Physics F I G U R E 145 The possible orbits, due to universal gravity, of a small mass around a single large mass (left) and a few recent examples of measured orbits (right), namely those of some extrasolar planets and of the Earth, all drawn around their respective central star, with distances given in astronomical units (© Geoffrey Marcy). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net respect to the central planet, paths are either elliptic, parabolic or hyperbolic. Can you determine the conditions for the energy and the angular momentum needed for these Challenge 342 s paths to appear? In practice, parabolic orbits do not exist in nature. (Some comets seem to approach this case when moving around the Sun; but almost all comets follow elliptical paths – as long as they are far from other planets.). Hyperbolic paths do exist; artificial satellites follow them when they are shot towards a planet, usually with the aim of changing the direction of the satellite’s journey across the Solar System. Why does the inverse square ‘law’ lead to conic sections? First, for two bodies, the total angular momentum 𝐿 is a constant: d𝜑 𝐿 = 𝑚𝑟2 𝜑̇ = 𝑚𝑟2 ( ) (51) d𝑡 and therefore the motion lies in a plane. Also the energy 𝐸 is a constant d𝑟 2 1 d𝜑 2 𝑚𝑀 𝐸 = 12 𝑚 ( ) + 2 𝑚 (𝑟 ) − 𝐺 . (52) d𝑡 d𝑡 𝑟 194 6 motion due to gravitation Challenge 343 e Together, the two equations imply that 𝐿2 1 𝑟= . (53) 𝐺𝑚2 𝑀 2𝐸𝐿2 1 + √1 + 2 3 2 cos 𝜑 𝐺𝑚𝑀 Now, any curve defined by the general expression 𝐶 𝐶 𝑟= or 𝑟 = (54) 1 + 𝑒 cos 𝜑 1 − 𝑒 cos 𝜑 is an ellipse for 0 < 𝑒 < 1, a parabola for 𝑒 = 1 and a hyperbola for 𝑒 > 1, one focus being at the origin. The quantity 𝑒, called the eccentricity, describes how squeezed the curve is. In other words, a body in orbit around a central mass follows a conic section. In all orbits, also the heavy mass moves. In fact, both bodies orbit around the common centre of mass. Both bodies follow the same type of curve – ellipse, parabola or hyperbola Motion Mountain – The Adventure of Physics Challenge 344 e – but the sizes of the two curves differ. If more than two objects move under mutual gravitation, many additional possibilities for motions appear. The classification and the motions are quite complex. In fact, this so-called many-body problem is still a topic of research, both for astronomers and for mathematicians. Let us look at a few observations. When several planets circle a star, they also attract each other. Planets thus do not move in perfect ellipses. The largest deviation is a perihelion shift, as shown in Figure 114. Page 152 It is observed for Mercury and a few other planets, including the Earth. Other deviations from elliptical paths appear during a single orbit. In 1846, the observed deviations of copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net the motion of the planet Uranus from the path predicted by universal gravity were used to predict the existence of another planet, Neptune, which was discovered shortly after- wards. Page 106 We have seen that mass is always positive and that gravitation is thus always attractive; there is no antigravity. Can gravity be used for levitation nevertheless, using more than two bodies? Yes; there are two examples.* The first are the geostationary satellites, which are used for easy transmission of television and other signals from and towards Earth. The Lagrangian libration points are the second example. Named after their discoverer, these are points in space near a two-body system, such as Moon–Earth or Earth–Sun, in which small objects have a stable equilibrium position. An overview is given in Fig- ure 147. Can you find their precise position, remembering to take rotation into account? Challenge 345 s There are three additional Lagrangian points on the Earth–Moon line (or Sun–planet Challenge 346 d line). How many of them are stable? There are thousands of asteroids, called Trojan asteroids, at and around the Lagrangian points of the Sun–Jupiter system. In 1990, a Trojan asteroid for the Mars–Sun system was discovered. Finally, in 1997, an ‘almost Trojan’ asteroid was found that follows the Earth on its way around the Sun (it is only transitionary and follows a somewhat more complex Ref. 156 orbit). This ‘second companion’ of the Earth has a diameter of 5 km. Similarly, on the Vol. III, page 226 * Levitation is discussed in detail in the section on electrodynamics. 6 motion due to gravitation 195 Motion Mountain – The Adventure of Physics F I G U R E 146 Geostationary satellites, seen here in the upper left quadrant, move against the other stars and show the location of the celestial Equator. (MP4 ﬁlm © Michael Kunze) geostationary copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net satellite planet (or Sun) fixed N L5 parabolic π/3 antenna Earth π/3 π/3 π/3 L4 moon (or planet) F I G U R E 147 Geostationary satellites (left) and the main stable Lagrangian points (right). main Lagrangian points of the Earth–Moon system a high concentration of dust has been observed. Astronomers know that many other objects follow irregular orbits, especially aster- Ref. 155 oids. For example, asteroid 2003 YN107 followed an irregular orbit, shown in Figure 148, that accompanied the Earth for a number of years. To sum up, the single equation 𝑎 = −𝐺𝑀𝑟/𝑟3 correctly describes a large number of phenomena in the sky. The first person to make clear that this expression describes 196 6 motion due to gravitation 0.2 0.15 0.1 0.05 y 0 Size of Moon's orbit Motion Mountain – The Adventure of Physics Exit 12 years later Start -0.05 -0.1 -0.15 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net x F I G U R E 148 An example of irregular orbit, partly measured and partly calculated, due to the gravitational attraction of several masses: the orbit of the temporary Earth quasi-satellite 2003 YN107 in geocentric coordinates. This asteroid, with a diameter of 20(10) m, became orbitally trapped near the Earth around 1995 and remained so until 2006. The black circle represents the Moon’s orbit around the Earth. (© Seppo Mikkola). everything happening in the sky was Pierre Simon Laplace in his famous treatise Traité de mécanique céleste. When Napoleon told him that he found no mention about the cre- ator in the book, Laplace gave a famous, one sentence summary of his book: Je n’ai pas eu besoin de cette hypothèse. ‘I had no need for this hypothesis.’ In particular, Laplace studied the stability of the Solar System, the eccentricity of the lunar orbit, and the ec- centricities of the planetary orbits, always getting full agreement between calculation and measurement. These results are quite a feat for the simple expression of universal gravitation; they also explain why it is called ‘universal’. But how accurate is the formula? Since astronomy allows the most precise measurements of gravitational motion, it also provides the most stringent tests. In 1849, Urbain Le Verrier concluded after intensive study that there was only one known example of a discrepancy between observation and universal gravity, namely one observation for the planet Mercury. (Nowadays a few more are known.) The 6 motion due to gravitation 197 Motion Mountain – The Adventure of Physics F I G U R E 149 Tides at Saint-Valéry en Caux on 20 September 2005 (© Gilles Régnier). point of least distance to the Sun of the orbit of planet Mercury, its perihelion, rotates around the Sun at a rate that is slightly less than that predicted: he found a tiny differ- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Ref. 157 ence, around 38 per century. (This was corrected to 43 per century in 1882 by Simon Newcomb.) Le Verrier thought that the difference was due to a planet between Mercury and the Sun, Vulcan, which he chased for many years without success. Indeed, Vulcan Vol. II, page 164 does not exist. The correct explanation of the difference had to wait for Albert Einstein. Tides Ref. 158 Why do physics texts always talk about tides? Because, as general relativity will show, tides prove that space is curved! It is thus useful to study them in a bit more detail. Fig- ure 149 how striking tides can be. Gravitation explains the sea tides as results of the at- traction of the ocean water by the Moon and the Sun. Tides are interesting; even though the amplitude of the tides is only about 0.5 m on the open sea, it can be up to 20 m at Challenge 347 s special places near the coast. Can you imagine why? The soil is also lifted and lowered by Ref. 57 the Sun and the Moon, by about 0.3 m, as satellite measurements show. Even the atmo- sphere is subject to tides, and the corresponding pressure variations can be filtered out Ref. 159 from the weather pressure measurements. Tides appear for any extended body moving in the gravitational field of another. To understand the origin of tides, picture a body in orbit, like the Earth, and imagine its components, such as the segments of Figure 150, as being held together by springs. Uni- versal gravity implies that orbits are slower the more distant they are from a central body. As a result, the segment on the outside of the orbit would like to be slower than the cent- 198 6 motion due to gravitation Sun before 𝑡1 deformed 𝑡=0 after spherical F I G U R E 150 Tidal deformations due to F I G U R E 151 The origin of tides. gravity. Motion Mountain – The Adventure of Physics ral one; but it is pulled by the rest of the body through the springs. In contrast, the inside segment would like to orbit more rapidly but is retained by the others. Being slowed down, the inside segments want to fall towards the Sun. In sum, both segments feel a pull away from the centre of the body, limited by the springs that stop the deformation. Therefore, extended bodies are deformed in the direction of the field inhomogeneity. For example, as a result of tidal forces, the Moon always has (roughly) the same face to the Earth. In addition, its radius in direction of the Earth is larger by about 5 m than the radius perpendicular to it. If the inner springs are too weak, the body is torn into copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net pieces; in this way a ring of fragments can form, such as the asteroid ring between Mars and Jupiter or the rings around Saturn. Let us return to the Earth. If a body is surrounded by water, it will form bulges in the direction of the applied gravitational field. In order to measure and compare the strength of the tides from the Sun and the Moon, we reduce tidal effects to their bare minimum. As shown in Figure 151, we can study the deformation of a body due to gravity by studying the arrangement of four bodies. We can study the free fall case because orbital motion and free fall are equivalent. Now, gravity makes some of the pieces approach and others diverge, depending on their relative positions. The figure makes clear that the strength of the deformation – water has no built-in springs – depends on the change of gravitational acceleration with distance; in other words, the relative acceleration that leads to the tides is proportional to the derivative of the gravitational acceleration. Page 453 Using the numbers from Appendix B, the gravitational accelerations from the Sun and the Moon measured on Earth are 𝐺𝑀Sun 𝑎Sun = = 5.9 mm/s2 𝑑2Sun 𝐺𝑀 𝑎Moon = 2 Moon = 0.033 mm/s2 (55) 𝑑Moon and thus the attraction from the Moon is about 178 times weaker than that from the Sun. 6 motion due to gravitation 199 Moon Earth’s rotation drives The Moon attracts the the bulge forward. tide bulge and thus slows down the rotation of the Earth. Earth Motion Mountain – The Adventure of Physics The bulge of the tide attracts the Moon and thus increases the Moon’s orbit radius. F I G U R E 152 The Earth, the Moon and the friction effects of the tides (not to scale). When two nearby bodies fall near a large mass, the relative acceleration is pro- portional to their distance, and follows 𝑑𝑎 = (𝑑𝑎/𝑑𝑟) 𝑑𝑟. The proportionality factor 𝑑𝑎/𝑑𝑟 = ∇𝑎, called the tidal acceleration (gradient), is the true measure of tidal effects. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 348 e Near a large spherical mass 𝑀, it is given by 𝑑𝑎 2𝐺𝑀 =− 3 (56) 𝑑𝑟 𝑟 which yields the values 𝑑𝑎Sun 2𝐺𝑀 = − 3 Sun = −0.8 ⋅ 10−13 /s2 𝑑𝑟 𝑑Sun 𝑑𝑎Moon 2𝐺𝑀 = − 3 Moon = −1.7 ⋅ 10−13 /s2 . (57) 𝑑𝑟 𝑑Moon In other words, despite the much weaker pull of the Moon, its tides are predicted to be over twice as strong as the tides from the Sun; this is indeed observed. When Sun, Moon and Earth are aligned, the two tides add up; these so-called spring tides are especially strong and happen every 14.8 days, at full and new moon. Tides lead to a pretty puzzle. Moon tides are much stronger than Sun tides. This im- Challenge 349 s plies that the Moon is much denser than the Sun. Why? Tides also produce friction, as shown in Figure 152. The friction leads to a slowing of the Earth’s rotation. Nowadays, the slowdown can be measured by precise clocks (even 200 6 motion due to gravitation F I G U R E 153 A spectacular result of tides: volcanism on Io (NASA). Motion Mountain – The Adventure of Physics Ref. 116 though short time variations due to other effects, such as the weather, are often larger). The results fit well with fossil results showing that 400 million years ago, in the Devonian Vol. II, page 231 period, a year had 400 days, and a day about 22 hours. It is also estimated that 900 million copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net years ago, each of the 481 days of a year were 18.2 hours long. The friction at the basis of this slowdown also results in an increase in the distance of the Moon from the Earth by Challenge 350 s about 3.8 cm per year. Are you able to explain why? As mentioned above, the tidal motion of the soil is also responsible for the triggering of earthquakes. Thus the Moon can have also dangerous effects on Earth. (Unfortunately, knowing the mechanism does not allow predicting earthquakes.) The most fascinating example of tidal effects is seen on Jupiter’s satellite Io. Its tides are so strong that they induce intense volcanic activity, as shown in Figure 153, with eruption plumes as high as 500 km. If tides are even stronger, they can destroy the body altogether, as happened to the body between Mars and Jupiter that formed the planetoids, or (possibly) to the moons that led to Saturn’s rings. In summary, tides are due to relative accelerations of nearby mass points. This has an Vol. II, page 136 important consequence. In the chapter on general relativity we will find that time multi- plied by the speed of light plays the same role as length. Time then becomes an additional dimension, as shown in Figure 154. Using this similarity, two free particles moving in the same direction correspond to parallel lines in space-time. Two particles falling side-by- side also correspond to parallel lines. Tides show that such particles approach each other. Vol. II, page 190 In other words, tides imply that parallel lines approach each other. But parallel lines can approach each other only if space-time is curved. In short, tides imply curved space-time and space. This simple reasoning could have been performed in the eighteenth century; however, it took another 200 years and Albert Einstein’s genius to uncover it. 6 motion due to gravitation 201 space 𝑡1 𝛼 𝑏 𝑡2 𝑀 time F I G U R E 154 Particles falling F I G U R E 155 Masses bend light. side-by-side approach over time. C an light fall? Motion Mountain – The Adventure of Physics “ Die Maxime, jederzeit selbst zu denken, ist die ” Aufklärung. Immanuel Kant* Towards the end of the seventeenth century people discovered that light has a finite velo- Vol. II, page 15 city – a story which we will tell in detail later. An entity that moves with infinite velocity cannot be affected by gravity, as there is no time to produce an effect. An entity with a finite speed, however, should feel gravity and thus fall. Does its speed increase when light reaches the surface of the Earth? For almost three centuries people had no means of detecting any such effect; so the question was not in- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net vestigated. Then, in 1801, the Prussian astronomer Johann Soldner (1776–1833) was the Ref. 160 first to put the question in a different way. Being an astronomer, he was used to measur- ing stars and their observation angles. He realized that light passing near a massive body would be deflected due to gravity. Soldner studied a body on a hyperbolic path, moving with velocity 𝑐 past a spher- ical mass 𝑀 at distance 𝑏 (measured from the centre), as shown in Figure 155. Soldner Challenge 351 ny deduced the deflection angle 2 𝐺𝑀 𝛼univ. grav. = . (58) 𝑏 𝑐2 The value of the angle is largest when the motion is just grazing the mass 𝑀. For light deflected by the mass of the Sun, the angle turns out to be at most a tiny 0.88 = 4.3 μrad. In Soldner’s time, this angle was too small to be measured. Thus the issue was forgotten. Had it been pursued, general relativity would have begun as an experimental science, and Vol. II, page 161 not as the theoretical effort of Albert Einstein! Why? The value just calculated is different from the measured value. The first measurement took place in 1919;** it found the correct dependence on the distance, but found a deflection of up to 1.75 , exactly double that of expression (58). The reason is not easy to find; in fact, it is due to the curvature of space, * The maxim to think at all times for oneself is the enlightenment. Challenge 352 s ** By the way, how would you measure the deflection of light near the bright Sun? 202 6 motion due to gravitation as we will see. In summary, light can fall, but the issue hides some surprises. Mass: inertial and gravitational Mass describes how an object interacts with others. In our walk, we have encountered two of its aspects. Inertial mass is the property that keeps objects moving and that offers resistance to a change in their motion. Gravitational mass is the property responsible for the acceleration of bodies nearby (the active aspect) or of being accelerated by objects nearby (the passive aspect). For example, the active aspect of the mass of the Earth de- termines the surface acceleration of bodies; the passive aspect of the bodies allows us to weigh them in order to measure their mass using distances only, e.g. on a scale or a balance. The gravitational mass is the basis of weight, the difficulty of lifting things.* Is the gravitational mass of a body equal to its inertial mass? A rough answer is given by the experience that an object that is difficult to move is also difficult to lift. The simplest experiment is to take two bodies of different masses and let them fall. If the acceleration is the same for all bodies, inertial mass is equal to (passive) gravitational mass, because Motion Mountain – The Adventure of Physics in the relation 𝑚𝑎 = ∇(𝐺𝑀𝑚/𝑟) the left-hand 𝑚 is actually the inertial mass, and the right-hand 𝑚 is actually the gravitational mass. Already in the seventeenth century Galileo had made widely known an even older argument showing without a single experiment that the gravitational acceleration is in- deed the same for all bodies. If larger masses fell more rapidly than smaller ones, then the following paradox would appear. Any body can be seen as being composed of a large fragment attached to a small fragment. If small bodies really fell less rapidly, the small fragment would slow the large fragment down, so that the complete body would have to fall less rapidly than the larger fragment (or break into pieces). At the same time, the body being larger than its fragment, it should fall more rapidly than that fragment. This copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net is obviously impossible: all masses must fall with the same acceleration. Many accurate experiments have been performed since Galileo’s original discussion. In all of them, the independence of the acceleration of free fall from mass and material Ref. 161 composition has been confirmed with the precision they allowed. In other words, exper- iments confirm: ⊳ Gravitational mass and inertial mass are equal. What is the origin of this mysterious equality? The equality of gravitational and inertial mass is not a mystery at all. Let us go back Page 100 to the definition of mass as a negative inverse acceleration ratio. We mentioned that the physical origin of the accelerations does not play a role in the definition because the origin does not appear in the expression. In other words, the value of the mass is by definition independent of the interaction. That means in particular that inertial mass, based on and measured with the electromagnetic interaction, and gravitational mass are identical by definition. The best proof of the equality of inertial and gravitational mass is illustrated in Fig- ure 156: it shows that the two concepts only differ by the viewpoint of the observer. In- ertial mass and gravitational mass describe the same observation. Challenge 353 ny * What are the weight values shown by a balance for a person of 85 kg juggling three balls of 0.3 kg each? 6 motion due to gravitation 203 Motion Mountain – The Adventure of Physics F I G U R E 156 The falling ball is in inertial motion for a falling observer and in gravitational motion for an observer on the ground. Therefore, inertial mass is the same as gravitational mass. We also note that we have not defined a separate concept of ‘passive gravitational mass’. (This concept is sometimes found in research papers.) The mass being accelerated by gravitation is the inertial mass. Worse, there is no way to define a ‘passive gravitational Challenge 354 s mass’ that differs from inertial mass. Try it! All methods that measure a passive gravita- tional mass, such as weighing an object, cannot be distinguished from the methods that copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net determine inertial mass from its reaction to acceleration. Indeed, all these methods use the same non-gravitational mechanisms. Bathroom scales are a typical example. Indeed, if the ‘passive gravitational mass’ were different from the inertial mass, we would have strange consequences. Not only is it hard to distinguish the two in an exper- iment; for those bodies for which it were different we would get into trouble with energy Challenge 355 e conservation. In fact, also assuming that (‘active’) ‘gravitational mass’ differs from inertial mass gets us into trouble. How could ‘gravitational mass’ differ from inertial mass? Would the difference depend on relative velocity, time, position, composition or on mass itself? No. Challenge 356 s Each of these possibilities contradicts either energy or momentum conservation. In summary, it is no wonder that all measurements confirm the equality of all mass types: there is no other option – as Galileo pointed out. The lack of other options is due to the fundamental equivalence of all mass definitions: ⊳ Mass ratios are acceleration ratios. Vol. II, page 157 The topic is usually rehashed in general relativity, with no new results, because the defin- ition of mass remains the same. Gravitational and inertial masses remain equal. In short: ⊳ Mass is a unique property of each body. 204 6 motion due to gravitation Another, deeper issue remains, though. What is the origin of mass? Why does it exist? This simple but deep question cannot be answered by classical physics. We will need some patience to find out. Curiosities and fun challenges ab ou t gravitation “ Fallen ist weder gefährlich noch eine Schande; ” Liegen bleiben ist beides.* Konrad Adenauer Cosmonauts on the International Space Station face two challenges: cancer-inducing cosmic radiation and the lack of gravity. The lack of gravity often leads to orientation problems and nausea in the first days, the so-called space sickness and motion sickness. When these disappear, the muscles start to reduce in volume by a few % per month, bones get weaker every week, the immune system is on permanent alarm state, blood gets pumped into the head more than usual and produces round ‘baby’ faces – easily seen on television – and strong headaches, legs loose blood and get thinner, body tem- Motion Mountain – The Adventure of Physics perature permanently increases by over one degree Celsius, the brain gets compressed by the blood and spinal fluid, and the eyesight deteriorates, because also the eyes get com- pressed. When cosmonauts return to Earth after six months in space, they have weak bones and muscles, and they are unable to walk and stand. They need a day or two to learn to do so again. Later, they often get hernias, and because of the bone reduction, kidney stones. Other health issues are likely to exist; but they have been kept confiden- tial by cosmonauts in order to maintain their image and their chances for subsequent missions. ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Gravity on the Moon is only one sixth of that on the Earth. Why does this imply that it is difficult to walk quickly and to run on the Moon (as can be seen in the TV images recorded there)? ∗∗ Understand and explain the following statement: a beam balance measures mass, a spring Challenge 357 e scale measures weight. ∗∗ Does the Earth have other satellites apart from the Moon and the artificial satellites shot into orbit up by rockets? Yes. The Earth has a number of mini-satellites and a large num- ber of quasi-satellites. An especially long-lived quasi-satellite, an asteroid called 2016 HO3, has a size of about 60 m and was discovered in 2016. As shown in Figure 157, it orbits the Earth and will continue to do so for another few hundred years, at a distance from 40 to 100 times that of the Moon. ∗∗ * ‘Falling is neither dangerous nor a shame; to keep lying is both.’ Konrad Adenauer (b. 1876 Köln, d. 1967 Rhöndorf ), West German Chancellor. 6 motion due to gravitation 205 Motion Mountain – The Adventure of Physics F I G U R E 157 The calculated orbit of the quasi-satellite 2016 HO3, a temporary companion of the Earth (courtesy NASA). Show that a sphere bouncing – without energy loss – down an inclined plane hits the plane in spots whose distances increase by a constant amount at every bounce. ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Is the acceleration due to gravity constant over time? Not really. Every day, it is estimated that 108 kg of material fall onto the Earth in the form of meteorites and asteroids. (Ex- amples can be seen in Figure 158 and Figure 159.) Nevertheless, it is unknown whether the mass of the Earth increases with time (due to the collection of meteorites and cosmic dust) or decreases (due to gas loss). If you find a way to settle the issue, publish it. ∗∗ Incidentally, discovering objects hitting the Earth is not at all easy. Astronomers like to point out that an asteroid as large as the one that led to the extinction of the dinosaurs could hit the Earth without any astronomer noticing in advance, if the direction is slightly unusual, such as from the south, where few telescopes are located. ∗∗ Several humans have survived free falls from aeroplanes for a thousand metres or more, even though they had no parachute. A minority of them even did so without any harm Challenge 358 s at all. How was this possible? ∗∗ Imagine that you have twelve coins of identical appearance, of which one is a forgery. The forged one has a different mass from the eleven genuine ones. How can you decide which is the forged one and whether it is lighter or heavier, using a simple balance only 206 6 motion due to gravitation Motion Mountain – The Adventure of Physics F I G U R E 158 A composite photograph of the Perseid meteor shower that is visible every year in mid August. In that month, the Earth crosses the cloud of debris stemming from comet Swift–Tuttle, and the source of the meteors appears to lie in the constellation of Perseus, because that is the direction in which the Earth is moving in mid August. The effect and the picture are thus similar to what is seen on Page 156 the windscreen when driving by car while it is snowing. (© Brad Goldpaint at goldpaintphotography. com). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 159 Two photographs, taken about a second apart, showing a meteor break-up (© Robert Mikaelyan). Challenge 359 e three times? You have nine identically-looking spheres, all of the same mass, except one, which is heavier. Can you determine which one, using the balance only two times? ∗∗ For a physicist, antigravity is repulsive gravity – it does not exist in nature. Nevertheless, 6 motion due to gravitation 207 F I G U R E 160 Brooms fall more rapidly than stones (© Luca Gastaldi). Motion Mountain – The Adventure of Physics the term ‘antigravity’ is used incorrectly by many people, as a short search on the in- ternet shows. Some people call any effect that overcomes gravity, ‘antigravity’. However, this definition implies that tables and chairs are antigravity devices. Following the defin- ition, most of the wood, steel and concrete producers are in the antigravity business. The internet definition makes absolutely no sense. ∗∗ Challenge 360 s What is the cheapest way to switch gravity off for 25 seconds? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ Do all objects on Earth fall with the same acceleration of 9.8 m/s2 , assuming that air resistance can be neglected? No; every housekeeper knows that. You can check this by yourself. As shown in Figure 160, a broom angled at around 35° hits the floor before a Challenge 361 s stone, as the sounds of impact confirm. Are you able to explain why? ∗∗ Also bungee jumpers are accelerated more strongly than 𝑔. For a bungee cord of mass 𝑚 and a jumper of mass 𝑀, the maximum acceleration 𝑎 is 𝑚 𝑚 𝑎 = 𝑔 (1 + (4 + )) . (59) 8𝑀 𝑀 Challenge 362 s Can you deduce the relation from Figure 161? ∗∗ Challenge 363 s Guess: What is the mass of a ball of cork with a radius of 1 m? ∗∗ Challenge 364 s Guess: One thousand 1 mm diameter steel balls are collected. What is the mass? 208 6 motion due to gravitation M 1000 km M Motion Mountain – The Adventure of Physics F I G U R E 161 The starting situation F I G U R E 162 An honest balance? for a bungee jumper. ∗∗ How can you use your observations made during your travels with a bathroom scale to Challenge 365 s show that the Earth is not flat? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Both the Earth and the Moon attract bodies. The centre of mass of the Earth–Moon system is 4800 km away from the centre of the Earth, quite near its surface. Why do Challenge 366 s bodies on Earth still fall towards the centre of the Earth? ∗∗ Does every spherical body fall with the same acceleration? No. If the mass of the object is comparable to that of the Earth, the distance decreases in a different way. Can you Challenge 367 e confirm this statement? Figure 162 shows a related puzzle. What then is wrong about Page 202 Galileo’s argument about the constancy of acceleration of free fall? ∗∗ What is the fastest speed that a human can achieve making use of gravitational accele- ration? There are various methods that try this; a few are shown in Figure 163. Terminal speed of free falling skydivers can be even higher, but no reliable record speed value ex- ists. The last word is not spoken yet, as all these records will be surpassed in the coming years. It is important to require normal altitude; at stratospheric altitudes, speed values Vol. II, page 136 can be four times the speed values at low altitude. ∗∗ It is easy to put a mass of a kilogram onto a table. Twenty kilograms is harder. A thousand 6 motion due to gravitation 209 Motion Mountain – The Adventure of Physics F I G U R E 163 Reducing air resistance increases the terminal speed: left, the 2007 speed skiing world record holder Simone Origone with 69.83 m/s and right, the 2007 speed world record holder for bicycles on snow Éric Barone with 61.73 m/s (© Simone Origone, Éric Barone). Challenge 368 s is impossible. However, 6 ⋅ 1024 kg is easy. Why? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Page 199 The friction between the Earth and the Moon slows down the rotation of both. The Moon stopped rotating millions of years ago, and the Earth is on its way to doing so as well. When the Earth stops rotating, the Moon will stop moving away from Earth. How far will Challenge 369 ny the Moon be from the Earth at that time? Afterwards however, even further in the future, the Moon will move back towards the Earth, due to the friction between the Earth–Moon system and the Sun. Even though this effect would only take place if the Sun burned for Challenge 370 s ever, which is known to be false, can you explain it? ∗∗ When you run towards the east, you lose weight. There are two different reasons for this: the ‘centrifugal’ acceleration increases so that the force with which you are pulled down diminishes, and the Coriolis force appears, with a similar result. Can you estimate the Challenge 371 ny size of the two effects? ∗∗ Laboratories use two types of ultracentrifuges: preparative ultracentrifuges isolate vir- uses, organelles and biomolecules, whereas analytical ultracentrifuges measure the shape and mass of macromolecules. The fastest commercially available models achieve 200 000 rpm, or 3.3 kHz, and a centrifugal acceleration of 106 ⋅ 𝑔. ∗∗ 210 6 motion due to gravitation F I G U R E 164 The four satellites of Jupiter discovered by Galileo Motion Mountain – The Adventure of Physics and their motion (© Robin Scagell). What is the relation between the time a stone takes falling through a distance 𝑙 and the Challenge 372 s time a pendulum takes swinging though half a circle of radius 𝑙? (This problem is due to Galileo.) How many digits of the number π can one expect to determine in this way? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Why can a spacecraft accelerate through the slingshot effect when going round a planet, Challenge 373 s despite momentum conservation? It is speculated that the same effect is also the reason for the few exceptionally fast stars that are observed in the galaxy. For example, the star Ref. 162 HE0457-5439 moves with 720 km/s, which is much higher than the 100 to 200 km/s of most stars in the Milky Way. It seems that the role of the accelerating centre was taken by a black hole. ∗∗ Ref. 163 The orbit of a planet around the Sun has many interesting properties. What is the hodo- Challenge 374 s graph of the orbit? What is the hodograph for parabolic and hyperbolic orbits? ∗∗ The Galilean satellites of Jupiter, shown in Figure 164, can be seen with small ama- teur telescopes. Galileo discovered them in 1610 and called them the Medicean satel- lites. (Today, they are named, in order of increasing distance from Jupiter, as Io, Europa, Ganymede and Callisto.) They are almost mythical objects. They were the first bodies found that obviously did not orbit the Earth; thus Galileo used them to deduce that the Earth is not at the centre of the universe. The satellites have also been candidates to be the first standard clock, as their motion can be predicted to high accuracy, so that the ‘standard time’ could be read off from their position. Finally, due to this high accuracy, in 1676, the speed of light was first measured with their help, as told in the section on 6 motion due to gravitation 211 Earth Moon Earth Moon Sun Motion Mountain – The Adventure of Physics F I G U R E 165 Which of the two Moon paths is correct? Vol. II, page 16 special relativity. ∗∗ A simple, but difficult question: if all bodies attract each other, why don’t or didn’t all Challenge 375 s stars fall towards each other? Indeed, the inverse square expression of universal gravity has a limitation: it does not allow one to make sensible statements about the matter in the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net universe. Universal gravity predicts that a homogeneous mass distribution is unstable; indeed, an inhomogeneous distribution is observed. However, universal gravity does not predict the average mass density, the darkness at night, the observed speeds of the distant galaxies, etc. In fact, ‘universal’ gravity does not explain or predict a single property of Vol. II, page 211 the universe. To do this, we need general relativity. ∗∗ The acceleration 𝑔 due to gravity at a depth of 3000 km is 10.05 m/s2 , over 2 % more Ref. 164 than at the surface of the Earth. How is this possible? Also, on the Tibetan plateau, 𝑔 is influenced by the material below it. ∗∗ When the Moon circles the Sun, does its path have sections concave towards the Sun, as Challenge 376 s shown at the right of Figure 165, or not, as shown on the left? (Independent of this issue, both paths in the diagram disguise that the Moon path does not lie in the same plane as the path of the Earth around the Sun.) ∗∗ You can prove that objects attract each other (and that they are not only attracted by the Earth) with a simple experiment that anybody can perform at home, as described on the www.fourmilab.ch/gravitation/foobar website. 212 6 motion due to gravitation ∗∗ It is instructive to calculate the escape velocity from the Earth, i.e., that velocity with which a body must be thrown so that it never falls back. It turns out to be around 11 km/s. (This was called the second cosmic velocity in the past; the first cosmic velocity was the name given to the lowest speed for an orbit, 7.9 km/s.) The exact value of the escape Challenge 377 e velocity depends on the latitude of the thrower, and on the direction of the throw. (Why?) What is the escape velocity from the Solar System? (It was once called the third cosmic velocity.) By the way, the escape velocity from our galaxy is over 500 km/s. What would happen if a planet or a system were so heavy that the escape velocity from it would be Challenge 378 s larger than the speed of light? ∗∗ Challenge 379 s What is the largest asteroid one can escape from by jumping? ∗∗ For bodies of irregular shape, the centre of gravity of a body is not the same as the centre Motion Mountain – The Adventure of Physics Challenge 380 s of mass. Are you able to confirm this? (Hint: Find and use the simplest example possible.) ∗∗ Can gravity produce repulsion? What happens to a small test body on the inside of a Challenge 381 ny large C-shaped mass? Is it pushed towards the centre of mass? ∗∗ A heavily disputed argument for the equality of inertial and gravitational mass was given Ref. 165 by Chubykalo, and Vlaev. The total kinetic energy 𝑇 of two bodies circling around their copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net common centre of mass, like the Earth and the Moon, is given by 𝑇 = 𝐺𝑚𝑀/2𝑅, where the two quantities 𝑚 and 𝑀 are the gravitational masses of the two bodies and 𝑅 their distance. From this expression, in which the inertial masses do not appear on the right side, they deduce that the inertial and gravitational mass must be proportional to each Challenge 382 s other. Can you see how? Is the reasoning correct? ∗∗ Ref. 166 The shape of the Earth is not a sphere. As a consequence, a plumb line usually does not Challenge 383 ny point to the centre of the Earth. What is the largest deviation in degrees? ∗∗ Owing to the slightly flattened shape of the Earth, the source of the Mississippi is about 20 km nearer to the centre of the Earth than its mouth; the water effectively runs uphill. Challenge 384 s How can this be? ∗∗ If you look at the sky every day at 6 a.m., the Sun’s position varies during the year. The result of photographing the Sun on the same film is shown in Figure 166. The curve, called the analemma, is due to two combined effects: the inclination of the Earth’s axis and the elliptical shape of the Earth’s orbit around the Sun. The top and the (hidden) bot- tom points of the analemma correspond to the solstices. How does the analemma look 6 motion due to gravitation 213 Motion Mountain – The Adventure of Physics F I G U R E 166 The analemma over Delphi, taken between January and December 2002 (© Anthony Ayiomamitis). Challenge 385 s if photographed every day at local noon? Why is it not a straight line pointing exactly south? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ The constellation in which the Sun stands at noon (at the centre of the time zone) is sup- posedly called the ‘zodiacal sign’ of that day. Astrologers say there are twelve of them, namely Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Sagittarius, Capri- cornus, Aquarius and Pisces and that each takes (quite precisely) a twelfth of a year or a twelfth of the ecliptic. Any check with a calendar shows that at present, the midday Sun is never in the zodiacal sign during the days usually connected to it. The relation has Page 152 shifted by about a month since it was defined, due to the precession of the Earth’s axis. A check with a map of the star sky shows that the twelve constellations do not have the same length and that on the ecliptic there are fourteen of them, not twelve. There is Ophiuchus or Serpentarius, the serpent bearer constellation, between Scorpius and Sagittarius, and Cetus, the whale, between Aquarius and Pisces. In fact, not a single astronomical state- Ref. 167 ment about zodiacal signs is correct. To put it clearly, astrology, in contrast to its name, is not about stars. (In German, the word ‘Strolch’, meaning ‘rogue’ or ‘scoundrel’, is derived from the word ‘astrologer’.) ∗∗ For a long time, it was thought that there is no additional planet in our Solar System out- Ref. 168 side Neptune and Pluto, because their orbits show no disturbances from another body. Today, the view has changed. It is known that there are only eight planets: Pluto is not 214 6 motion due to gravitation Sedna Jupiter Kuiper Belt Mars Earth Venus Uranus Mercury Saturn Jupiter Asteroids Pluto Inner Outer Solar System Solar System Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Inner extent of Oort Cloud Orbit of Sedna F I G U R E 167 The orbit of Sedna in comparison with the orbits of the planets in the Solar System (NASA). a planet, but the first of a set of smaller objects in the so-called Kuiper belt. Kuiper belt objects are regularly discovered; over 1000 are known today. In 2003, two major Kuiper objects were discovered; one, called Sedna, is almost as Ref. 169 large as Pluto, the other, called Eris, is even larger than Pluto and has a moon. Both have strongly elliptical orbits (see Figure 167). Since Pluto and Eris, like the asteroid Ceres, have cleaned their orbit from debris, these three objects are now classified as dwarf planets. Outside the Kuiper belt, the Solar System is surrounded by the so-called Oort cloud. In contrast to the flattened Kuiper belt, the Oort cloud is spherical in shape and has a radius of up to 50 000 AU, as shown in Figure 167 and Figure 168. The Oort cloud consists of a huge number of icy objects consisting of mainly of water, and to a lesser degree, of methane and ammonia. Objects from the Oort cloud that enter the inner Solar System become comets; in the distant past, such objects have brought water onto the Earth. 6 motion due to gravitation 215 Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 168 The Kuiper belt, containing mainly planetoids, and the Oort cloud orbit, containing comets, around the Solar System (NASA, JPL, Donald Yeoman). ∗∗ In astronomy new examples of motion are regularly discovered even in the present cen- tury. Sometimes there are also false alarms. One example was the alleged fall of mini comets on the Earth. They were supposedly made of a few dozen kilograms of ice, hitting Ref. 170 the Earth every few seconds. It is now known not to happen. ∗∗ Universal gravity allows only elliptical, parabolic or hyperbolic orbits. It is impossible for a small object approaching a large one to be captured. At least, that is what we have learned so far. Nevertheless, all astronomy books tell stories of capture in our Solar Sys- tem; for example, several outer satellites of Saturn have been captured. How is this pos- Challenge 386 s sible? ∗∗ How would a tunnel have to be shaped in order that a stone would fall through it without touching the walls? (Assume constant density.) If the Earth did not rotate, the tunnel 216 6 motion due to gravitation would be a straight line through its centre, and the stone would fall down and up again, in an oscillating motion. For a rotating Earth, the problem is much more difficult. What Challenge 387 s is the shape when the tunnel starts at the Equator? ∗∗ The International Space Station circles the Earth every 90 minutes at an altitude of about 380 km. You can see where it is from the website www.heavens-above.com. By the way, whenever it is just above the horizon, the station is the third brightest object in the night Challenge 388 e sky, superseded only by the Moon and Venus. Have a look at it. ∗∗ Is it true that the centre of mass of the Solar System, its barycentre, is always inside the Challenge 389 s Sun? Even though the Sun or a star move very little when planets move around them, this motion can be detected with precision measurements making use of the Doppler Vol. II, page 31 effect for light or radio waves. Jupiter, for example, produces a speed change of 13 m/s in the Sun, the Earth 1 m/s. The first planets outside the Solar System, around the pulsar Motion Mountain – The Adventure of Physics PSR1257+12 and around the normal G-type star Pegasi 51, were discovered in this way, in 1992 and 1995. In the meantime, several thousand so-called exoplanets have been discovered with this and other methods. Some have even masses comparable to that of the Earth. This research also showed that exoplanets are more numerous than stars, and that earth-like planets are rare. ∗∗ Not all points on the Earth receive the same number of daylight hours during a year. The Challenge 390 d effects are difficult to spot, though. Can you find one? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ Can the phase of the Moon have a measurable effect on the human body, for example Challenge 391 s through tidal effects? ∗∗ There is an important difference between the heliocentric system and the old idea that all planets turn around the Earth. The heliocentric system states that certain planets, such as Mercury and Venus, can be between the Earth and the Sun at certain times, and behind the Sun at other times. In contrast, the geocentric system states that they are al- ways in between. Why did such an important difference not immediately invalidate the geocentric system? And how did the observation of phases, shown in Figure 169 and Challenge 392 s Figure 170, invalidate the geocentric system? ∗∗ The strangest reformulation of the description of motion given by 𝑚𝑎 = ∇𝑈 is the almost Ref. 171 absurd looking equation ∇𝑣 = d𝑣/d𝑠 (60) where 𝑠 is the motion path length. It is called the ray form of the equation of motion. Can Challenge 393 s you find an example of its application? 6 motion due to gravitation 217 Motion Mountain – The Adventure of Physics F I G U R E 169 The phases of the Moon and of Venus, as observed from Athens in summer 2007 (© Anthony Ayiomamitis). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 170 Universal gravitation also explains the observations of Venus, the evening and morning star. In particular, universal gravitation, and the elliptical orbits it implies, explains its phases and its change of angular size. The pictures shown here were taken in 2004 and 2005. The observations can easily be made with a binocular or a small telescope (© Wah!; ﬁlm available at apod.nasa.gov/apod/ ap060110.html). ∗∗ Seen from Neptune, the size of the Sun is the same as that of Jupiter seen from the Earth Challenge 394 s at the time of its closest approach. True? ∗∗ Ref. 172 The gravitational acceleration for a particle inside a spherical shell is zero. The vanishing of gravity in this case is independent of the particle shape and its position, and independ- Challenge 395 s ent of the thickness of the shell. Can you find the argument using Figure 171? This works only because of the 1/𝑟2 dependence of gravity. Can you show that the result does not 218 6 motion due to gravitation d𝑚 𝑟 𝑚 𝑅 d𝑀 F I G U R E 171 The vanishing of gravitational force inside a spherical shell of matter. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 172 Le Sage’s own illustration of his model, showing the smaller density of ‘ultramondane corpuscules’ between the attracting bodies and the higher density outside them (© Wikimedia) hold for non-spherical shells? Note that the vanishing of gravity inside a spherical shell usually does not hold if other matter is found outside the shell. How could one eliminate Challenge 396 s the effects of outside matter? ∗∗ What is gravity? This simple question has a long history. In 1690, Nicolas Fatio de Duillier Ref. 173 and in 1747, Georges-Louis Le Sage proposed an explanation for the 1/𝑟2 dependence. Le Sage argued that the world is full of small particles – he called them ‘corpuscules ultra- mondains’ – flying around randomly and hitting all objects. Single objects do not feel the hits, since they are hit continuously and randomly from all directions. But when two objects are near to each other, they produce shadows for part of the flux to the other body, resulting in an attraction, as shown in Figure 172. Can you show that such an attraction Challenge 397 e has a 1/𝑟2 dependence? However, Le Sage’s proposal has a number of problems. First, the argument only Challenge 398 e works if the collisions are inelastic. (Why?) However, that would mean that all bodies 6 motion due to gravitation 219 Ref. 2 would heat up with time, as Jean-Marc Lévy-Leblond explains. Secondly, a moving body in free space would be hit by more or faster particles in the front than in the back; as a result, the body should be decelerated. Finally, gravity would depend on size, but in a strange way. In particular, three bodies lying on a line should not produce shadows, as no such shadows are observed; but the naive model predicts such shadows. Despite all criticisms, the idea that gravity is due to particles has regularly resurfaced in physics research ever since. In the most recent version, the hypothetical particles are called gravitons. On the other hand, no such particles have ever been observed. We will understand the origin of gravitation in the final part of our mountain ascent. ∗∗ Challenge 399 ny For which bodies does gravity decrease as you approach them? ∗∗ Could one put a satellite into orbit using a cannon? Does the answer depend on the Challenge 400 s direction in which one shoots? Motion Mountain – The Adventure of Physics ∗∗ Two old computer users share experiences. ‘I threw my Pentium III and Pentium IV out of the window.’ ‘And?’ ‘The Pentium III was faster.’ ∗∗ Challenge 401 s How often does the Earth rise and fall when seen from the Moon? Does the Earth show phases? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 402 ny What is the weight of the Moon? How does it compare with the weight of the Alps? ∗∗ If a star is made of high-density material, the speed of a planet orbiting near to it could Challenge 403 s be greater than the speed of light. How does nature avoid this strange possibility? ∗∗ What will happen to the Solar System in the future? This question is surprisingly hard to answer. The main expert of this topic, French planetary scientist Jacques Laskar, simu- Ref. 174 lated a few hundred million years of evolution using computer-aided calculus. He found Page 424 that the planetary orbits are stable, but that there is clear evidence of chaos in the evolu- tion of the Solar System, at a small level. The various planets influence each other in subtle and still poorly understood ways. Effects in the past are also being studied, such as the energy change of Jupiter due to its ejection of smaller asteroids from the Solar System, or the energy gains of Neptune. There is still a lot of research to be done in this field. ∗∗ One of the open problems of the Solar System is the description of planet distances dis- covered in 1766 by Johann Daniel Titius (1729–1796) and publicized by Johann Elert 220 6 motion due to gravitation TA B L E 27 An unexplained property of nature: planet distances from the Sun and the values resulting from the Titius–Bode rule. Planet 𝑛 predicted measured d i s ta nc e i n AU Mercury −∞ 0.4 0.4 Venus 0 0.7 0.7 Earth 1 1.0 1.0 Mars 2 1.6 1.5 Planetoids 3 2.8 2.2 to 3.2 Jupiter 4 5.2 5.2 Saturn 5 10.0 9.5 Uranus 6 19.6 19.2 Neptune 7 38.8 30.1 Pluto 8 77.2 39.5 Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 173 The motion of the planetoids compared to that of the planets (Shockwave animation © Hans-Christian Greier) Bode (1747–1826). Titius discovered that planetary distances 𝑑 from the Sun can be ap- proximated by 𝑑 = 𝑎 + 2𝑛 𝑏 with 𝑎 = 0.4 AU , 𝑏 = 0.3 AU (61) where distances are measured in astronomical units and 𝑛 is the number of the planet. The resulting approximation is compared with observations in Table 27. Interestingly, the last three planets, as well as the planetoids, were discovered after Bode’s and Titius’ deaths; the rule had successfully predicted Uranus’ distance, as well as that of the planetoids. Despite these successes – and the failure for the last two planets – nobody has yet found a model for the formation of the planets that explains Titius’ rule. The large satellites of Jupiter and of Uranus have regular spacing, but not according 6 motion due to gravitation 221 TA B L E 28 The orbital periods known to the Babylonians. B ody Period Saturn 29 a Jupiter 12 a Mars 687 d Sun 365 d Venus 224 d Mercury 88 d Moon 29 d to the Titius–Bode rule. Explaining or disproving the rule is one of the challenges that remain in classical Motion Mountain – The Adventure of Physics Ref. 175 mechanics. Some researchers maintain that the rule is a consequence of scale invari- Ref. 176 ance, others maintain that it is an accident or even a red herring. The last interpretation is also suggested by the non-Titius–Bode behaviour of practically all extrasolar planets. The issue is not closed. ∗∗ Around 3000 years ago, the Babylonians had measured the orbital times of the seven celestial bodies that move across the sky. Ordered from longest to shortest, they wrote them down in Table 28. Six of the celestial bodies are visible in the beautiful Figure 174. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net The Babylonians also introduced the week and the division of the day into 24 hours. They dedicated every one of the 168 hours of the week to a celestial body, following the order of Table 28. They also dedicated the whole day to that celestial body that corres- ponds to the first hour of that day. The first day of the week was dedicated to Saturn; Challenge 404 e the present ordering of the other days of the week then follows from Table 28. This story Ref. 177 was told by Cassius Dio (c. 160 to c. 230). Towards the end of Antiquity, the ordering was taken up by the Roman empire. In Germanic languages, including English, the Latin names of the celestial bodies were replaced by the corresponding Germanic gods. The order Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday and Friday is thus a consequence of both the astronomical measurements and the astrological superstitions of the ancients. ∗∗ In 1722, the great mathematician Leonhard Euler made a mistake in his calculation that led him to conclude that if a tunnel, or better, a deep hole were built from one pole of the Earth to the other, a stone falling into it would arrive at the Earth’s centre and then immediately turn and go back up. Voltaire made fun of this conclusion for many years. Can you correct Euler and show that the real motion is an oscillation from one pole to the other, and can you calculate the time a pole-to-pole fall would take (assuming Challenge 405 s homogeneous density)? What would be the oscillation time for an arbitrary straight surface-to-surface tunnel 222 6 motion due to gravitation Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 174 These are the six celestial bodies that are visible at night with the naked eye and whose positions vary over the course of the year. The nearly vertical line connecting them is the ecliptic, the narrow stripe around it the zodiac. Together with the Sun, the seven celestial bodies were used to name the days of the week. (© Alex Cherney) 6 motion due to gravitation 223 F I G U R E 175 The solar eclipse of 11 August 1999, photographed by Jean-Pierre Haigneré, member of the Mir 27 crew, and the (enhanced) solar eclipse of 29 March 2006 (© CNES and Laurent Laveder/PixHeaven.net). Motion Mountain – The Adventure of Physics Earth wire copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 176 A wire attached to the Earth’s Equator. Challenge 406 s of length 𝑙, thus not going from pole to pole? The previous challenges circumvented the effects of the Earth’s rotation. The topic Ref. 178 becomes much more interesting if rotation is included. What would be the shape of a Challenge 407 s tunnel so that a stone falling through it never touches the wall? ∗∗ Figure 175 shows a photograph of a solar eclipse taken from the Russian space station Mir and a photograph taken at the centre of the shadow from the Earth. Indeed, a global view of a phenomenon can be quite different from a local one. What is the speed of the Challenge 408 s shadow? ∗∗ In 2005, satellite measurements have shown that the water in the Amazon river presses down the land up to 75 mm more in the season when it is full of water than in the season Ref. 179 when it is almost empty. 224 6 motion due to gravitation ∗∗ Imagine that a wire existed that does not break. How long would such a wire have to be so that, when attached to the Equator, it would stand upright in the air, as shown in Challenge 409 s Figure 176? Could one build an elevator into space in this way? ∗∗ Usually there are roughly two tides per day. But there are places, such as on the coast of Vietnam, where there is only one tide per day. See www.jason.oceanobs.com/html/ Challenge 410 ny applications/marees/marees_m2k1_fr.html. Why? ∗∗ It is sufficient to use the concept of centrifugal force to show that the rings of Saturn cannot be made of massive material, but must be made of separate pieces. Can you find Challenge 411 s out how? ∗∗ Motion Mountain – The Adventure of Physics Why did Mars lose its atmosphere? Nobody knows. It has recently been shown that the solar wind is too weak for this to happen. This is one of the many open riddles of the solar system. ∗∗ All bodies in the Solar System orbit the Sun in the same direction. All? No; there are ex- ceptions. One intriguing asteroid that orbits the Sun near Jupiter in the wrong direction was discovered in 2015: it has a size of 3 km. For an animation of its astonishing orbit, opposite to all Trojan asteroids, see www.astro.uwo.ca/~wiegert. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ The observed motion due to gravity can be shown to be the simplest possible, in the fol- lowing sense. If we measure change of a falling object with the expression ∫ 𝑚𝑣2 /2 − Page 253 𝑚𝑔ℎ d𝑡, then a constant acceleration due to gravity minimizes the change in every ex- Challenge 412 e ample of fall. Can you confirm this? ∗∗ Motion due to gravity is fun: think about roller coasters. If you want to know more about how they are built, visit www.vekoma.com. ∗∗ Gravity and air friction lead to interesting effects. What is the shape of a rope hanging from a helicopter when the helicopter is flying? What is the shape when there is a weight at the end of the rope? A small parachute? (The internet has the answer.) “ The scientific theory I like best is that the rings ” of Saturn are made of lost airline luggage. Mark Russel 6 motion due to gravitation 225 Summary on gravitation Spherical bodies of mass 𝑀 attract other bodies at a distance 𝑟 by inducing an accele- ration towards them given by 𝑎 = 𝐺𝑀/𝑟2 . This expression, universal gravitation, de- scribes snowboarders, skiers, paragliders, athletes, couch potatoes, pendula, stones, can- ons, rockets, tides, eclipses, planet shapes, planet motion and much more. Universal grav- itation is the first example of a unified description: it describes how everything falls. By the acceleration it produces, gravitation limits the appearance of uniform motion in nature. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Chapter 7 C L A S SIC A L M E C HA N IC S , F ORC E A N D T H E PR E DIC TA BI L I T Y OF MOT ION A ll those types of motion in which the only permanent property of body is mass define the field of mechanics. The same name is given lso to the experts studying the field. We can think of mechanics as the ath- letic part of physics.** Both in athletics and in mechanics only lengths, times and masses are measured – and of interest at all. Motion Mountain – The Adventure of Physics More specifically, our topic of investigation so far is called classical mechanics, to dis- tinguish it from quantum mechanics. The main difference is that in classical physics ar- bitrary small values are assumed to exist, whereas this is not the case in quantum physics. Classical mechanics is often also called Galilean physics or Newtonian physics.*** Classical mechanics states that motion is predictable: it thus states that there are no surprises in motion. Is this correct in all cases? Is predictability valid in the presence of friction? Of free will? Are there really no surprises in nature? These issues merit a discussion; they will accompany us for a stretch of our adventure. We know that there is more to the world than gravity. Simple observations make this point: floors and friction. Neither can be due to gravity. Floors do not fall, and thus are copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net not described by gravity; and friction is not observed in the skies, where motion is purely due to gravity.**** Also on Earth, friction is unrelated to gravity, as you might want to Challenge 413 e check yourself. There must be another interaction responsible for friction. We shall study it in the third volume. But a few issues merit a discussion right away. ** This is in contrast to the actual origin of the term ‘mechanics’, which means ‘machine science’. It derives from the Greek μηκανή, which means ‘machine’ and even lies at the origin of the English word ‘machine’ itself. Sometimes the term ‘mechanics’ is used for the study of the motion of solid bodies only, excluding, e.g., hydrodynamics. This use fell out of favour in physics in the twentieth century. *** The basis of classical mechanics, the description of motion using only space and time, is called kinemat- ics. An example is the description of free fall by 𝑧(𝑡) = 𝑧0 + 𝑣0 (𝑡 − 𝑡0 ) − 12 𝑔(𝑡 − 𝑡0 )2 . The other, main part of classical mechanics is the description of motion as a consequence of interactions between bodies; it is called dynamics. An example of dynamics is the formula of universal gravity. The distinction between kinematics and dynamics can also be made in relativity, thermodynamics and electrodynamics. **** This is not completely correct: in the 1980s, the first case of gravitational friction was discovered: the Vol. II, page 174 emission of gravity waves. We discuss it in detail in the chapter on general relativity. The discovery does not change the main point, however. 7 classical mechanics, force and the predictability of motion 227 Motion Mountain – The Adventure of Physics F I G U R E 177 The parabola shapes formed by accelerated water beams show that motion in everyday life is predictable (© Oase GmbH). TA B L E 29 Some force values in nature. O b s e r va t i o n Force Value measured in a magnetic resonance force microscope 820 zN Force needed to rip a DNA molecule apart by pulling at its two ends 600 pN copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Maximum force exerted by human bite 2.1 kN Force exerted by quadriceps up to 3 kN Typical peak force exerted by sledgehammer 5 kN Force sustained by 1 cm2 of a good adhesive up to 10 kN Force needed to tear a good rope used in rock climbing 30 kN Maximum force measurable in nature 3.0 ⋅ 1043 N Should one use force? Power? “ The direct use of physical force is so poor a solution [...] that it is commonly employed only ” by small children and great nations. David Friedman Everybody has to take a stand on this question, even students of physics. Indeed, many types of forces are used and observed in daily life. One speaks of muscular, gravitational, psychic, sexual, satanic, supernatural, social, political, economic and many others. Physi- cists see things in a simpler way. They call the different types of forces observed between objects interactions. The study of the details of all these interactions will show that, in everyday life, they are of electrical or gravitational origin. For physicists, all change is due to motion. The term force then also takes on a more 228 7 classical mechanics, force and the predictability of motion restrictive definition. (Physical) force is defined as the change of momentum with time, i.e., as d𝑝 𝐹= . (62) d𝑡 A few measured values are listed in Figure 29. Ref. 87 A horse is running so fast that the hooves touch the ground only 20 % of the time. Challenge 414 s What is the load carried by its legs during contact? Force is the change of momentum. Since momentum is conserved, we can also say that force measures the flow of momentum. As we will see in detail shortly, whenever a force accelerates a body, momentum flows into it. Indeed, momentum can be imagined to Ref. 87 be some invisible and intangible substance. Force measures how much of this substance flows into or out of a body per unit time. ⊳ Force is momentum flow. Motion Mountain – The Adventure of Physics The conservation of momentum is due to the conservation of this liquid. Like any liquid, momentum flows through a surface. Using the Galilean definition of linear momentum 𝑝 = 𝑚𝑣, we can rewrite the defin- ition of force (for constant mass) as 𝐹 = 𝑚𝑎 , (63) where 𝐹 = 𝐹(𝑡, 𝑥) is the force acting on an object of mass 𝑚 and where 𝑎 = 𝑎(𝑡, 𝑥) = d𝑣/d𝑡 = d2 𝑥/d𝑡2 is the acceleration of the same object, that is to say, its change of ve- locity.* The expression states in precise terms that force is what changes the velocity of copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net masses. The quantity is called ‘force’ because it corresponds in many, but not all aspects to everyday muscular force. For example, the more force is used, the further a stone can be thrown. Equivalently, the more momentum is pumped into a stone, the further it can be thrown. As another example, the concept of weight describes the flow of momentum due to gravity. ⊳ Gravitation constantly pumps momentum into massive bodies. Sand in an hourglass is running, and the hourglass is on a scale. Is the weight shown Challenge 415 s on the scale larger, smaller or equal to the weight when the sand has stopped falling? Forces are measured with the help of deformations of bodies. Everyday force values can be measured by measuring the extension of a spring. Small force values, of the order of 1 nN, can be detected by measuring the deflection of small levers with the help of a reflected laser beam. * This equation was first written down by the mathematician and physicist Leonhard Euler (b. 1707 Basel, d. 1783 St. Petersburg) in 1747, 20 years after the death of Newton, to whom it is usually and falsely ascribed. It was Euler, one of the greatest mathematicians of all time (and not Newton), who first understood that this definition of force is useful in every case of motion, whatever the appearance, be it for point particles or Ref. 28 extended objects, and be they rigid, deformable or fluid bodies. Surprisingly and in contrast to frequently- Vol. II, page 83 made statements, equation (63) is even correct in relativity. 7 classical mechanics, force and the predictability of motion 229 +p Flow of +p Motion Mountain – The Adventure of Physics F I G U R E 178 The pulling child pumps momentum into the chariot. In fact, some momentum ﬂows back to the ground due to dynamic friction (not drawn). However, whenever the concept of force is used, it should be remembered that physical force is different from everyday force or everyday effort. Effort is probably best approxim- ated by the concept of (physical) power, usually abbreviated 𝑃, and defined (for constant force) as d𝑊 𝑃= = 𝐹𝑣 (64) d𝑡 copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net in which (physical) work 𝑊 is defined as 𝑊 = 𝐹𝑠, where 𝑠 is the distance along which the force acts. Physical work is a form of energy, as you might want to check. Work, as a form of energy, has to be taken into account when the conservation of energy is checked. With the definition of work just given you can solve the following puzzles. What hap- Challenge 416 s pens to the electricity consumption of an escalator if you walk on it instead of standing Challenge 417 d still? What is the effect of the definition of power for the salary of scientists? A man who Challenge 418 s walks carrying a heavy rucksack is hardly doing any work; why then does he get tired? When students in exams say that the force acting on a thrown stone is least at the Ref. 180 highest point of the trajectory, it is customary to say that they are using an incorrect view, namely the so-called Aristotelian view, in which force is proportional to velocity. Sometimes it is even said that they are using a different concept of state of motion. Critics then add, with a tone of superiority, how wrong all this is. This is an example of intellec- tual disinformation. Every student knows from riding a bicycle, from throwing a stone or from pulling an object that increased effort results in increased speed. The student is right; those theoreticians who deduce that the student has a mistaken concept of force are wrong. In fact, instead of the physical concept of force, the student is just using the everyday version, namely effort. Indeed, the effort exerted by gravity on a flying stone is least at the highest point of the trajectory. Understanding the difference between physical force and everyday effort is the main hurdle in learning mechanics.* * This stepping stone is so high that many professional physicists do not really take it themselves; this is 230 7 classical mechanics, force and the predictability of motion Description with Description with forces at one single momentum flow point Alternatively: +p -p Motion Mountain – The Adventure of Physics Flow of -p Flow of +p copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net +p Flow of +p F I G U R E 179 The two equivalent descriptions of situations with zero net force, i.e., with a closed momentum ﬂow. Compression occurs when momentum ﬂow and momentum point in the same direction; extension occurs when momentum ﬂow and momentum point in opposite directions. confirmed by the innumerable comments in papers that state that physical force is defined using mass, and, at the same time, that mass is defined using force (the latter part of the sentence being a fundamental mistake). 7 classical mechanics, force and the predictability of motion 231 Often the flow of momentum, equation (62), is not recognized as the definition of force. This is mainly due to an everyday observation: there seem to be forces without any associated acceleration or change in momentum, such as in a string under tension or in water at high pressure. When one pushes against a tree, as shown in Figure 179, there is no motion, yet a force is applied. If force is momentum flow, where does the momentum go? It flows into the slight deformations of the arms and the tree. In fact, when one starts pushing and thus deforming, the associated momentum change of the molecules, the atoms, or the electrons of the two bodies can be observed. After the de- formation is established a continuous and equal flow of momentum is going on in both directions. Ref. 87 Because force is net momentum flow, the concept of force is not really needed in the description of motion. But sometimes the concept is practical. This is the case in everyday life, where it is useful in situations where net momentum values are small or negligible. For example, it is useful to define pressure as force per area, even though it is actually a momentum flow per area. At the microscopic level, momentum alone suffices for the description of motion. Motion Mountain – The Adventure of Physics In the section title we asked about the usefulness of force and power. Before we can answer conclusively, we need more arguments. Through its definition, the concepts of force and power are distinguished clearly from ‘mass’, ‘momentum’, ‘energy’ and from each other. But where do forces originate? In other words, which effects in nature have the capacity to accelerate bodies by pumping momentum into objects? Table 30 gives an overview. Forces, surfaces and conservation We saw that force is the change of momentum. We also saw that momentum is conserved. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net How do these statements come together? The answer is the same for all conserved quant- ities. We imagine a closed surface that is the boundary of a volume in space. Conserva- tion implies that the conserved quantity enclosed inside the surface can only change by flowing through that surface.* All conserved quantities in nature – such as energy, linear momentum, electric charge, and angular momentum – can only change by flowing through surfaces. In particular, whenever the momentum of a body changes, this happens through a surface. Momentum change is due to momentum flow. In other words, the concept of force always implies a surface through which momentum flows. * Mathematically, the conservation of a quantity 𝑞 is expressed with the help of the volume density 𝜌 = 𝑞/𝑉, the current 𝐼 = 𝑞/𝑡, and the flow or flux 𝑗 = 𝜌𝑣, so that 𝑗 = 𝑞/𝐴𝑡. Conservation then implies d𝑞 ∂𝜌 =∫ d𝑉 = − ∫ 𝑗d𝐴 = −𝐼 (65) d𝑡 𝑉 ∂𝑡 𝐴=∂𝑉 or, equivalently, ∂𝜌 + ∇𝑗 = 0 . (66) ∂𝑡 This is the continuity equation for the quantity 𝑞. All this only states that a conserved quantity in a closed volume 𝑉 can only change by flowing through the surface 𝐴. This is a typical example of how complex mathematical expressions can obfuscate the simple physical content. 232 7 classical mechanics, force and the predictability of motion Motion Mountain – The Adventure of Physics F I G U R E 180 Friction-based processes (courtesy Wikimedia). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ⊳ Force is the flow of momentum through a surface. Ref. 299 This point is essential in understanding physical force. Every force requires a surface for its definition. To refine your own concept of force, you can search for the relevant surface when a Challenge 419 e rope pulls a chariot, or when an arm pushes a tree, or when a car accelerates. It is also helpful to compare the definition of force with the definition of power: both quantities are flows through surfaces. As a result, we can say: ⊳ A motor is a momentum pump. Friction and motion Every example of motion, from the motion that lets us choose the direction of our gaze to the motion that carries a butterfly through the landscape, can be put into one of the two left-most columns of Table 30. Physically, those two columns are separated by the following criterion: in the first class, the acceleration of a body can be in a different direc- tion from its velocity. The second class of examples produces only accelerations that are exactly opposed to the velocity of the moving body, as seen from the frame of reference 7 classical mechanics, force and the predictability of motion 233 TA B L E 30 Selected processes and devices changing the motion of bodies. S i t uat i o n s t h at c a n S i t uat i o n s t h at Motors and l e a d t o ac ce l e r at i on o nly l e a d t o a c t uat o r s d e c e l e r at i o n piezoelectricity quartz under applied voltage thermoluminescence walking piezo tripod collisions satellite in planet encounter car crash rocket motor growth of mountains meteorite crash swimming of larvae magnetic effects compass needle near magnet electromagnetic braking electromagnetic gun magnetostriction transformer losses linear motor current in wire near magnet electric heating galvanometer electric effects Motion Mountain – The Adventure of Physics rubbed comb near hair friction between solids electrostatic motor bombs fire muscles, sperm flagella cathode ray tube electron microscope Brownian motor light levitating objects by light light bath stopping atoms (true) light mill solar sail for satellites light pressure inside stars solar cell elasticity bow and arrow trouser suspenders ultrasound motor copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net bent trees standing up again pillow, air bag bimorphs osmosis water rising in trees salt conservation of food osmotic pendulum electro-osmosis tunable X-ray screening heat & pressure freezing champagne bottle surfboard water resistance hydraulic engines tea kettle quicksand steam engine barometer parachute air gun, sail earthquakes sliding resistance seismometer attraction of passing trains shock absorbers water turbine nuclei radioactivity plunging into the Sun supernova explosion biology bamboo growth decreasing blood vessel molecular motors diameter gravitation falling emission of gravity waves pulley 234 7 classical mechanics, force and the predictability of motion of the braking medium. Such a resisting force is called friction, drag or a damping. All Challenge 420 e examples in the second class are types of friction. Just check. Some examples of processes based on friction are given in Figure 180. Challenge 421 s Here is a puzzle on cycling: does side wind brake – and why? Friction can be so strong that all motion of a body against its environment is made impossible. This type of friction, called static friction or sticking friction, is common and important: without it, turning the wheels of bicycles, trains or cars would have no effect. Without static friction, wheels driven by a motor would have no grip. Similarly, not a single screw would stay tightened and no hair clip would work. We could neither run nor walk in a forest, as the soil would be more slippery than polished ice. In fact not only our own motion, but all voluntary motion of living beings is based on friction. The same is the case for all self-moving machines. Without static friction, the propellers in ships, aeroplanes and helicopters would not have any effect and the wings of aeroplanes would Challenge 422 s produce no lift to keep them in the air. (Why?) In short, static friction is necessary whenever we or an engine want to move against the environment. Motion Mountain – The Adventure of Physics Friction, sport, machines and predictabilit y Once an object moves through its environment, it is hindered by another type of friction; it is called dynamic friction and acts between all bodies in relative motion.* Without Ref. 181 dynamic friction, falling bodies would always rebound to the same height, without ever coming to a stop; neither parachutes nor brakes would work; and even worse, we would have no memory, as we will see later. All motion examples in the second column of Table 30 include friction. In these ex- amples, macroscopic energy is not conserved: the systems are dissipative. In the first copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net column, macroscopic energy is constant: the systems are conservative. The first two columns can also be distinguished using a more abstract, mathematical criterion: on the left are accelerations that can be derived from a potential, on the right, decelerations that can not. As in the case of gravitation, the description of any kind of motion is much simplified by the use of a potential: at every position in space, one needs only the single value of the potential to calculate the trajectory of an object, instead of the three values of the acceleration or the force. Moreover, the magnitude of the velocity of an object at any point can be calculated directly from energy conservation. The processes from the second column cannot be described by a potential. These are the cases where it is best to use force if we want to describe the motion of the system. For example, the friction or drag force 𝐹 due to wind resistance of a body is roughly given by 1 𝐹 = 𝑐w 𝜌𝐴𝑣2 (67) 2 where 𝐴 is the area of its cross-section and 𝑣 its velocity relative to the air, 𝜌 is the density of air. The drag coefficient 𝑐w is a pure number that depends on the shape of the moving * There might be one exception. Recent research suggest that maybe in certain crystalline systems, such as tungsten bodies on silicon, under ideal conditions gliding friction can be extremely small and possibly even Ref. 182 vanish in certain directions of motion. This so-called superlubrication is presently a topic of research. 7 classical mechanics, force and the predictability of motion 235 typical passenger aeroplane cw = 0.03 typical sports car or van cw = 0.44 modern sedan cw = 0.28 dolphin and penguin cw = 0.035 soccer ball turbulent (above c.10 m/s) cw = 0.2 laminar (below c.10 m/s) cw = 0.45 Motion Mountain – The Adventure of Physics F I G U R E 181 Shapes and air/water resistance. object. A few examples are given in Figure 181. The formula is valid for all fluids, not only for air, below the speed of sound, as long as the drag is due to turbulence. This is usually the case in air and in water. (At very low velocities, when the fluid motion is not Page 361 turbulent but laminar, drag is called viscous and follows an (almost) linear relation with speed.) You may check that drag, or aerodynamic resistance cannot be derived from a Challenge 423 e potential.* The drag coefficient 𝑐w is a measured quantity. Calculating drag coefficients with com- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net puters, given the shape of the body and the properties of the fluid, is one of the most difficult tasks of science; the problem is still not solved. An aerodynamic car has a value between 0.25 and 0.3; many sports cars share with vans values of 0.44 and higher, and ra- cing car values can be as high as 1, depending on the amount of force that is used to keep the car fastened to the ground. The lowest known values are for dolphins and penguins.** Ref. 184 Wind resistance is also of importance to humans, in particular in athletics. It is estim- ated that 100 m sprinters spend between 3 % and 6 % of their power overcoming drag. This leads to varying sprint times 𝑡w when wind of speed 𝑤 is involved, related by the expression 𝑡0 𝑤𝑡w 2 = 1.03 − 0.03 (1 − ) , (68) 𝑡w 100 m * Such a statement about friction is correct only in three dimensions, as is the case in nature; in the case of Challenge 424 s a single dimension, a potential can always be found. ** It is unclear whether there is, in nature, a smallest possible value for the drag coefficient. The topic of aerodynamic shapes is also interesting for fluid bodies. They are kept together by surface tension. For example, surface tension keeps the wet hairs of a soaked brush together. Surface tension also determines the shape of rain drops. Experiments show that their shape is spherical for drops smaller than Vol. V, page 308 2 mm diameter, and that larger rain drops are lens shaped, with the flat part towards the bottom. The usual Ref. 183 tear shape is not encountered in nature; something vaguely similar to it appears during drop detachment, but never during drop fall. 236 7 classical mechanics, force and the predictability of motion where the more conservative estimate of 3 % is used. An opposing wind speed of −2 m/s gives an increase in time of 0.13 s, enough to change a potential world record into an ‘only’ excellent result. (Are you able to deduce the 𝑐w value for running humans from Challenge 425 ny the formula?) Likewise, parachuting exists due to wind resistance. Can you determine how the speed of a falling body, with or without a parachute, changes with time, assuming constant Challenge 426 s shape and drag coefficient? In contrast, static friction has different properties. It is proportional to the force press- Ref. 185 ing the two bodies together. Why? Studying the situation in more detail, sticking friction is found to be proportional to the actual contact area. It turns out that putting two solids into contact is rather like turning Switzerland upside down and putting it onto Austria; the area of contact is much smaller than that estimated macroscopically. The import- ant point is that the area of actual contact is proportional to the normal force, i.e., the force component that is perpendicular to the surface. The study of what happens in that contact area is still a topic of research; researchers are investigating the issues using in- struments such as atomic force microscopes, lateral force microscopes and triboscopes. Motion Mountain – The Adventure of Physics These efforts resulted in computer hard discs which last longer, as the friction between the disc and the reading head is a central quantity in determining the lifetime. All forms of friction are accompanied by an increase in the temperature of the mov- ing body. The reason became clear after the discovery of atoms. Friction is not observed in few – e.g. 2, 3, or 4 – particle systems. Friction only appears in systems with many particles, usually millions or more. Such systems are called dissipative. Both the tem- perature changes and friction itself are due to the motion of large numbers of micro- scopic particles against each other. This motion is not included in the Galilean descrip- tion. When it is included, friction and energy loss disappear, and potentials can then be copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net used throughout. Positive accelerations – of microscopic magnitude – then also appear, and motion is found to be conserved. In short, all motion is conservative on a microscopic scale. On a microscopic scale it is thus possible and most practical to describe all motion without the concept of force.* The moral of the story is twofold: First, one should use force and power only in one situation: in the case of friction, and only when one does not want to go into the details.** Secondly, friction is not an obstacle to predictability. Motion remains predictable. * The first scientist who eliminated force from the description of nature was Heinrich Rudolf Hertz (b. 1857 Hamburg, d. 1894 Bonn), the famous discoverer of electromagnetic waves, in his textbook on mechanics, Die Prinzipien der Mechanik, Barth, 1894, republished by Wissenschaftliche Buchgesellschaft, 1963. His idea was strongly criticized at that time; only a generation later, when quantum mechanics quietly got rid of the concept for good, did the idea become commonly accepted. (Many have speculated about the role Hertz would have played in the development of quantum mechanics and general relativity, had he not died so young.) In his book, Hertz also formulated the principle of the straightest path: particles follow geodesics. This same description is one of the pillars of general relativity, as we will see later on. ** But the cost is high; in the case of human relations the evaluation should be somewhat more discerning, Ref. 186 as research on violence has shown. 7 classical mechanics, force and the predictability of motion 237 “ Et qu’avons-nous besoin de ce moteur, quand l’étude réfléchie de la nature nous prouve que le mouvement perpétuel est la première de ses ” lois ?* Donatien de Sade Justine, ou les malheurs de la vertu. C omplete states – initial conditions “ ” Quid sit futurum cras, fuge quaerere ...** Horace, Odi, lib. I, ode 9, v. 13. Let us continue our exploration of the predictability of motion. We often describe the motion of a body by specifying the time dependence of its position, for example as 𝑥(𝑡) = 𝑥0 + 𝑣0 (𝑡 − 𝑡0 ) + 12 𝑎0 (𝑡 − 𝑡0 )2 + 16 𝑗0 (𝑡 − 𝑡0 )3 + ... . (69) The quantities with an index 0, such as the starting position 𝑥0 , the starting velocity Motion Mountain – The Adventure of Physics 𝑣0 , etc., are called initial conditions. Initial conditions are necessary for any description of motion. Different physical systems have different initial conditions. Initial conditions thus specify the individuality of a given system. Initial conditions also allow us to distin- guish the present situation of a system from that at any previous time: initial conditions specify the changing aspects of a system. Equivalently, they summarize the past of a sys- tem. Page 27 Initial conditions are thus precisely the properties we have been seeking for a descrip- tion of the state of a system. To find a complete description of states we thus need only a complete description of initial conditions, which we can thus righty call also initial states. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net It turns out that for gravitation, as for all other microscopic interactions, there is no need for initial acceleration 𝑎0 , initial jerk 𝑗0 , or higher-order initial quantities. In nature, acce- leration and jerk depend only on the properties of objects and their environment; they do not depend on the past. For example, the expression 𝑎 = 𝐺𝑀/𝑟2 of universal gravity, giving the acceleration of a small body near a large one, does not depend on the past, but only on the environment. The same happens for the other fundamental interactions, as we will find out shortly. Page 87 The complete state of a moving mass point is thus described by specifying its position and its momentum at all instants of time. Thus we have now achieved a complete de- scription of the intrinsic properties of point objects, namely by their mass, and of their states of motion, namely by their momentum, energy, position and time. For extended rigid objects we also need orientation and angular momentum. This is the full list for rigid objects; no other state observables are needed. Can you specify the necessary state observables in the cases of extended elastic bod- Challenge 427 s ies and of fluids? Can you give an example of an intrinsic property that we have so far Challenge 428 s missed? * ‘And whatfor do we need this motor, when the reasoned study of nature proves to us that perpetual motion is the first of its laws?’ Ref. 85 ** ‘What future will be tomorrow, never ask ...’ Horace is Quintus Horatius Flaccus (65–8 bce), the great Roman poet. 238 7 classical mechanics, force and the predictability of motion The set of all possible states of a system is given a special name: it is called the phase space. We will use the concept repeatedly. Like any space, it has a number of dimensions. Challenge 429 s Can you specify this number for a system consisting of 𝑁 point particles? It is interesting to recall an older challenge and ask again: does the universe have initial Challenge 430 s conditions? Does it have a phase space? Given that we now have a description of both properties and states for point objects, extended rigid objects and deformable bodies, can we predict all motion? Not yet. There are situations in nature where the motion of an object depends on characteristics other Challenge 431 s than its mass; motion can depend on its colour (can you find an example?), on its tem- perature, and on a few other properties that we will soon discover. And for each intrinsic property there are state observables to discover. Each additional intrinsic property is the basis of a field of physical enquiry. Speed was the basis for mechanics, temperature is the basis for thermodynamics, charge is the basis for electrodynamics, etc. We must there- fore conclude that as yet we do not have a complete description of motion. “ An optimist is somebody who thinks that the ” future is uncertain. Motion Mountain – The Adventure of Physics Anonymous Do surprises exist? Is the fu ture determined? “ Die Ereignisse der Zukunft können wir nicht aus den gegenwärtigen erschließen. Der Glaube ” an den Kausalnexus ist ein Aberglaube.* Ludwig Wittgenstein, Tractatus, 5.1361 “ ” Freedom is the recognition of necessity. Friedrich Engels (1820–1895) copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net If, after climbing a tree, we jump down, we cannot halt the jump in the middle of the trajectory; once the jump has begun, it is unavoidable and determined, like all passive motion. However, when we begin to move an arm, we can stop or change its motion from a hit to a caress. Voluntary motion does not seem unavoidable or predetermined. Challenge 432 e Which of these two cases is the general one? Let us start with the example that we can describe most precisely so far: the fall of a body. Once the gravitational potential 𝜑 acting on a particle is given and taken into account, we can use the expression 𝑎(𝑥) = −∇𝜑 = −𝐺𝑀𝑟/𝑟3 , (70) and we can use the state at a given time, given by initial conditions such as 𝑥(𝑡0 ) and 𝑣(𝑡0 ) , (71) to determine the motion of the particle in advance. Indeed, with these two pieces of information, we can calculate the complete trajectory 𝑥(𝑡). * ‘We cannot infer the events of the future from those of the present. Belief in the causal nexus is supersti- tion.’ Our adventure, however, will confirm the everyday observation that this statement is wrong. 7 classical mechanics, force and the predictability of motion 239 An equation that has the potential to predict the course of events is called an evolution equation. Equation (70), for example, is an evolution equation for the fall of the object. (Note that the term ‘evolution’ has different meanings in physics and in biology.) An evolution equation embraces the observation that not all types of change are observed in nature, but only certain specific cases. Not all imaginable sequences of events are ob- served, but only a limited number of them. In particular, equation (70) embraces the idea that from one instant to the next, falling objects change their motion based on the gravitational potential acting on them. Evolution equations do not exist only for motion due to gravity, but for motion due to all forces in nature. Given an evolution equation and initial state, the whole motion of a system is thus uniquely fixed, a property of motion often called determinism. For example, astronomers can calculate the position of planets with high precision for thousands of years in advance. Let us carefully distinguish determinism from several similar concepts, to avoid mis- understandings. Motion can be deterministic and at the same time be unpredictable in Vol. V, page 46 practice. The unpredictability of motion can have four origins: Motion Mountain – The Adventure of Physics 1. an impracticably large number of particles involved, including situations with fric- tion, 2. insufficient information about initial conditions, and 3. the mathematical complexity of the evolution equations, 4. strange shapes of space-time. For example, in case of the weather the first three conditions are fulfilled at the same time. It is hard to predict the weather over periods longer than about a week or two. (In 1942, Hitler made once again a fool of himself across Germany by requesting a precise weather copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net forecast for the following twelve months.) Despite the difficulty of prediction, weather change is still deterministic. As another example, near black holes all four origins apply together. We will discuss black holes in the section on general relativity. Despite being unpredictable, motion is deterministic near black holes. Motion can be both deterministic and time random, i.e., with different outcomes in Page 127 similar experiments. A roulette ball’s motion is deterministic, but it is also random.* As Vol. IV, page 157 we will see later, quantum systems fall into this category, as do all examples of irreversible motion, such as a drop of ink spreading out in clear water. Also the fall of a die is both deterministic and random. In fact, studies on how to predict the result of a die throw Ref. 188 with the help of a computer are making rapid progress; these studies also show how to throw a die in order to increase the odds to get a specific desired result. In all such cases the randomness and the irreproducibility are only apparent; they disappear when the description of states and initial conditions in the microscopic domain are included. In short, determinism does not contradict (macroscopic) irreversibility. However, on the microscopic scale, deterministic motion is always reversible. A final concept to be distinguished from determinism is acausality. Causality is the requirement that a cause must precede the effect. This is trivial in Galilean physics, but * Mathematicians have developed a large number of tests to determine whether a collection of numbers may be called random; roulette results pass all these tests – in honest casinos only, however. Such tests typically check the equal distribution of numbers, of pairs of numbers, of triples of numbers, etc. Other tests are the Ref. 187 𝜒2 test, the Monte Carlo test(s), and the gorilla test. 240 7 classical mechanics, force and the predictability of motion becomes of importance in special relativity, where causality implies that the speed of light is a limit for the spreading of effects. Indeed, it seems impossible to have deterministic motion (of matter and energy) which is acausal, in other words, faster than light. Can you Challenge 433 s confirm this? This topic will be looked at more deeply in the section on special relativity. Saying that motion is ‘deterministic’ means that it is fixed in the future and also in the past. It is sometimes stated that predictions of future observations are the crucial test for a successful description of nature. Owing to our often impressive ability to influence the future, this is not necessarily a good test. Any theory must, first of all, describe past observations correctly. It is our lack of freedom to change the past that results in our lack of choice in the description of nature that is so central to physics. In this sense, the term ‘initial condition’ is an unfortunate choice, because in fact, initial conditions summarize the past of a system.* The central ingredient of a deterministic description is that all motion can be reduced to an evolution equation plus one specific state. This state can be either initial, intermediate, or final. Deterministic motion is uniquely specified into the past and into the future. To get a clear concept of determinism, it is useful to remind ourselves why the concept Motion Mountain – The Adventure of Physics of ‘time’ is introduced in our description of the world. We introduce time because we observe first that we are able to define sequences in observations, and second, that un- restricted change is impossible. This is in contrast to films, where one person can walk through a door and exit into another continent or another century. In nature we do not observe metamorphoses, such as people changing into toasters or dogs into tooth- brushes. We are able to introduce ‘time’ only because the sequential changes we observe Challenge 434 s are extremely restricted. If nature were not reproducible, time could not be used. In short, determinism expresses the observation that sequential changes are restricted to a single possibility. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Since determinism is connected to the use of the concept of time, new questions arise whenever the concept of time changes, as happens in special relativity, in general relativ- ity and in theoretical high energy physics. There is a lot of fun ahead. In summary, every description of nature that uses the concept of time, such as that of everyday life, that of classical physics and that of quantum mechanics, is intrinsically and inescapably deterministic, since it connects observations of the past and the future, eliminating alternatives. In short, ⊳ The use of time implies determinism, and vice versa. When drawing metaphysical conclusions, as is so popular nowadays when discussing Vol. V, page 46 quantum theory, one should never forget this connection. Whoever uses clocks but denies determinism is nurturing a split personality!** The future is determined. * The problems with the term ‘initial conditions’ become clear near the big bang: at the big bang, the uni- verse has no past, but it is often said that it has initial conditions. This contradiction will be explored later in our adventure. ** That can be a lot of fun though. 7 classical mechanics, force and the predictability of motion 241 Free will “ ” You do have the ability to surprise yourself. Richard Bandler and John Grinder The idea that motion is determined often produces fear, because we are taught to asso- ciate determinism with lack of freedom. On the other hand, we do experience freedom in our actions and call it free will. We know that it is necessary for our creativity and for our happiness. Therefore it seems that determinism is opposed to happiness. But what precisely is free will? Much ink has been consumed trying to find a pre- cise definition. One can try to define free will as the arbitrariness of the choice of initial conditions. However, initial conditions must themselves result from the evolution equa- tions, so that there is in fact no freedom in their choice. One can try to define free will from the idea of unpredictability, or from similar properties, such as uncomputability. But these definitions face the same simple problem: whatever the definition, there is no way to prove experimentally that an action was performed freely. The possible defini- tions are useless. In short, because free will cannot be defined, it cannot be observed. Motion Mountain – The Adventure of Physics (Psychologists also have a lot of additional data to support this conclusion, but that is another topic.) No process that is gradual – in contrast to sudden – can be due to free will; gradual processes are described by time and are deterministic. In this sense, the question about free will becomes one about the existence of sudden changes in nature. This will be a recurring topic in the rest of this walk. Can nature surprise us? In everyday life, nature does not. Sudden changes are not observed. Of course, we still have to investigate this question in other domains, in the very small and in the very large. Indeed, we will change our opinion several times during our adventure, but the conclusion remains. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net We note that the lack of surprises in everyday life is built deep into our nature: evolu- tion has developed curiosity because everything that we discover is useful afterwards. If nature continually surprised us, curiosity would make no sense. Many observations contradict the existence of surprises: in the beginning of our walk we defined time using the continuity of motion; later on we expressed this by saying that time is a consequence of the conservation of energy. Conservation is the opposite of surprise. By the way, a challenge remains: can you show that time would not be definable Challenge 435 s even if surprises existed only rarely? In summary, so far we have no evidence that surprises exist in nature. Time exists be- cause nature is deterministic. Free will cannot be defined with the precision required by physics. Given that there are no sudden changes, there is only one consistent conclusion: free will is a feeling, in particular of independence of others, of independence from fear and of accepting the consequences of one’s actions.* Free will is a strange name for a * That free will is a feeling can also be confirmed by careful introspection. Indeed, the idea of free will always arises after an action has been started. It is a beautiful experiment to sit down in a quiet environment, with the intention to make, within an unspecified number of minutes, a small gesture, such as closing a hand. If Challenge 436 e you carefully observe, in all detail, what happens inside yourself around the very moment of decision, you find either a mechanism that led to the decision, or a diffuse, unclear mist. You never find free will. Such an experiment is a beautiful way to experience deeply the wonders of the self. Experiences of this kind might also be one of the origins of human spirituality, as they show the connection everybody has with the rest of nature. 242 7 classical mechanics, force and the predictability of motion Ref. 189 feeling of satisfaction. This solves the apparent paradox; free will, being a feeling, exists as a human experience, even though all objects move without any possibility of choice. There is no contradiction. Ref. 190 Even if human action is determined, it is still authentic. So why is determinism so frightening? That is a question everybody has to ask themselves. What difference does Challenge 437 e determinism imply for your life, for the actions, the choices, the responsibilities and the pleasures you encounter?* If you conclude that being determined is different from being free, you should change your life! Fear of determinism usually stems from refusal to take the world the way it is. Paradoxically, it is precisely the person who insists on the existence of free will who is running away from responsibility. Summary on predictability Despite difficulties to predict specific cases, all motion we encountered so far is both deterministic and predictable. Even friction is predictable, in principle, if we take into account the microscopic details of matter. Motion Mountain – The Adventure of Physics In short, classical mechanics states that the future is determined. In fact, we will dis- cover that all motion in nature, even in the domains of quantum theory and general relativity, is predictable. Motion is predictable. This is not a surprising result. If motion were not predictable, we could not have introduced the concepts of ‘motion’ and ‘time’ in the first place. We can only talk and think about motion because it is predictable. From predictability to global descriptions of motion “ ” Πλεῖν ἀνάγκε, ζῆν οὐκ ἀνάγκη.** Pompeius copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Physicists aim to talk about motion with the highest precision possible. Predictability is an aspect of precision. The highest predictability – and thus the highest precision – is possible when motion is described as globally as possible. All over the Earth – even in Australia – people observe that stones fall ‘down’. This ancient observation led to the discovery of universal gravity. To find it, all that was neces- sary was to look for a description of gravity that was valid globally. The only additional observation that needs to be recognized in order to deduce the result 𝑎 = 𝐺𝑀/𝑟2 is the variation of gravity with height. In short, thinking globally helps us to make our description of motion more precise and our predictions more useful. How can we describe motion as globally as possible? It turns out that there are six approaches to this question, each of which will be helpful on our way to the top of Motion Mountain. We first give an overview; then we explore each of them. 1. Action principles or variational principles, the first global approach to motion, arise Challenge 438 s * If nature’s ‘laws’ are deterministic, are they in contrast with moral or ethical ‘laws’? Can people still be held responsible for their actions? ** Navigare necesse, vivere non necesse. ‘To navigate is necessary, to live is not.’ Gnaeus Pompeius Magnus Ref. 191 (106–48 bce) is cited in this way by Plutarchus (c. 45 to c. 125). 7 classical mechanics, force and the predictability of motion 243 A B F I G U R E 182 What shape of rail allows the F I G U R E 183 Can motion be described in a black stone to glide most rapidly from manner common to all observers? point A to the lower point B? when we overcome a fundamental limitation of what we have learned so far. When we predict the motion of a particle from its current acceleration with an evolution Page 238 equation, we are using the most local description of motion possible. We use the acce- Motion Mountain – The Adventure of Physics leration of a particle at a certain place and time to determine its position and motion just after that moment and in the immediate neighbourhood of that place. Evolution equations thus have a mental ‘horizon’ of radius zero. The contrast to evolution equations are variational principles. A famous example is illustrated in Figure 182. The challenge is to find the path that allows the fastest possible gliding motion from a high point to a distant low point. The sought path is Challenge 439 d the brachistochrone, from ancient Greek for ‘shortest time’, This puzzle asks about a property of motion as a whole, for all times and all positions. The global approach required by questions such as this one will lead us to a description of motion which is copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net simple, precise and fascinating: the so-called principle of cosmic laziness, also known as the principle of least action. 2. Relativity, the second global approach to motion, emerges when we compare the vari- ous descriptions of the same system produced by all possible observers. For example, the observations by somebody falling from a cliff – as shown in Figure 183 – a pas- senger in a roller coaster, and an observer on the ground will usually differ. The re- lationships between these observations, the so-called symmetry transformations, lead us to a global description of motion, valid for everybody. Later, this approach will lead us to Einstein’s special and general theory of relativity. 3. Mechanics of extended and rigid bodies, rather than mass points, is required to un- derstand the objects, plants and animals of everyday life. For such bodies, we want to understand how all parts of them move. As an example, the counter-intuitive res- Challenge 440 e ult of the experiment in Figure 184 shows why this topic is worthwhile. The rapidly rotating wheel suspended on only one end of the axis remains almost horizontal, but slowly rotates around the rope. In order to design machines, it is essential to understand how a group of rigid bod- ies interact with one another. For example, take the Peaucellier-Lipkin linkage shown Ref. 192 in Figure 185. A joint F is fixed on a wall. Two movable rods lead to two opposite corners of a movable rhombus, whose rods connect to the other two corners C and P. This mechanism has several astonishing properties. First of all, it implicitly defines a 244 7 classical mechanics, force and the predictability of motion bicycle wheel rotating rapidly rope on rigid axis rope a b b F C P a b b F I G U R E 184 What happens when one F I G U R E 185 A famous mechanism, the rope is cut? Peaucellier-Lipkin linkage, consists of (grey) rods and (red) joints and allows drawing a straight line with a compass: ﬁx point F, put a pencil into joint P, and then move C with a compass along a circle. Motion Mountain – The Adventure of Physics circle of radius 𝑅 so that one always has the relation 𝑟C = 𝑅2 /𝑟P between the distances of joints C and P from the centre of this circle. This is called an inversion at a circle. Challenge 441 s Can you find this special circle? Secondly, if you put a pencil in joint P, and let joint C follow a certain circle, the pencil P draws a straight line. Can you find that circle? Challenge 442 s The mechanism thus allows drawing a straight line with the help of a compass. Ref. 193 A famous machine challenge is to devise a wooden carriage, with gearwheels that connect the wheels to an arrow, with the property that, whatever path the carriage Challenge 443 d takes, the arrow always points south (see Figure 187). The solution to this puzzle will even be useful in helping us to understand general relativity, as we will see. Such a copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Vol. II, page 206 wagon allows measuring the curvature of a surface and of space. Another important machine part is the differential gearbox. Without it, cars could Challenge 444 e not follow bends on the road. Can you explain it to your friends? Also nature uses machine parts. In 2011, screws and nuts were found in a joint Ref. 194 of a weevil beetle, Trigonopterus oblongus. In 2013, the first example of biological gears have been discovered: in young plant hoppers of the species Issus coleoptratus, Ref. 195 toothed gears ensure that the two back legs jump synchronously. Figure 186 shows some details. You might enjoy the video on this discovery available at www.youtube. com/watch?v=Q8fyUOxD2EA. Another interesting example of rigid motion is the way that human movements, such as the general motions of an arm, are composed from a small number of basic Ref. 196 motions. All these examples are from the fascinating field of engineering; unfortu- nately, we will have little time to explore this topic in our hike. 4. The next global approach to motion is the description of non-rigid extended bod- ies. For example, fluid mechanics studies the flow of fluids (like honey, water or air) around solid bodies (like spoons, ships, sails or wings). The aim is to understand how all parts of the fluid move. Fluid mechanics thus describes how insects, birds Ref. 197 and aeroplanes fly,* why sailing-boats can sail against the wind, what happens when * The mechanisms of insect flight are still a subject of active research. Traditionally, fluid dynamics has 7 classical mechanics, force and the predictability of motion 245 Motion Mountain – The Adventure of Physics F I G U R E 186 The gears found in young plant hoppers (© Malcolm Burrows). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net carriage N W E path S F I G U R E 187 A south-pointing carriage: whatever the path it follows, the arrow on it always points south. a hard-boiled egg is made to spin on a thin layer of water, or how a bottle full of wine Challenge 445 s can be emptied in the fastest way possible. As well as fluids, we can study the behaviour of deformable solids. This area of re- search is called continuum mechanics. It deals with deformations and oscillations of concentrated on large systems, like boats, ships and aeroplanes. Indeed, the smallest human-made object that can fly in a controlled way – say, a radio-controlled plane or helicopter – is much larger and heavier than many flying objects that evolution has engineered. It turns out that controlling the flight of small things requires more knowledge and more tricks than controlling the flight of large things. There is more about this topic on page 278 in Volume V. 246 7 classical mechanics, force and the predictability of motion possible? possible? F I G U R E 188 How and where does a falling F I G U R E 189 Why do hot-air balloons brick chimney break? stay inﬂated? How can you measure the weight of a bicycle rider using only a ruler? Motion Mountain – The Adventure of Physics F I G U R E 190 Why do marguerites – or ox-eye daisies, Leucanthemum vulgare – usually have around 21 (left and centre) or around 34 (right) petals? (© Anonymous, Giorgio Di Iorio and Thomas Lüthi) extended structures. It seeks to explain, for example, why bells are made in particu- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 446 s lar shapes; how large bodies – such as the falling chimneys shown in Figure 188 – or small bodies – such as diamonds – break when under stress; and how cats can turn themselves the right way up as they fall. During the course of our journey we will repeatedly encounter issues from this field, which impinges even upon general relativity and the world of elementary particles. 5. Statistical mechanics is the study of the motion of huge numbers of particles. Statist- ical mechanics is yet another global approach to the study of motion. The concepts needed to describe gases, such as temperature, entropy and pressure (see Figure 189), are essential tools of this discipline. In particular, the concepts of statistical physics help us to understand why some processes in nature do not occur backwards. These concepts will also help us take our first steps towards the understanding of black holes. 6. The last global approach to motion, self-organization, involves all of the above- mentioned viewpoints at the same time. Such an approach is needed to understand everyday experience, and life itself. Why does a flower form a specific number of petals, as shown in Figure 190? How does an embryo differentiate in the womb? What makes our hearts beat? How do mountains ridges and cloud patterns emerge? How do stars and galaxies evolve? How are sea waves formed by the wind? All these phenomena are examples of self-organization processes; life scientists speak of growth processes. Whatever we call them, all these processes are charac- 7 classical mechanics, force and the predictability of motion 247 terized by the spontaneous appearance of patterns, shapes and cycles. Self-organized processes are a common research theme across many disciplines, including biology, chemistry, medicine, geology and engineering. We will now explore the six global approaches to motion. We will begin with the first approach, namely, the description of motion using a variational principle. This beautiful method for describing, understanding and predicting motion was the result of several centuries of collective effort, and is the highlight of Galilean physics. Variational prin- ciples also provide the basis for all the other global approaches just mentioned. They are also needed for all the further descriptions of motion that we will explore afterwards. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Chapter 8 M E A SU R I NG C HA NG E W I T H AC T ION M otion can be described by numbers. Take a single particle that oves. The expression (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) describes how, during its otion, position changes with time. The description of particle motion is com- pleted by stating how the velocity (𝑣𝑥 (𝑡), 𝑣𝑦 (𝑡), 𝑣𝑧 (𝑡)) changes over time. Realizing that these two expressions fully describe the behaviour of a moving point particle was a Motion Mountain – The Adventure of Physics milestone in the development of modern physics. The next milestone of modern physics is achieved by answering a short but hard ques- Page 20 tion. If motion is a type of change, as the Greek already said, ⊳ How can we measure the amount of change? Physicists took almost two centuries of attempts to uncover the way to measure change. In fact, change can be measured by a single number. Due to the long search, the quantity that measures change has a strange name: it is called (physical) action,** usually abbre- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net viated 𝑆. To remember the connection of ‘action’ with change, just think about a Holly- wood film: a lot of action means a large amount of change. Introducing physical action as a measure of change is important because it provides the first and also the most useful global description of motion. In fact, we already know enough to define action straight away. Imagine taking two snapshots of a system at different times. How could you define the amount of change that occurred in between? When do things change a lot, and when do they change only a little? First of all, a system with many moving parts shows a lot of ** Note that this ‘action’ is not the same as the ‘action’ appearing in statements such as ‘every action has an equal and opposite reaction’. This other usage, coined by Newton for certain forces, has not stuck; therefore the term has been recycled. After Newton, the term ‘action’ was first used with an intermediate meaning, before it was finally given the modern meaning used here. This modern meaning is the only meaning used in this text. Another term that has been recycled is the ‘principle of least action’. In old books it used to have a different meaning from the one in this chapter. Nowadays, it refers to what used to be called Hamilton’s principle in the Anglo-Saxon world, even though it is (mostly) due to others, especially Leibniz. The old names and meanings are falling into disuse and are not continued here. Behind these shifts in terminology is the story of an intense two-centuries-long attempt to describe mo- tion with so-called extremal or variational principles: the objective was to complete and improve the work initiated by Leibniz. These principles are only of historical interest today because all are special cases of the Ref. 198 principle of least action described here. 8 measuring change with action 249 F I G U R E 191 Giuseppe Lagrangia/Joseph Lagrange (1736 –1813). Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 192 Physical action measures change: an example of process with large action value (© Christophe Blanc). change. So it makes sense that the action of a system composed of independent subsys- tems should be the sum of the actions of these subsystems. Secondly, systems with high energy or speed, such as the explosions shown in Fig- ure 192, show larger change than systems at lower energy or speed. Indeed, we already Page 111 introduced energy as the quantity that measures how much a system changes over time. Thirdly, change often – but not always – builds up over time; in other cases, recent change can compensate for previous change, as in a pendulum, when the system can return back to the original state. Change can thus increase or decrease with time. Finally, for a system in which motion is stored, transformed or shifted from one subsys- tem to another, especially when kinetic energy is stored or changed to potential energy, change diminishes over time. 250 8 measuring change with action TA B L E 31 Some action values for changes and processes either observed or imagined. System and process A pproxim at e a c t i o n va l u e Smallest measurable action 1.1 ⋅ 10−34 Js Light Smallest blackening of photographic film < 10−33 Js Photographic flash c. 10−17 Js Electricity Electron ejected from atom or molecule c. 10−33 Js Current flow in lightning bolt c. 104 Js Mechanics and materials Tearing apart two neighbouring iron atoms c. 10−33 Js Breaking a steel bar c. 101 Js Tree bent by the wind from one side to the other c. 500 Js Motion Mountain – The Adventure of Physics Making a white rabbit vanish by ‘real’ magic c. 100 PJs Hiding a white rabbit c. 0.1 Js Car crash c. 2 kJs Driving car stops within the blink of an eye c. 20 kJs Levitating yourself within a minute by 1 m c. 40 kJs Large earthquake c. 1 PJs Driving car disappears within the blink of an eye c. 1 ZJs Sunrise c. 0.1 ZJs Chemistry copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Atom collision in liquid at room temperature c. 10−33 Js Smelling one molecule c. 10−31 Js Burning fuel in a cylinder in an average car engine explosion c. 104 Js Held versus dropped glass c. 0.8 Js Life Air molecule hitting eardrum c. 10−32 Js Ovule fertilization c. 10−20 Js Cell division c. 10−15 Js Fruit fly’s wing beat c. 10−10 Js Flower opening in the morning c. 1 nJs Getting a red face c. 10 mJs Maximum brain change in a minute c. 5 Js Person walking one body length c. 102 Js Birth c. 2 kJs Change due to a human life c. 1 EJs Nuclei, stars and more Single nuclear fusion reaction in star c. 10−15 Js Explosion of gamma-ray burster c. 1046 Js Universe after one second has elapsed undefinable 8 measuring change with action 251 𝐿 𝐿(𝑡) = 𝑇 − 𝑈 average 𝐿 integral ∫ 𝐿(𝑡)d𝑡 F I G U R E 193 Deﬁning a total change or action 𝑡 Δ𝑡 𝑡m as an accumulation (addition, or integral) of 𝑡i 𝑡f small changes or actions over time (simpliﬁed elapsed time for clarity). Motion Mountain – The Adventure of Physics All the mentioned properties, taken together, imply: ⊳ The natural measure of change is the average difference between kinetic and potential energy multiplied by the elapsed time. This quantity has all the right properties: it is the sum of the corresponding quantities for all subsystems if these are independent; it generally increases with time; and the quantity Challenge 447 e decreases if the system transforms motion into potential energy. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Thus the (physical) action 𝑆, measuring the change in a (physical) system, is defined as 𝑡f 𝑡f 𝑆 = 𝐿 Δ𝑡 = 𝑇 − 𝑈 (𝑡f − 𝑡i ) = ∫ (𝑇 − 𝑈) d𝑡 = ∫ 𝐿 d𝑡 , (72) 𝑡i 𝑡i Page 186 where 𝑇 is the kinetic energy, 𝑈 the potential energy we already know, 𝐿 is the difference between these, and the overbar indicates a time average. The quantity 𝐿 is called the Lagrangian (function) of the system,* describes what is being added over time, whenever things change. The sign ∫ is a stretched ‘S’, for ‘sum’, and is pronounced ‘integral of’. In intuitive terms it designates the operation – called integration – of adding up the values of a varying quantity in infinitesimal time steps d𝑡. The initial and the final times are written below and above the integration sign, respectively. Figure 193 illustrates the idea: the integral is simply the size of the dark area below the curve 𝐿(𝑡). * It is named for Giuseppe Lodovico Lagrangia (b. 1736 Torino, d. 1813 Paris), better known as Joseph Louis Lagrange. He was the most important mathematician of his time; he started his career in Turin, then worked for 20 years in Berlin, and finally for 26 years in Paris. Among other things he worked on number theory and analytical mechanics, where he developed most of the mathematical tools used nowadays for calculations in classical mechanics and classical gravitation. He applied them successfully to many motions in the solar system. 252 8 measuring change with action Challenge 448 s Mathematically, the integral of the Lagrangian, i.e., of the curve 𝐿(𝑡), is defined as 𝑡f f ∫ 𝐿(𝑡) d𝑡 = lim ∑ 𝐿(𝑡m)Δ𝑡 = 𝐿 ⋅ (𝑡f − 𝑡i ) . (73) 𝑡i Δ𝑡→0 m=i In other words, the integral is the limit, as the time slices get smaller, of the sum of the areas of the individual rectangular strips that approximate the function. Since the ∑ sign also means a sum, and since an infinitesimal Δ𝑡 is written d𝑡, we can understand the nota- tion used for integration. Integration is a sum over slices. The notation was developed by Gottfried Wilhelm Leibniz to make exactly this point. Physically speaking, the integral of the Lagrangian measures the total effect that 𝐿 builds up over time. Indeed, action is called ‘effect’ in some languages, such as German. The effect that builds up is the total change in the system. In short, ⊳ The integral of the Lagrangian, the action, measures the total change that Motion Mountain – The Adventure of Physics occurs in a system. Physical action is total change. Action, or change, is the integral of the Lagrangian over time. The unit of action, and thus of change, is the unit of energy, the Joule, times the unit of time, the second. ⊳ Change is measured in Js. A large value means a big change. Table 31 shows some action values observed in nature. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net To understand the definition of action in more detail, we start with the simplest possible case: a system for which the potential energy vanishes, such as a particle moving freely. Obviously, the higher the kinetic energy is, the more change there is in a given time. Also, if we observe the particle at two instants, the more distant they are the larger the change. The change of a free particle accumulates with time. This is as expected. Next, we explore a single particle moving in a potential. For example, a falling stone loses potential energy in exchange for a gain in kinetic energy. The more kinetic energy is stored into potential energy, the less change there is. Hence the minus sign in the defini- tion of 𝐿. If we explore a particle that is first thrown up in the air and then falls, the curve for 𝐿(𝑡) first is below the times axis, then above. We note that the definition of integra- tion makes us count the grey surface below the time axis negatively. Change can thus be negative, and be compensated by subsequent change, as expected. To measure change for a system made of several independent components, we simply add all the kinetic energies and subtract all the potential energies. This technique allows us to define action values for gases, liquids and solid matter. In short, action is an additive quantity. Even if the components interact, we still get a sensible result. In summary, physical action measures, in a single number, the change observed in a system between two instants of time. Action, or change, is measured in Js. Physical action quantifies the change due to a physical process. This is valid for all observations, i.e., for all processes and all systems: an explosion of a firecracker, a caress of a loved one or a colour change of computer display. We will discover later that describing change with a 8 measuring change with action 253 F I G U R E 194 The minimum of a curve has vanishing slope. single number is also possible in relativity and quantum theory: any change going on in Page 20 any system of nature, be it transport, transformation or growth, can be measured with a single number. Motion Mountain – The Adventure of Physics The principle of least action “ The optimist thinks this is the best of all ” possible worlds, and the pessimist knows it. Robert Oppenheimer We now have a precise measure of change. This, as it turns out, allows a simple, global and powerful description of motion. In nature, the change happening between two instants is always the smallest possible. ⊳ In nature, action is minimal. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net This is the essence of the famous principle of least action. It is valid for every example of motion.* Of all possible motions, nature always chooses for which the change is minimal. Let us study a few examples. The simple case of a free particle, when no potentials are involved, the principle of least action implies that the particle moves in a straight line with constant velocity. All Challenge 449 e other paths would lead to larger actions. Can you verify this? When gravity is present, a thrown stone flies along a parabola – or more precisely, along an ellipse – because any other path, say one in which the stone makes a loop in the Challenge 450 e air, would imply a larger action. Again you might want to verify this for yourself. All observations support the simple and basic statement: things always move in a way that produces the smallest possible value for the action. This statement applies to the full path and to any of its segments. Bertrand Russell called it the ‘law of cosmic laziness’. We could also call it the principle of maximal efficiency of nature. It is customary to express the idea of minimal change in a different way. The action varies when the path is varied. The actual path is the one with the smallest action. You will * In fact, in some macroscopic situations the action can be a saddle point, so that the snobbish form of the Ref. 199 principle is that the action is ‘stationary’. In contrast to what is often heard, the action is never a maximum. Moreover, for motion on small (infinitesimal) scales, the action is always a minimum. The mathematical condition of vanishing variation, given below, encompasses all these details. 254 8 measuring change with action recall from school that at a minimum the derivative of a quantity vanishes: a minimum has a horizontal slope. This relation is shown in Figure 194. In the present case, we do not vary a quantity, but a complete path; hence we do not speak of a derivative or slope, but of a variation. It is customary to write the variation of action as 𝛿𝑆. The principle of least action thus states: ⊳ The actual trajectory between specified end points satisfies 𝛿𝑆 = 0. Mathematicians call this a variational principle. Note that the end points have to be spe- cified: we have to compare motions with the same initial and final situations. Before discussing the principle of least action further, we check that it is indeed equi- valent to the evolution equation.* To do this, we can use a standard procedure, part of * For those interested, here are a few comments on the equivalence of Lagrangians and evolution equations. First of all, Lagrangians do not exist for non-conservative, or dissipative systems. We saw that there is no Page 234 potential for any motion involving friction (and more than one dimension); therefore there is no action in Motion Mountain – The Adventure of Physics these cases. One approach to overcome this limitation is to use a generalized formulation of the principle of least action. Whenever there is no potential, we can express the work variation 𝛿𝑊 between different trajectories 𝑥𝑖 as 𝛿𝑊 = ∑ 𝑚𝑖 𝑥̈𝑖 𝛿𝑥𝑖 . (74) 𝑖 Motion is then described in the following way: 𝑡f ⊳ The actual trajectory satisfies ∫ (𝛿𝑇 + 𝛿𝑊)d𝑡 = 0 provided 𝛿𝑥(𝑡i ) = 𝛿𝑥(𝑡f ) = 0 . (75) 𝑡i The quantity being varied has no name; it represents a generalized notion of change. You might want to Challenge 451 e check that it leads to the correct evolution equations. Thus, although proper Lagrangian descriptions exist copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net only for conservative systems, for dissipative systems the principle can be generalized and remains useful. Many physicists will prefer another approach. What a mathematician calls a generalization is a special case for a physicist: the principle (75) hides the fact that all results from the usual principle of minimal action, if we include the complete microscopic details. There is no friction in the microscopic domain. Friction is an approximate, macroscopic concept. Nevertheless, more mathematical viewpoints are useful. For example, they lead to interesting limitations for the use of Lagrangians. These limitations, which apply only if the world is viewed as purely classical – which it isn’t – were discovered about a hundred years ago. In those times, computers were not available, and the exploration of new calculation techniques was important. Here is a summary. The coordinates used in connection with Lagrangians are not necessarily the Cartesian ones. Generalized coordinates are especially useful when there are constraints on the motion. This is the case for a pendulum, where the weight always has to be at the same distance from the suspension, or for an ice skater, where the Ref. 200 skate has to move in the direction in which it is pointing. Generalized coordinates may even be mixtures of positions and momenta. They can be divided into a few general types. Generalized coordinates are called holonomic–scleronomic if they are related to Cartesian coordinates in a fixed way, independently of time: physical systems described by such coordinates include the pendulum and a particle in a potential. Coordinates are called holonomic–rheonomic if the dependence involves time. An example of a rheonomic system would be a pendulum whose length depends on time. The two terms Page 392 rheonomic and scleronomic are due to Ludwig Boltzmann. These two cases, which concern systems that are only described by their geometry, are grouped together as holonomic systems. The term is due to Heinrich Vol. III, page 99 Hertz. The more general situation is called anholonomic, or nonholonomic. Lagrangians work well only for holo- nomic systems. Unfortunately, the meaning of the term ‘nonholonomic’ has changed. Nowadays, the term is also used for certain rheonomic systems. The modern use calls nonholonomic any system which involves 8 measuring change with action 255 the so-called calculus of variations. The condition 𝛿𝑆 = 0 implies that the action, i.e., the area under the curve in Figure 193, is a minimum. A little bit of thinking shows that if the Lagrangian is of the form 𝐿(𝑥n, 𝑣n ) = 𝑇(𝑣n ) − 𝑈(𝑥n), then the minimum area is achieved Challenge 452 e when d ∂𝑇 ∂𝑈 ( )=− (76) d𝑡 ∂𝑣n ∂𝑥n where n counts all coordinates of all particles.* For a single particle, these Lagrange’s Challenge 453 e equations of motion reduce to 𝑚𝑎 = −∇𝑈 . (78) This is the evolution equation: it says that the acceleration times the mass of a particle is the gradient of the potential energy 𝑈. The principle of least action thus implies the Challenge 454 s equation of motion. (Can you show the converse, which is also correct?) In other words, all systems evolve in such a way that the change or action is as small as possible. Nature is economical. Nature is maximally efficient. Or: Nature is lazy. Nature Motion Mountain – The Adventure of Physics is thus the opposite of a Hollywood thriller, in which the action is maximized; nature is more like a wise old man who keeps his actions to a minimum. The principle of minimal action states that the actual trajectory is the one for which the average of the Lagrangian over the whole trajectory is minimal (see Figure 193). Challenge 455 e Nature is a Dr. Dolittle. Can you verify this? This viewpoint allows one to deduce Lag- range’s equations (76) directly. The principle of least action distinguishes the actual trajectory from all other ima- ginable ones. This observation lead Leibniz to his famous interpretation that the actual world is the ‘best of all possible worlds.’** We may dismiss this as metaphysical specula- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net tion, but we should still be able to feel the fascination of the issue. Leibniz was so excited about the principle of least action because it was the first time that actual observations were distinguished from all other imaginable possibilities. For the first time, the search for reasons why things are the way they are became a part of physical investigation. Could the world be different from what it is? In the principle of least action, we have a hint of velocities. Therefore, an ice skater or a rolling disc is often called a nonholonomic system. Care is thus necessary to decide what is meant by nonholonomic in any particular context. Even though the use of Lagrangians, and of action, has its limitations, these need not bother us at the microscopic level, because microscopic systems are always conservative, holonomic and scleronomic. At the fundamental level, evolution equations and Lagrangians are indeed equivalent. * The most general form for a Lagrangian 𝐿(𝑞n , 𝑞ṅ , 𝑡), using generalized holonomic coordinates 𝑞n , leads to Lagrange equations of the form d ∂𝐿 ∂𝐿 ( )= . (77) d𝑡 ∂𝑞ṅ ∂𝑞n In order to deduce these equations, we also need the relation 𝛿𝑞 ̇ = d/d𝑡(𝛿𝑞). This relation is valid only for holonomic coordinates introduced in the previous footnote and explains their importance. We remark that the Lagrangian for a moving system is not unique; however, the study of how the various Ref. 201 Lagrangians for a given moving system are related is not part of this walk. By the way, the letter 𝑞 for position and 𝑝 for momentum were introduced in physics by the mathem- atician Carl Jacobi (b. 1804 Potsdam, d. 1851 Berlin). ** This idea was ridiculed by the influential philosopher Voltaire (b. 1694 Paris, d. 1778 Paris) in his lucid writings, notably in the brilliant book Candide, written in 1759, and still widely available. 256 8 measuring change with action a negative answer. Leibniz also deduced from the result that gods cannot choose their Challenge 456 s actions. (What do you think?) L agrangians and motion “ ” Never confuse movement with action. Ref. 202 Ernest Hemingway Systems evolve by minimizing change. Change, or action, is the time integral of the Lag- rangian. As a way to describe motion, the Lagrangian has several advantages over the evolution equation. First of all, the Lagrangian is usually more compact than writing the corresponding evolution equations. For example, only one Lagrangian is needed for one system, however many particles it includes. One makes fewer mistakes, especially sign mistakes, as one rapidly learns when performing calculations. Just try to write down the evolution equations for a chain of masses connected by springs; then compare the effort Challenge 457 e with a derivation using a Lagrangian. (The system is often studied because it behaves in many aspects like a chain of atoms.) We will encounter another example shortly: David Motion Mountain – The Adventure of Physics Hilbert took only a few weeks to deduce the equations of motion of general relativity us- ing a Lagrangian, whereas Albert Einstein had worked for ten years searching for them directly. In addition, the description with a Lagrangian is valid with any set of coordinates de- scribing the objects of investigation. The coordinates do not have to be Cartesian; they can be chosen as we prefer: cylindrical, spherical, hyperbolic, etc. These so-called gen- eralized coordinates allow one to rapidly calculate the behaviour of many mechanical systems that are in practice too complicated to be described with Cartesian coordinates. For example, for programming the motion of robot arms, the angles of the joints provide copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net a clearer description than Cartesian coordinates of the ends of the arms. Angles are non- Cartesian coordinates. They simplify calculations considerably: the task of finding the most economical way to move the hand of a robot from one point to another is solved much more easily with angular variables. More importantly, the Lagrangian allows us to quickly deduce the essential proper- ties of a system, namely, its symmetries and its conserved quantities. We will develop this Page 276 important idea shortly, and use it regularly throughout our walk. Finally, the Lagrangian formulation can be generalized to encompass all types of in- teractions. Since the concepts of kinetic and potential energy are general, the principle of least action can be used in electricity, magnetism and optics as well as mechanics. The principle of least action is central to general relativity and to quantum theory, and allows us to easily relate both fields to classical mechanics. As the principle of least action became well known, people applied it to an ever-increa- Ref. 198 sing number of problems. Today, Lagrangians are used in everything from the study of elementary particle collisions to the programming of robot motion in artificial intelli- gence. (Table 32 shows a few examples.) However, we should not forget that despite its remarkable simplicity and usefulness, the Lagrangian formulation is equivalent to the Challenge 458 s evolution equations. It is neither more general nor more specific. In particular, it is not an explanation for any type of motion, but only a different view of it. In fact, the search for a new physical ‘law’ of motion is just the search for a new Lagrangian. This makes 8 measuring change with action 257 TA B L E 32 Some Lagrangians. System Lagrangian Q ua nt i t i e s Free, non-relativistic 𝐿 = 12 𝑚𝑣2 mass 𝑚, speed 𝑣 = d𝑥/d𝑡 mass point Particle in potential 𝐿 = 12 𝑚𝑣2 − 𝑚𝜑(𝑥) gravitational potential 𝜑 Mass on spring 𝐿 = 12 𝑚𝑣2 − 12 𝑘𝑥2 elongation 𝑥, spring constant 𝑘 Mass on frictionless 𝐿 = 12 𝑚𝑣2 − 𝑘(𝑥2 + 𝑦2 ) spring constant 𝑘, table attached to spring coordinates 𝑥, 𝑦 Chain of masses and 𝐿 = 12 𝑚 ∑ 𝑣𝑖2 − 12 𝑚𝜔2 ∑𝑖,𝑗 (𝑥𝑖 − 𝑥𝑗 )2 coordinates 𝑥𝑖 , lattice springs (simple model of frequency 𝜔 atoms in a linear crystal) Free, relativistic mass 𝐿 = −𝑐2 𝑚√1 − 𝑣2 /𝑐2 mass 𝑚, speed 𝑣, speed of point light 𝑐 Motion Mountain – The Adventure of Physics sense, as the description of nature always requires the description of change. Change in nature is always described by actions and Lagrangians. The principle of least action states that the action is minimal when the end points of Ref. 203 the motion, and in particular the time between them, are fixed. It is less well known that the reciprocal principle also holds: if the action value – the change value – is kept fixed, Challenge 459 ny the elapsed time for the actual motion is maximal. Can you show this? Even though the principle of least action is not an explanation of motion, the principle somehow calls for such an explanation. We need some patience, though. Why nature fol- lows the principle of least action, and how it does so, will become clear when we explore copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net quantum theory. Why is motion so often b ounded? Looking around ourselves on Earth or in the sky, we find that matter is not evenly distrib- uted. Matter tends to be near other matter: it is lumped together in aggregates. Figure 195 Ref. 204 shows a typical example. Some major examples of aggregates are listed in Figure 196 and Table 33. All aggregates have mass and size. In the mass–size diagram of Figure 196, both scales are logarithmic. We note three straight lines: a line 𝑚 ∼ 𝑙 extending from the Planck mass* upwards, via black holes, to the universe itself; a line 𝑚 ∼ 1/𝑙 extending from the Planck mass downwards, to the lightest possible aggregate; and the usual matter line with 𝑚 ∼ 𝑙3 , extending from atoms upwards, via everyday objects, the Earth to the Sun. The first of the lines, the black hole limit, is explained by general relativity; the last two, the aggregate limit and the common matter line, by quantum theory.** The aggregates outside the common matter line also show that the stronger the inter- action that keeps the components together, the smaller the aggregate. But why is matter mainly found in lumps? * The Planck mass is given by 𝑚Pl = √ℏ𝑐/𝐺 = 21.767(16) μg. ** Figure 196 suggests that domains beyond physics exist; we will discover later on that this is not the case, as mass and size are not definable in those domains. 258 8 measuring change with action Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 195 Motion in the universe is bounded. (© Mike Hankey) First of all, aggregates form because of the existence of attractive interactions between objects. Secondly, they form because of friction: when two components approach, an aggregate can only be formed if the released energy can be changed into heat. Thirdly, aggregates have a finite size because of repulsive effects that prevent the components from collapsing completely. Together, these three factors ensure that in the universe, bound motion is much more common than unbound, ‘free’ motion. Only three types of attraction lead to aggregates: gravity, the attraction of electric charges, and the strong nuclear interaction. Similarly, only three types of repulsion are observed: rotation, pressure, and the Pauli exclusion principle (which we will encounter Vol. IV, page 136 later on). Of the nine possible combinations of attraction and repulsion, not all appear in nature. Can you find out which ones are missing from Figure 196 and Table 33, and Challenge 460 s why? Together, attraction, friction and repulsion imply that change and action are minim- 8 measuring change with action 259 universe mass [kg] galaxy 1040 black holes star cluster lim ce: Sun Beyond nature and science: beyond Planck length limit ol ien it k h sc Earth e neutron ac d bl an 1020 Beyond nature and science: undefined star e e th t u r nd na mountain yo nd ine be eyo B ter l mat human 100 mon Planck mass com cell Motion Mountain – The Adventure of Physics heavy DNA 10-20 nucleus uranium muon hydrogen proton electron Aggregates m 10-40 neutrino icr os co pi ca gg Elementary re ga particles copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net lightest te lim 10-60 imaginable it aggregate 10-40 10-20 100 1020 size [m] F I G U R E 196 Elementary particles and aggregates found in nature. ized when objects come and stay together. The principle of least action thus implies the stability of aggregates. By the way, the formation history also explains why so many ag- Challenge 461 s gregates rotate. Can you tell why? But why does friction exist at all? And why do attractive and repulsive interactions exist? And why is it – as it would appear from the above – that in some distant past matter was not found in lumps? In order to answer these questions, we must first study another global property of motion: symmetry. TA B L E 33 Some major aggregates observed in nature. A g g r e g at e Size Obs. Constituents (diameter) num. Gravitationally bound aggregates 260 8 measuring change with action A g g r e g at e Size Obs. Constituents (diameter) num. Matter across universe c. 100 Ym 1 superclusters of galaxies, hydrogen and helium atoms Quasar 1012 to 1014 m 20 ⋅ 106 baryons and leptons Supercluster of galaxies c. 3 Ym 107 galaxy groups and clusters 9 Galaxy cluster c. 60 Zm 25 ⋅ 10 10 to 50 galaxies Galaxy group or cluster c. 240 Zm 50 to over 2000 galaxies Our local galaxy group 50 Zm 1 c. 40 galaxies General galaxy 0.5 to 2 Zm 3.5 ⋅ 1012 1010 to 3 ⋅ 1011 stars, dust and gas clouds, probably star systems Our galaxy 1.0(0.1) Zm 1 1011 stars, dust and gas clouds, solar systems 5 Interstellar clouds up to 15 Em ≫ 10 hydrogen, ice and dust Solar System 𝑎 unknown > 400 star, planets Motion Mountain – The Adventure of Physics Our Solar System 30 Pm 1 Sun, planets (Pluto’s orbit’s diameter: 11.8 Tm), moons, planetoids, comets, asteroids, dust, gas Oort cloud 6 to 30 Pm 1 comets, dust Kuiper belt 60 Tm 1 planetoids, comets, dust Star 𝑏 10 km to 100 Gm 1022±1 ionized gas: protons, neutrons, electrons, neutrinos, photons Our star, the Sun 1.39 Gm copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Planet 𝑎 (Jupiter, Earth) 143 Mm, 12.8 Mm 8+ > solids, liquids, gases; in particular, 400 heavy atoms Planetoids (Varuna, etc) 50 to 1 000 km > 100 solids (est. 109 ) Moons 10 to 1 000 km > 50 solids Neutron stars 10 km > 1000 mainly neutrons Electromagnetically bound aggregates 𝑐 Dwarf planets, minor 1 m to 2400 km > 106 (109 estimated) solids, usually planets, asteroids 𝑑 monolithic Comets 10 cm to 50 km > 109 (1012 possible) ice and dust Mountains, solids, liquids, 1 nm to > 100 km n.a. molecules, atoms gases, cheese Animals, plants, kefir 5 μm to 1 km 1026±2 organs, cells brain, human 0.2 m 1010 neurons and other cell types Cells: 1031±1 organelles, membranes, molecules smallest (Nanoarchaeum c. 400 nm molecules equitans) amoeba c. 600 μm molecules 8 measuring change with action 261 A g g r e g at e Size Obs. Constituents (diameter) num. largest (whale nerve, c. 30 m molecules single-celled plants) Molecules: 1078±2 atoms H2 c. 50 pm 1072±2 atoms DNA (human) 2 m (total per cell) 1021 atoms Atoms, ions 30 pm to 300 pm 1080±2 electrons and nuclei Aggregates bound by the weak interaction 𝑐 None Aggregates bound by the strong interaction 𝑐 Nucleus 0.9 to > 7 f m 1079±2 nucleons Nucleon (proton, neutron) 0.9 f m 1080±2 quarks Mesons c. 1 f m n.a. quarks Motion Mountain – The Adventure of Physics Neutron stars: see further up 𝑎. Only in 1994 was the first evidence found for objects circling stars other than our Sun; of over 5000 extrasolar planets found so far, most are found around F, G and K stars, including neutron stars. For example, Ref. 205 three objects circle the pulsar PSR 1257+12, and a matter ring circles the star β Pictoris. The objects seem to be dark stars, brown dwarfs or large gas planets like Jupiter. Due to the limitations of observation systems, none of the systems found so far form solar systems of the type we live in. In fact, only a few Earth-like planets have been found so far. 𝑏. The Sun is among the brightest 7 % of stars. Of all stars, 80 %, are red M dwarfs, 8 % are orange K dwarfs, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net and 5 % are white D dwarfs: these are all faint. Almost all stars visible in the night sky belong to the bright 7 %. Some of these are from the rare blue O class or blue B class (such as Spica, Regulus and Rigel); 0.7 % consist of the bright, white A class (such as Sirius, Vega and Altair); 2 % are of the yellow–white F class (such as Canopus, Procyon and Polaris); 3.5 % are of the yellow G class (like Alpha Centauri, Capella or the Sun). Exceptions include the few visible K giants, such as Arcturus and Aldebaran, and the rare M supergiants, Vol. II, page 250 such as Betelgeuse and Antares. More on stars later on. Vol. V, page 342 𝑐. For more details on microscopic aggregates, see the table of composites. Ref. 206 𝑑. It is estimated that there are up to 1020 small Solar System bodies (asteroids, meteoroids, planetoids or minor planets) that are heavier than 100 kg. Incidentally, no asteroids between Mercury and the Sun – the hypothetical Vulcanoids – have been found so far. Curiosities and fun challenges ab ou t L agrangians The principle of least action as a mathematical description is due to Leibniz. He under- stood its validity in 1707. It was then rediscovered and named by Maupertuis in 1746, Page 136 who wrote: Lorsqu’il arrive quelque changement dans la Nature, la quantité d’action né- cessaire pour ce changement est la plus petite qu’il soit possible.* * ‘When some change occurs in Nature, the quantity of action necessary for this change is the smallest that is possible.’ 262 8 measuring change with action Samuel König, the first scientist to state publicly and correctly in 1751 that the principle was due to Leibniz, and not to Maupertuis, was expelled from the Prussian Academy of Sciences for stating so. This was due to an intrigue of Maupertuis, who was the president of the academy at the time. The intrigue also made sure that the bizarre term ‘action’ was retained. Despite this disgraceful story, Leibniz’ principle quickly caught on, and was then used and popularized by Euler, Lagrange and finally by Hamilton. ∗∗ The basic idea of the principle of least action, that nature is as lazy as possible, is also called lex parismoniae. This general idea is already expressed by Ptolemy, and later by Fermat, Malebranche, and ’s Gravesande. But Leibniz was the first to understand its validity and mathematical usefulness for the description of all motion. ∗∗ When Lagrange published his book Mécanique analytique, in 1788, it formed one of the high points in the history of mechanics and established the use of variational principles. Motion Mountain – The Adventure of Physics He was proud of having written a systematic exposition of mechanics without a single figure. Obviously the book was difficult to read and was not a sales success at all. There- fore his methods took another generation to come into general use. ∗∗ Given that action is the basic quantity describing motion, we can define energy as action per unit time, and momentum as action per unit distance. The energy of a system thus describes how much it changes over time, and the momentum describes how much it Challenge 462 s changes over distance. What are angular momentum and rotational energy? copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ∗∗ In Galilean physics, the Lagrangian is the difference between kinetic and potential en- ergy. Later on, this definition will be generalized in a way that sharpens our understand- ing of this distinction: the Lagrangian becomes the difference between a term for free particles and a term due to their interactions. In other words, every particle motion is a continuous compromise between what the particle would do if it were free and what other particles want it to do. In this respect, particles behave a lot like humans beings. ∗∗ ‘In nature, effects of telekinesis or prayer are impossible, as in most cases the change inside the brain is much smaller than the change claimed in the outside world.’ Is this Challenge 463 s argument correct? ∗∗ Challenge 464 ny How is action measured? What is the best device or method to measure action? ∗∗ Challenge 465 s Explain: why is 𝑇 + 𝑈 constant, whereas 𝑇 − 𝑈 is minimal? ∗∗ 8 measuring change with action 263 𝛼 air water 𝛽 F I G U R E 197 Refraction of light is due to travel-time optimization. In nature, the sum 𝑇 + 𝑈 of kinetic and potential energy is constant during motion (for closed systems), whereas the action is minimal. Is it possible to deduce, by combining Challenge 466 s these two facts, that systems tend to a state with minimum potential energy? Motion Mountain – The Adventure of Physics ∗∗ Another minimization principle can be used to understand the construction of animal Ref. 207 bodies, especially their size and the proportions of their inner structures. For example, the heart pulse and breathing frequency both vary with animal mass 𝑚 as 𝑚−1/4 , and the dissipated power varies as 𝑚3/4 . It turns out that such exponents result from three properties of living beings. First, they transport energy and material through the organ- ism via a branched network of vessels: a few large ones, and increasingly many smaller ones. Secondly, the vessels all have the same minimum size. And thirdly, the networks are optimized in order to minimize the energy needed for transport. Together, these rela- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net tions explain many additional scaling rules; they might also explain why animal lifespan scales as 𝑚−1/4 , or why most mammals have roughly the same number of heart beats in Page 125 a lifetime. A competing explanation, using a different minimization principle, states that quarter powers arise in any network built in order that the flow arrives to the destination by the Ref. 208 most direct path. ∗∗ The minimization principle for the motion of light is even more beautiful: light always takes the path that requires the shortest travel time. It was known long ago that this idea describes exactly how light changes direction when it moves from air to water, and effect illustrated in Figure 197. In water, light moves more slowly; the speed ratio between air and water is called the refractive index of water. The refractive index, usually abbreviated 𝑛, is material-dependent. The value for water is about 1.3. This speed ratio, together with the minimum-time principle, leads to the ‘law’ of refraction, a simple relation between Challenge 467 s the sines of the two angles. Can you deduce it? ∗∗ Can you confirm that all the mentioned minimization principles – that for the growth of trees, that for the networks inside animals, that for the motion of light – are special Challenge 468 s cases of the principle of least action? In fact, this is the case for all known minimization 264 8 measuring change with action principles in nature. Each of them, like the principle of least action, is a principle of least change. ∗∗ In Galilean physics, the value of the action depends on the speed of the observer, but not on his position or orientation. But the action, when properly defined, should not depend on the observer. All observers should agree on the value of the observed change. Only special relativity will fulfil the requirement that action be independent of the observer’s Challenge 469 s speed. How will the relativistic action be defined? ∗∗ What is the amount of change accumulated in the universe since the big bang? Measuring all the change that is going on in the universe presupposes that the universe is a physical Challenge 470 s system. Is this the case? ∗∗ Motion Mountain – The Adventure of Physics One motion for which action is particularly well minimized in nature is dear to us: Ref. 209 walking. Extensive research efforts try to design robots which copy the energy saving functioning and control of human legs. For an example, see the website by Tao Geng at cswww.essex.ac.uk/tgeng/research.html. ∗∗ Challenge 471 d Can you prove the following integration challenge? 𝜑 π 𝜑 ∫ sec 𝑡 d𝑡 = ln tan ( + ) (79) copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net 0 4 2 ∗∗ What is the shape of the ideal halfpipe for skateboarding? What does ‘ideal’ imply? Challenge 472 s Which requirement leads to a cycloid? Which requirement speaks against a cycloid? ∗∗ Page 125 As mentioned above, animal death is a physical process and occurs when an animal has consumed or metabolized around 1 GJ/kg. Show that the total action of an animal scales Challenge 473 e as 𝑀5/4 . Summary on action Systems move by minimizing change. Change, or action, is the time average of kinetic energy minus potential energy. The statement ‘motion minimizes change’ expresses mo- tion’s predictability and its continuity. The statement also implies that all motion is as simple as possible. Systems move by minimizing change. Equivalently, systems move by maximizing the elapsed time between two situations. Both statements show that nature is lazy. Systems move by minimizing change. In the next chapters, we show that this state- ment implies the observer-invariance, conservation, mirror-invariance, reversibility and Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net 265 8 measuring change with action relativity of everyday motion. Chapter 9 MOT ION A N D SYM M ET RY “ ” Am Anfang war die Symmetrie.** Ref. 210 Werner Heisenberg T he second way to describe motion globally is to describe it in such a way hat all observers agree. Now, whenever an observation stays exactly Motion Mountain – The Adventure of Physics he same when switching from one observer to another, we call the observation invariant or absolute or symmetric. Whenever an observation changes when switching Page 242 from one observer to another, we call it relative. To explore relativity thus means to explore invariance and symmetry. ⊳ Symmetry is invariance after change. Change of observer, or change of point of view, is one such possible change; another pos- sibility can be some change operated on the system under observation itself. For example, a forget-me-not flower, shown in Figure 198, is symmetrical because it looks the same copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net after turning around it, or after turning it, by 72 degrees; many fruit tree flowers have the same symmetry. One also says that under certain changes of viewpoint the flower has an invariant property, namely its shape. If many such viewpoints are possible, one talks about a high symmetry, otherwise a low symmetry. For example, a four-leaf clover has a higher symmetry than a three-leaf one. In physics, the viewpoints are often called frames of reference. Whenever we speak about symmetry in flowers, in everyday life, in architecture or in the arts we usually mean mirror symmetry, rotational symmetry or some combination. These are geometric symmetries. Like all symmetries, geometric symmetries imply in- variance under specific change operations. The complete list of geometric symmetries is Ref. 211 known for a long time. Table 34 gives an overview of the basic types. Figure 199 and Fig- ure 200 give some important examples. Additional geometric symmetries include colour symmetries, where colours are exchanged, and spin groups, where symmetrical objects do not contain only points but also spins, with their special behaviour under rotations. Also combinations with scale symmetry, as they appear in fractals, and variations on curved backgrounds are extensions of the basic table. Challenge 474 e ** ‘In the beginning, there was symmetry.’ Do you agree with this statement? It has led many researchers astray during the search for the unification of physics. Probably, Heisenberg meant to say that in the be- ginning, there was simplicity. However, there are many conceptual and mathematical differences between symmetry and simplicity. 9 motion and symmetry 267 F I G U R E 198 Forget-me-not, also called Myosotis (Boraginaceae), has ﬁve-fold symmetry (© Markku Savela). TA B L E 34 The classiﬁcation and the number of simple geometric symmetries. Motion Mountain – The Adventure of Physics D i m e n s i o nR e p e t i t i o n Tr a n s l at i o n s types 0 1 2 3 point line plane s pa c e groups groups groups groups 1 1 row 2 2 n.a. n.a. 2 5 nets or plane lattice 2 (cyclic, 7 friezes 17 wall-papers n.a. types (square, oblique, dihedral) or copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net hexagonal, 10 rosette rectangular, centred groups rectangular) (C1 , C2 , C3 , C4 , C6 , D1 , D2 , D3 , D4 , D6 ) 3 14 (Bravais) lattices (3 32 crystal 75 rods 80 layers 230 crystal cubic, 2 tetragonal, 4 groups, also structures, orthorhombic, 1 called crystal- also called hexagonal, 1 trigonal, lographic space groups, 2 monoclinic, 1 point groups Fedorov triclinic type) groups or crystallo- graphic groups A high symmetry means that many possible changes leave an observation invariant. At first sight, not many objects or observations in nature seem to be symmetrical: after all, in the nature around us, geometric symmetry is more the exception than the rule. But this is a fallacy. On the contrary, we can deduce that nature as a whole is symmetric Challenge 475 s from the simple fact that we have the ability to talk about it! Moreover, the symmetry of nature is considerably higher than that of a forget-me-not or of any other symmetry from 268 9 motion and symmetry The 17 wallpaper patterns and a way to identify them quickly. Is the maximum rotation order 1, 2, 3, 4 or 6? Is there a mirror (m)? Is there an indecomposable glide reﬂection (g)? Is there a rotation axis on a mirror? Is there a rotation axis not on a mirror? oo pg K ** pm A *o cm M 0 2222 p1 p2 T S2222 y *632 n n 22o y g? n p6m g? pgg D632 y P22 n g? Motion Mountain – The Adventure of Physics m? y y n 1 *2222 632 m? m? 6 max 2 n pmm n rotation y p6 g? D2222 order y 3 4 S632 22* y an axis on m? m? n pmg n a mirror? an axis not y D22 y on a mirror? n y 3*3 2*22 an axis not p31m on a mirror? cmm copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net n y n D222 D33 *333 442 p3m1 p4 D333 *442 S442 333 p3 S333 4*2 p4g D42 p4m D442 Every pattern is identiﬁed according to three systems of notation: 442 The Conway-Thurston notation. p4 The International Union of Crystallography notation. S442 The Montesinos notation, as in his book “Classical Tesselations and Three Manifolds” F I G U R E 199 The full list of possible symmetries of wallpaper patterns, the so-called wallpaper groups, their usual names, and a way to distinguish them (© Dror Bar-Natan). 9 motion and symmetry 269 Crystal system Crystall class or crystal group Triclinic system (three axes, none at right angles) C1 Ci Monoclinic system (two axes at right angles, a C2 Cs or C1h C2h third not) Orthorhombic system (three unequal axes at right angles) D2 C2v D2h Motion Mountain – The Adventure of Physics Tetragonal system (three axes at right angles, one unequal) C4 S4 C4h D4 C4v D2d D4h Trigonal system (three equal axes at 120 degrees, a copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net fourth at right angles with threefold C3 S6 D3 C3v D3d symmetry) Hexagonal system (three equal axes at 120 degrees, a fourth at right angles with sixfold C6 C3h C6h D6 C6v D3h D6h symmetry) Cubic or isometric system (three equal axes at right angles) T Th O Td Oh F I G U R E 200 The full list of possible symmetries of units cells in crystals, the crystallographic point groups or crystal groups or crystal classes (© Jonathan Goss, after Neil Ashcroft and David Mermin). 270 9 motion and symmetry Table 34. A consequence of this high symmetry is, among others, the famous expression 𝐸0 = 𝑐2 𝑚. Why can we think and talk ab ou t the world? “ The hidden harmony is stronger than the ” apparent. Ref. 212 Heraclitus of Ephesus, about 500 bce Why can we understand somebody when he is talking about the world, even though we are not in his shoes? We can for two reasons: because most things look similar from different viewpoints, and because most of us have already had similar experiences be- forehand. ‘Similar’ means that what we and what others observe somehow correspond. In other words, many aspects of observations do not depend on viewpoint. For example, the num- ber of petals of a flower has the same value for all observers. We can therefore say that this quantity has the highest possible symmetry. We will see below that mass is another such Motion Mountain – The Adventure of Physics example. Observables with the highest possible symmetry are called scalars in physics. Other aspects change from observer to observer. For example, the apparent size varies with the distance of observation. However, the actual size is observer-independent. In general terms, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net ⊳ Any type of viewpoint-independence is a form of symmetry. The observation that two people looking at the same thing from different viewpoints can understand each other proves that nature is symmetric. We start to explore the details of this symmetry in this section and we will continue during most of the rest of our hike. In the world around us, we note another general property: not only does the same phenomenon look similar to different observers, but different phenomena look similar to the same observer. For example, we know that if fire burns the finger in the kitchen, it will do so outside the house as well, and also in other places and at other times. Nature shows reproducibility. Nature shows no surprises. In fact, our memory and our thinking Challenge 476 s are only possible because of this basic property of nature. (Can you confirm this?) As we will see, reproducibility leads to additional strong restrictions on the description of nature. Without viewpoint-independence and reproducibility, talking to others or to one- self would be impossible. Even more importantly, we will discover that viewpoint- independence and reproducibility do more than determine the possibility of talking to each other: they also fix much (but not all) of the content of what we can say to each other. In other words, we will see that most of our description of nature follows logically, almost without choice, from the simple fact that we can talk about nature to our friends. 9 motion and symmetry 271 Viewpoints “ Toleranz ... ist der Verdacht der andere könnte ” Recht haben.* Kurt Tucholsky (b. 1890 Berlin, d. 1935 Göteborg), German writer “ Toleranz – eine Stärke, die man vor allem dem ” politischen Gegner wünscht.** Wolfram Weidner (b. 1925) German journalist When a young human starts to meet other people in childhood, he quickly finds out that certain experiences are shared, while others, such as dreams, are not. Learning to make this distinction is one of the adventures of human life. In these pages, we concentrate on a section of the first type of experiences: physical observations. However, even between these, distinctions are to be made. In daily life we are used to assuming that weights, volumes, lengths and time intervals are independent of the viewpoint of the observer. We can talk about these observed quantities to anybody, and there are no disagreements Motion Mountain – The Adventure of Physics over their values, provided they have been measured correctly. However, other quantities do depend on the observer. Imagine talking to a friend after he jumped from one of the trees along our path, while he is still falling downwards. He will say that the forest floor is approaching with high speed, whereas the observer below will maintain that the floor is stationary. Obviously, the difference between the statements is due to their different viewpoints. The velocity of an object (in this example that of the forest floor or of the friend himself) is thus a less symmetric property than weight or size. Not all observers agree on the value. of velocity, nor even on its direction. In the case of viewpoint-dependent observations, understanding each other is still possible with the help of a little effort: each observer can imagine observing from the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net point of view of the other, and check whether the imagined result agrees with the state- ment of the other.*** If the statement thus imagined and the actual statement of the other observer agree, the observations are consistent, and the difference in statements is due only to the different viewpoints; otherwise, the difference is fundamental, and they cannot agree or talk. Using this approach, you can even argue whether human feelings, Challenge 477 s judgements, or tastes arise from fundamental differences or not. The distinction between viewpoint-independent – or invariant – quantities and viewpoint-dependent – or relative –quantities is an essential one. Invariant quantities, such as mass or shape, describe intrinsic properties, and relative quantities, depending on the observer, make up the state of the system. Therefore, in order to find a complete description of the state of a physical system, we must answer the following questions: — Which viewpoints are possible? — How are descriptions transformed from one viewpoint to another? — Which observables do these symmetries admit? * ‘Tolerance ... is the suspicion that the other might be right.’ ** ‘Tolerance – a strength one mainly wishes to political opponents.’ *** Humans develop the ability to imagine that others can be in situations different from their own at the Ref. 213 age of about four years. Therefore, before the age of four, humans are unable to conceive special relativity; afterwards, they can. 272 9 motion and symmetry — What do these results tell us about motion? So far, in our exploration of motion we have first of all studied viewpoints that differ in location, in orientation, in time and, most importantly, in motion. With respect to each other, observers can be at rest, can be rotated, can move with constant speed or can even accelerate. These ‘concrete’ changes of viewpoint are those we will study first. In this case, the requirement of consistency of observations made by different observers is Page 156 called the principle of relativity. The symmetries associated with this type of invariance Page 281 are also called external symmetries. They are listed in Table 36. A second class of fundamental changes of viewpoint concerns ‘abstract’ changes. Viewpoints can differ by the mathematical description used: such changes are called Vol. III, page 85 changes of gauge. They will be introduced first in the section on electrodynamics. Again, it is required that all statements be consistent across different mathematical descriptions. This requirement of consistency is called the principle of gauge invariance. The associated symmetries are called internal symmetries. The third class of changes, whose importance may not be evident from everyday life, Motion Mountain – The Adventure of Physics is that of the behaviour of a system under the exchange of its parts. The associated in- variance is called permutation symmetry. It is a discrete symmetry, and we will encounter Vol. IV, page 112 it as a fundamental principle when we explore quantum theory. The three consistency requirements just described are called ‘principles’ because these basic statements are so strong that they almost completely determine the ‘laws’ of physics – i.e., the description of motion – as we will see shortly. Later on we will discover that looking for a complete description of the state of objects will also yield a complete description of their intrinsic properties. But enough of introduction: let us come to the heart of the topic. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Symmetries and groups Because we are looking for a description of motion that is complete, we need to under- stand and describe the full set of symmetries of nature. But what is symmetry? A system is said to be symmetric or to possess a symmetry if it appears identical when observed from different viewpoints. We also say that the system possesses an invariance under change from one viewpoint to the other. Viewpoint changes are equivalent to sym- metry operations or transformations of a system. A symmetry is thus a set of transform- ations that leaves a system invariant. However, a symmetry is more than a set: the suc- cessive application of two symmetry operations is another symmetry operation. In other terms, a symmetry is a set 𝐺 = {𝑎, 𝑏, 𝑐, ...} of elements, the transformations, together with a binary operation ∘ called concatenation or multiplication and pronounced ‘after’ or ‘times’, in which the following properties hold for all elements 𝑎, 𝑏 and 𝑐: associativity, i.e., (𝑎 ∘ 𝑏) ∘ 𝑐 = 𝑎 ∘ (𝑏 ∘ 𝑐) a neutral element 𝑒 exists such that 𝑒 ∘ 𝑎 = 𝑎 ∘ 𝑒 = 𝑎 an inverse element 𝑎−1 exists such that 𝑎−1 ∘ 𝑎 = 𝑎 ∘ 𝑎−1 = 𝑒 . (80) Any set that fulfils these three defining properties, or axioms, is called a (mathematical) group. Historically, the notion of group was the first example of a mathematical struc- 9 motion and symmetry 273 Motion Mountain – The Adventure of Physics F I G U R E 201 A ﬂower of Crassula ovata showing three ﬁvefold multiplets: petals, stems and buds (© J.J. Harrison) ture which was defined in a completely abstract manner.* Can you give an example of a Challenge 478 s group taken from daily life? Groups appear frequently in physics and mathematics, be- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Ref. 214 cause symmetries are almost everywhere, as we will see.** Can you list the symmetry Challenge 480 s operations of the pattern of Figure 202? Multiplets Looking at a symmetric and composed system such as the ones shown in Figure 201 or Challenge 481 e Figure 202, we notice that each of its parts, for example each red patch, belongs to a set of similar objects, called a multiplet. ⊳ Each part or component of a symmetric system can be classified according to what type of multiplet it belongs to. * The term ‘group’ is due to Evariste Galois (b. 1811 Bourg-la-Reine, d. 1832 Paris), its structure to Augustin- Louis Cauchy (b. 1789 Paris, d. 1857 Sceaux) and the axiomatic definition to Arthur Cayley (b. 1821 Rich- mond upon Thames, d. 1895 Cambridge). ** In principle, mathematical groups need not be symmetry groups; but it can be proven that all groups can be seen as transformation groups on some suitably defined mathematical space, so that in mathematics we can use the terms ‘symmetry group’ and ‘group’ interchangeably. A group is called Abelian if its concatenation operation is commutative, i.e., if 𝑎 ∘ 𝑏 = 𝑏 ∘ 𝑎 for all pairs of elements 𝑎 and 𝑏. In this case, the concatenation is sometimes called addition. Do rotations form an Abelian Challenge 479 e group? A subset 𝐺1 ⊂ 𝐺 of a group 𝐺 can itself be a group; one then calls it a subgroup and often says sloppily that 𝐺 is larger than 𝐺1 or that 𝐺 is a higher symmetry group than 𝐺1 . 274 9 motion and symmetry Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Copyright © 1990 Christoph Schiller F I G U R E 202 A Hispano–Arabic ornament from the Governor’s Palace in Sevilla (© Christoph Schiller). For some of the coloured patches in Figure 202 we need four objects to make up a full multiplet, whereas for others we need two, or only one, as in the case of the central star. Taken as a whole, each multiplet has (at least) the symmetry properties of the whole system. Therefore we have two challenges to solve. First of all, we need to find all symmetries of nature. Secondly, throughout our adventure, we need to determine the full multiplet for every part of nature that we observe. Above all, we will need to determine the mul- tiplets for the smallest parts found in nature, the elementary particles. ⊳ A multiplet is a set of parts or components that transform into each other under all symmetry transformations. 9 motion and symmetry 275 R epresentations Mathematicians often call abstract multiplets representations. By specifying to which multiplet or representation a part or component belongs, we describe in which way the component is part of the whole system. Let us see how this classification is achieved. In mathematical language, symmetry transformations are often described by matrices. For example, in the plane, a reflection along the first diagonal is represen- ted by the matrix 0 1 𝐷(refl) = ( ) , (81) 1 0 since every point (𝑥, 𝑦) becomes transformed to (𝑦, 𝑥) when multiplied by the matrix Challenge 482 e 𝐷(refl). Therefore, for a mathematician a representation of a symmetry group 𝐺 is an assignment of a matrix 𝐷(𝑎) to each group element 𝑎 such that the representation of the concatenation of two elements 𝑎 and 𝑏 is the product of the representations 𝐷 of the elements: 𝐷(𝑎 ∘ 𝑏) = 𝐷(𝑎)𝐷(𝑏) . (82) Motion Mountain – The Adventure of Physics For example, the matrix of equation (81), together with the corresponding matrices for all the other symmetry operations, have this property.* For every symmetry group, the construction and classification of all possible repres- entations is an important task. It corresponds to the classification of all possible mul- tiplets a symmetric system can be made of. Therefore, if we understand the classification of all multiplets and parts which can appear in Figure 202, we will also understand how to classify all possible parts of which an object or an example of motion can be composed! A representation 𝐷 is called unitary if all matrices 𝐷(𝑎) are unitary.** All representa- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net tions appearing in physics, with only a handful of exceptions, are unitary: this term is the * There are some obvious, but important, side conditions for a representation: the matrices 𝐷(𝑎) must be invertible, or non-singular, and the identity operation of 𝐺 must be mapped to the unit matrix. In even more compact language one says that a representation is a homomorphism from 𝐺 into the group of non-singular or invertible matrices. A matrix 𝐷 is invertible if its determinant det 𝐷 is not zero. In general, if a mapping 𝑓 from a group 𝐺 to another 𝐺 satisfies 𝑓(𝑎 ∘𝐺 𝑏) = 𝑓(𝑎) ∘𝐺 𝑓(𝑏) , (83) the mapping 𝑓 is called an homomorphism. A homomorphism 𝑓 that is one-to-one (injective) and onto (surjective) is called an isomorphism. If a representation is also injective, it is called faithful, true or proper. In the same way as groups, more complex mathematical structures such as rings, fields and associative algebras may also be represented by suitable classes of matrices. A representation of the field of complex Vol. IV, page 223 numbers is given later on. ** The transpose 𝐴𝑇 of a matrix 𝐴 is defined element-by-element by (𝐴𝑇 )ik = 𝐴 ki . The complex conjugate 𝐴∗ of a matrix 𝐴 is defined by (𝐴∗ )ik = (𝐴 ik )∗ . The adjoint 𝐴† of a matrix 𝐴 is defined by 𝐴† = (𝐴𝑇 )∗ . A matrix is called symmetric if 𝐴𝑇 = 𝐴, orthogonal if 𝐴𝑇 = 𝐴−1 , Hermitean or self-adjoint (the two are synonymous in all physical applications) if 𝐴† = 𝐴 (Hermitean matrices have real eigenvalues), and unitary if 𝐴† = 𝐴−1 . Unitary matrices have eigenvalues of norm one. Multiplication by a unitary matrix is a one-to- one mapping; since the time evolution of physical systems is a mapping from one time to another, evolution is always described by a unitary matrix. An antisymmetric or skew-symmetric matrix is defined by 𝐴𝑇 = −𝐴, an anti-Hermitean matrix by 𝐴† = −𝐴 and an anti-unitary matrix by 𝐴† = −𝐴−1 . All the corresponding mappings are one-to-one. A matrix is singular, and the corresponding vector transformation is not one-to-one, if det 𝐴 = 0. 276 9 motion and symmetry most restrictive because it specifies that the corresponding transformations are one-to- one and invertible, which means that one observer never sees more or less than another. Obviously, if an observer can talk to a second one, the second one can also talk to the first. Unitarity is a natural property of representations in natural systems. The final important property of a multiplet, or representation, concerns its structure. If a multiplet can be seen as composed of sub-multiplets, it is called reducible, else irre- ducible; the same is said about representations. The irreducible representations obviously cannot be decomposed any further. For example, the (almost perfect) symmetry group of Figure 202, commonly called D4 , has eight elements. It has the general, faithful, unitary Challenge 483 e and irreducible matrix representation cos 𝑛π/2 − sin 𝑛π/2 −1 0 1 0 0 1 0 −1 ( ) 𝑛 = 0..3, ( ),( ),( ),( ). (84) sin 𝑛π/2 cos 𝑛π/2 0 1 0 −1 1 0 −1 0 The representation is an octet. The complete list of possible irreducible representations of Challenge 484 e the group D4 also includes singlets, doublets and quartets. Can you find them all? These Motion Mountain – The Adventure of Physics representations allow the classification of all the white and black ribbons that appear in the figure, as well as all the coloured patches. The most symmetric elements are singlets, and the least symmetric ones are members of the quartets. The complete system is always a singlet as well. With these concepts we are now ready to talk about motion and moving systems with improved precision. The symmetries and vo cabulary of motion Every day we experience that we are able to talk to each other about motion. It must copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net therefore be possible to find an invariant quantity describing it. We already know it: it is the action, the measure of change. For example, lighting a match is a change. The mag- nitude of the change is the same whether the match is lit here or there, in one direction or another, today or tomorrow. Indeed, the (Galilean) action is a number whose value is the same for each observer at rest, independent of his orientation or the time at which he makes his observation. In the case of the Arabic pattern of Figure 202, the symmetry allows us to deduce the list of multiplets, or representations, that can be its building blocks. This approach must be possible for a moving system as well. Table 35 shows how. In the case of the Arabic pattern, from the various possible observation viewpoints, we deduced the classification of the ribbons into singlets, doublets, etc. For a moving system, the building blocks, cor- responding to the ribbons, are the (physical) observables. Since we observe that nature is symmetric under many different changes of viewpoint, we can classify all observables. To do so, we first need to take the list of all viewpoint transformations and then deduce the list of all their representations. Our everyday life shows that the world stays unchanged after changes in position, orientation and instant of observation. We also speak of space translation invariance, rotation invariance and time translation invariance. These transformations are different from those of the Arabic pattern in two respects: they are continuous and they are un- bounded. As a result, their representations will generally be continuously variable and 9 motion and symmetry 277 TA B L E 35 Correspondences between the symmetries of an ornament, a ﬂower and motion. System H i s pa n o – A r - Flower Motion abic pat t e r n Structure and set of ribbons and set of petals, stem motion path and components patches observables System pattern symmetry flower symmetry symmetry of Lagrangian symmetry Mathematical D4 C5 in Galilean relativity: description of the position, orientation, symmetry group instant and velocity changes Invariants number of multiplet petal number number of coordinates, elements magnitude of scalars, vectors and tensors Motion Mountain – The Adventure of Physics Representations multiplet types of multiplet types of tensors, including scalars of the elements components and vectors components Most symmetric singlet part with circular scalar representation symmetry Simplest faithful quartet quintet vector representation Least symmetric quartet quintet no limit (tensor of infinite copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net representation rank) without bounds: they will be quantities, or magnitudes. In other words, ⊳ Because the continuity of observation change, observables must be con- structed with (real) numbers. In this way, we have deduced why numbers are necessary for any description of motion.* Since observers can differ in orientation, representations will be mathematical objects possessing a direction. To cut a long story short, the symmetry under change of obser- vation position, orientation or instant leads to the result that all observables are either ‘scalars’, ‘vectors’ or higher-order ‘tensors.’** ⊳ A scalar is an observable quantity which stays the same for all observers. A scalar corresponds to a singlet. Examples are the mass or the charge of an object, the distance between two points, the distance of the horizon, and many others. The possible * Only scalars, in contrast to vectors and higher-order tensors, may also be quantities that only take a dis- Challenge 485 e crete set of values, such as +1 or −1 only. In short, only scalars may be discrete observables. ** Later on, spinors will be added to, and complete, this list. 278 9 motion and symmetry values of a scaler are (usually) continuous, unbounded and without direction. Other ex- amples of scalars are the potential at a point and the temperature at a point. Velocity is obviously not a scalar; nor is the coordinate of a point. Can you find more examples and Challenge 486 s counter-examples? Energy is a puzzling observable. It is a scalar if only changes of place, orientation and instant of observation are considered. But energy is not a scalar if changes of observer speed are included. Nobody ever searched for a generalization of energy that is a scalar also for moving observers. Only Albert Einstein discovered it, completely by accident. Vol. II, page 65 More about this issue will be told shortly. ⊳ Any quantity which has a magnitude and a direction and which ‘stays the same’ with respect to the environment when changing viewpoint is a vector. For example, the arrow between two fixed points on the floor is a vector. Its length is the same for all observers; its direction changes from observer to observer, but not with respect to its environment. On the other hand, the arrow between a tree and the place Motion Mountain – The Adventure of Physics where a rainbow touches the Earth is not a vector, since that place does not stay fixed when the observer changes. Mathematicians say that vectors are directed entities staying invariant under coordin- ate transformations. Velocities of objects, accelerations and field strength are examples Challenge 487 e of vectors. (Can you confirm this?) The magnitude of a vector is a scalar: it is the same for any observer. By the way, a famous and baffling result of nineteenth-century exper- iments is that the velocity of a light beam is not a vector like the velocity of a car; the velocity of a light beam is not a vector for Galilean transformations.* This mystery will Vol. II, page 15 be solved shortly. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Tensors are generalized vectors. As an example, take the moment of inertia of an Ref. 215 object. It specifies the dependence of the angular momentum on the angular velocity. Page 117 For any object, doubling the magnitude of angular velocity doubles the magnitude of angular momentum; however, the two vectors are not parallel to each other if the object Page 164 is not a sphere. In general, if any two vector quantities are proportional, in the sense that doubling the magnitude of one vector doubles the magnitude of the other, but without the two vectors being parallel to each other, then the proportionality ‘factor’ is a (second order) tensor. Like all proportionality factors, tensors have a magnitude. In addition, tensors have a direction and a shape: they describe the connection between the vectors they relate. Just as vectors are the simplest quantities with a magnitude and a direction, so tensors are the simplest quantities with a magnitude, a direction and a shape, i.e., a direction depending on a second, chosen direction. Just as vectors can be visualized as oriented arrows, symmetric tensors – but not non-symmetric ones – can be visualized as Challenge 489 s oriented ellipsoids.** Can you name another example of a tensor? * Galilean transformations are changes of viewpoints from one observer to a second one, moving with re- spect to the first. ‘Galilean transformation’ is just a term for what happens in everyday life, where velocities add and time is the same for everybody. The term, introduced in 1908 by Philipp Frank, is mostly used as a contrast to the Lorentz transformation that is so common in special relativity. ** A rank-𝑛 tensor is the proportionality factor between a rank-1 tensor – i.e., a vector – and a rank-(𝑛 − 1) tensor. Vectors and scalars are rank 1 and rank 0 tensors. Scalars can be pictured as spheres, vectors as arrows, and symmetric rank-2 tensors as ellipsoids. A general, non-symmetric rank-2 tensor can be split 9 motion and symmetry 279 Let us get back to the description of motion. Table 35 shows that in physical sys- tems – like in a Hispano-Arabic ornament – we always have to distinguish between the symmetry of the whole Lagrangian – corresponding to the symmetry of the complete ornament – and the representation of the observables – corresponding to the ribbon multiplets. Since the action must be a scalar, and since all observables must be tensors, Lagrangians contain sums and products of tensors only in combinations forming scal- ars. Lagrangians thus contain only scalar products or generalizations thereof. In short, Lagrangians always look like 𝐿 = 𝛼 𝑎𝑖 𝑏𝑖 + 𝛽 𝑐𝑗𝑘 𝑑𝑗𝑘 + 𝛾 𝑒𝑙𝑚𝑛 𝑓𝑙𝑚𝑛 + ... (85) where the indices attached to the variables 𝑎, 𝑏, 𝑐 etc. always come in matching pairs to be summed over. (Therefore summation signs are usually simply left out.) The Greek letters represent constants. For example, the action of a free point particle in Galilean physics was given as 𝑚 𝑆 = ∫ 𝐿 d𝑡 = ∫ 𝑣2 d𝑡 Motion Mountain – The Adventure of Physics (86) 2 which is indeed of the form just mentioned. We will encounter many other cases during our study of motion.* Galileo already understood that motion is also invariant under changes of viewpoints uniquely into a symmetric and an antisymmetric tensor. An antisymmetric rank-2 tensor corresponds to a polar vector. Tensors of higher rank correspond to more and more complex shapes. A vector has the same length and direction for every observer; a tensor (of rank 2) has the same determ- inant, the same trace, and the same sum of diagonal subdeterminants for all observers. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net A vector is described mathematically by a list of components; a tensor (of rank 2) is described by a matrix of components. The rank or order of a tensor thus gives the number of indices the observable has. Can you Challenge 488 e show this? * By the way, is the usual list of possible observation viewpoints – namely different positions, different observation instants, different orientations, and different velocities – also complete for the action (86)? Sur- Ref. 216 prisingly, the answer is no. One of the first who noted this fact was Niederer, in 1972. Studying the quantum theory of point particles, he found that even the action of a Galilean free point particle is invariant under some additional transformations. If the two observers use the coordinates (𝑡, 𝑥) and (𝜏, 𝜉), the action (86) Challenge 490 ny is invariant under the transformations 𝑟𝑥 + 𝑥0 + 𝑣𝑡 𝛼𝑡 + 𝛽 𝜉= and 𝜏= with 𝑟𝑇 𝑟 = 1 and 𝛼𝛿 − 𝛽𝛾 = 1 . (87) 𝛾𝑡 + 𝛿 𝛾𝑡 + 𝛿 where 𝑟 describes the rotation from the orientation of one observer to the other, 𝑣 the velocity between the two observers, and 𝑥0 the vector between the two origins at time zero. This group contains two important special cases of transformations: The connected, static Galilei group 𝜉 = 𝑟𝑥 + 𝑥0 + 𝑣𝑡 and 𝜏=𝑡 𝑥 𝛼𝑡 + 𝛽 The transformation group SL(2,R) 𝜉 = and 𝜏= (88) 𝛾𝑡 + 𝛿 𝛾𝑡 + 𝛿 The latter, three-parameter group includes spatial inversion, dilations, time translation and a set of time- dependent transformations such as 𝜉 = 𝑥/𝑡, 𝜏 = 1/𝑡 called expansions. Dilations and expansions are rarely mentioned, as they are symmetries of point particles only and do not apply to everyday objects and systems. They will return to be of importance later on, however. 280 9 motion and symmetry Page 156 with different velocities. However, the action just given does not reflect this. It took some years to find out the correct generalization: it is given by the theory of special relativity. But before we study it, we need to finish the present topic. R eproducibility, conservation and Noether ’ s theorem “ I will leave my mass, charge and momentum to ” science. Graffito The reproducibility of observations, i.e., the symmetry under change of instant of time or ‘time translation invariance’, is a case of viewpoint-independence. (That is not obvi- Challenge 491 ny ous; can you find its irreducible representations?) The connection has several important consequences. We have seen that symmetry implies invariance. It turns out that for con- tinuous symmetries, such as time translation symmetry, this statement can be made more precise: Motion Mountain – The Adventure of Physics ⊳ For any continuous symmetry of the Lagrangian there is an associated con- served constant of motion and vice versa. The exact formulation of this connection is the theorem of Emmy Noether.* She found the result in 1915 when helping Albert Einstein and David Hilbert, who were both strug- gling and competing at constructing general relativity. However, the result applies to any Ref. 217 type of Lagrangian. Noether investigated continuous symmetries depending on a continuous parameter 𝑏. A viewpoint transformation is a symmetry if the action 𝑆 does not depend on the value of 𝑏. For example, changing position as copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net 𝑥 → 𝑥 + 𝑏 (89) leaves the action 𝑆0 = ∫ 𝑇(𝑣) − 𝑈(𝑥) d𝑡 (90) invariant, since 𝑆(𝑏) = 𝑆0 . This situation implies that ∂𝑇 = 𝑝 = const . (91) ∂𝑣 In short, symmetry under change of position implies conservation of momentum. The converse is also true. Challenge 492 e In the case of symmetry under shift of observation instant, we find 𝑇 + 𝑈 = const . (92) * Emmy Noether (b. 1882 Erlangen, d. 1935 Bryn Mawr), mathematician. The theorem is only a sideline in her career which she dedicated mostly to number theory. The theorem also applies to gauge symmetries, where it states that to every gauge symmetry corresponds an identity of the equation of motion, and vice versa. 9 motion and symmetry 281 In other words, time translation invariance implies constant energy. Again, the converse is also correct. The conserved quantity for a continuous symmetry is sometimes called the Noether charge, because the term charge is used in theoretical physics to designate conserved extensive observables. So, energy and momentum are Noether charges. ‘Electric charge’, ‘gravitational charge’ (i.e., mass) and ‘topological charge’ are other common examples. Challenge 493 s What is the conserved charge for rotation invariance? We note that the expression ‘energy is conserved’ has several meanings. First of all, it means that the energy of a single free particle is constant in time. Secondly, it means that the total energy of any number of independent particles is constant. Finally, it means that the energy of a system of particles, i.e., including their interactions, is constant in time. Collisions are examples of the latter case. Noether’s theorem makes all of these points at Challenge 494 e the same time, as you can verify using the corresponding Lagrangians. But Noether’s theorem also makes, or rather repeats, an even stronger statement: if energy were not conserved, time could not be defined. The whole description of nature requires the existence of conserved quantities, as we noticed when we introduced the Motion Mountain – The Adventure of Physics Page 27 concepts of object, state and environment. For example, we defined objects as permanent entities, that is, as entities characterized by conserved quantities. We also saw that the Page 238 introduction of time is possible only because in nature there are ‘no surprises’. Noether’s theorem describes exactly what such a ‘surprise’ would have to be: the non-conservation of energy. However, energy jumps have never been observed – not even at the quantum level. Since symmetries are so important for the description of nature, Table 36 gives an overview of all the symmetries of nature that we will encounter. Their main properties are also listed. Except for those marked as ‘approximate’, an experimental proof of in- copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net correctness of any of them would be a big surprise indeed – and guarantee eternal fame. Various speculations about additional symmetries exist; so far, all these speculations and quests for even more eternal fame have turned out to be mistaken. The list of symmet- ries is also the full list of universal statements, i.e., of statements about all observations, that scientists make. For example, when it is said that ‘‘all stones fall down’’ the state- ment implies the existence of time and space translation invariance. For philosophers interested in logical induction, the list is thus important also from this point of view. TA B L E 36 The known symmetries of nature, with their properties; also the complete list of logical inductions used in physics. Symmetry Type S pa c e G r o u p Pos - Con- Va - Main [num- of ac-topo- sible served cuum/ effect ber of tion logy rep- qua nt - m at - pa r a - r e s e nt- ity/ ter is met- ations charge sym- ers] metric Geometric or space-time, external, symmetries Time and space R × R3 space, not scalars, momentum yes/yes allow translation [4 par.] time compact vectors, and energy everyday 282 9 motion and symmetry TA B L E 36 (Continued) The known symmetries of nature, with their properties; also the complete list of logical inductions used in physics. Symmetry Type S pa c e G r o u p Pos - Con- Va - Main [num- of ac-topo- sible served cuum/ effect ber of tion logy rep- qua nt - m at - pa r a - r e s e nt- ity/ ter is met- ations charge sym- ers] metric Rotation SO(3) space 𝑆2 tensors angular yes/yes communi- [3 par.] momentum cation Galilei boost R3 [3 par.] space, not scalars, velocity of yes/for relativity time compact vectors, centre of low of motion tensors mass speeds Lorentz homogen- space- not tensors, energy- yes/yes constant eous Lie time compact spinors momentum light speed SO(3,1) 𝑇𝜇𝜈 Motion Mountain – The Adventure of Physics [6 par.] Poincaré inhomo- space- not tensors, energy- yes/yes ISL(2,C) geneous time compact spinors momentum Lie 𝑇𝜇𝜈 [10 par.] Dilation R+ [1 par.] space- ray 𝑛-dimen. none yes/no massless invariance time continuum particles Special R4 [4 par.] space- R4 𝑛-dimen. none yes/no massless conformal time continuum particles invariance copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Conformal [15 par.] space- involved massless none yes/no light cone invariance time tensors, invariance spinors Dynamic, interaction-dependent symmetries: gravity 1/𝑟2 gravity SO(4) config. as SO(4) vector pair perihelion yes/yes closed [6 par.] space direction orbits Diffeomorphism [∞ par.] space- involved space- local yes/no perihelion invariance time times energy– shift momentum Dynamic, classical and quantum-mechanical motion symmetries Parity (‘spatial’) discrete Hilbert discrete even, odd P-parity yes/no mirror inversion P or phase world space exists Motion (‘time’) discrete Hilbert discrete even, odd T-parity yes/no reversibil- inversion T or phase ity space 9 motion and symmetry 283 TA B L E 36 (Continued) The known symmetries of nature, with their properties; also the complete list of logical inductions used in physics. Symmetry Type S pa c e G r o u p Pos - Con- Va - Main [num- of ac-topo- sible served cuum/ effect ber of tion logy rep- qua nt - m at - pa r a - r e s e nt- ity/ ter is met- ations charge sym- ers] metric Charge global, Hilbert discrete even, odd C-parity yes/no anti- conjugation C antilinear, or phase particles anti- space exist Hermitean CPT discrete Hilbert discrete even CPT-parity yes/yes makes field or phase theory space possible Motion Mountain – The Adventure of Physics Dynamic, interaction-dependent, gauge symmetries Electromagnetic [∞ par.] space of unim- unim- electric yes/yes massless classical gauge fields portant portant charge light invariance Electromagnetic Abelian Hilbert circle S1 fields electric yes/yes massless q.m. gauge inv. Lie U(1) space charge photon [1 par.] Electromagnetic Abelian space of circle S1 abstract abstract yes/no none duality Lie U(1) fields [1 par.] copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Weak gauge non- Hilbert as 𝑆𝑈(3) particles weak no/ Abelian space charge approx. Lie SU(2) [3 par.] Colour gauge non- Hilbert as 𝑆𝑈(3) coloured colour yes/yes massless Abelian space quarks gluons Lie SU(3) [8 par.] Chiral discrete fermions discrete left, right helicity approxi- ‘massless’ symmetry mately fermions𝑎 Permutation symmetries Particle discrete Fock discrete fermions none n.a./yes Gibbs’ exchange space and paradox etc. bosons Vol. III, page 320 For details about the connection between symmetry and induction, see later on. The explanation of the terms in the table will be completed in the rest of the walk. The real numbers are denoted as 𝑅. 𝑎. Only approximate; ‘massless’ means that 𝑚 ≪ 𝑚Pl , i.e., that 𝑚 ≪ 22 μg. 284 9 motion and symmetry Parit y inversion and motion reversal The symmetries in Table 36 include two so-called discrete symmetries that are important for the discussion of motion. The first symmetry is parity invariance for objects or processes under spatial inversion. The symmetry is also called mirror invariance or right-left symmetry. Both objects and processes can be mirror symmetric. A single glove or a pair of scissors are not mirror- symmetric. How far can you throw a stone with your other hand? Most people have a preferred hand, and the differences are quite pronounced. Does nature have such a right- left preference? In everyday life, the answer is clear: everything that exists or happens in one way can also exist or happen in its mirrored way. Numerous precision experiments have tested mirror invariance; they show that ⊳ Every process due to gravitation, electricity or magnetism can also happen in a mirrored way. Motion Mountain – The Adventure of Physics There are no exceptions. For example, there are people with the heart on the right side; there are snails with left-handed houses; there are planets that rotate the other way. Astro- nomy and everyday life – which are governed by gravity and electromagnetic processes – are mirror-invariant. We also say that gravitation and electromagnetism are parity in- variant. Later we will discover that certain rare processes not due to gravity or electro- Vol. V, page 245 magnetism, but to the weak nuclear interaction, violate parity. Mirror symmetry has two representations: ‘+ or singlet’, such as mirror-symmetric objects, and ‘− or doublet’, such as handed objects. Because of mirror symmetry, scalar quantities can thus be divided into true scalars, like temperature, and pseudo-scalars, like magnetic flux or magnetic charge. True scalars do not change sign under mirror copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net reflection, whereas pseudo-scalars do. In the same way, one distinguishes true vectors, or polar vectors, such as velocity, from pseudo-vectors, or axial vectors, like angular velocity, angular momentum, torque, vorticity and the magnetic field. (Can you find an example Challenge 495 ny of a pseudo-tensor?) The other discrete symmetry is motion reversal. It is sometimes also called, falsely, ‘time reversal’. Can phenomena happen backwards? Does reverse motion trace out the forward path? This question is not easy. Exploring motion due to gravitation shows that such motion can always also happen in the reverse direction. (Also for motion reversal, observables belong either to a + or to a − representation.) In the case of motion due to electricity and magnetism, such as the behaviour of atoms in gases, solids and liquids, the question is more involved. Can broken objects be made to heal? We will discuss the issue in the section on thermodynamics, but we will reach the same conclusion, despite the appearance of the contrary: ⊳ Every motion due to gravitation, electricity or magnetism can also happen in the reverse direction. Motion reversion is a symmetry for all processes due to gravitation and electromagnetic interaction. Everyday motion is reversible. And again, certain even rarer nuclear pro- cesses will provide exceptions. 9 motion and symmetry 285 Interaction symmetries In nature, when we observe a system, we can often neglect the environment. Many pro- cesses occur independently of what happens around them. This independence is a phys- ical symmetry. Given the independence of observations from the details occurring in the environment, we deduce that interactions between systems and the environment de- crease with distance. In particular, we deduce that gravitational attraction, electric attrac- tion and repulsion, and also magnetic attraction and repulsion must vanish for distant Challenge 496 e bodies. Can you confirm this? Gauge symmetry is also an interaction symmetry. We will encounter them in our exploration of quantum physics. In a sense, these symmetries are more specific cases of the general decrease of interactions with distance. Curiosities and fun challenges ab ou t symmetry Right-left symmetry is an important property in everyday life; for example, humans prefer faces with a high degree of right-left symmetry. Humans also prefer that objects on Motion Mountain – The Adventure of Physics the walls have shapes that are right-left symmetric. It turns out that the eye and the brain has symmetry detectors built in. They detect deviations from perfect right-left symmetry. ∗∗ What is the path followed by four turtles starting on the four angles of a square, if each of them continuously walks, at constant speed, towards the next one? How long is the Challenge 497 s distance they travel? ∗∗ copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 498 s What is the symmetry of a simple oscillation? And of a wave? ∗∗ Challenge 499 s For what systems is motion reversal a symmetry transformation? ∗∗ Challenge 500 s What is the symmetry of a continuous rotation? ∗∗ A sphere has a tensor for the moment of inertia that is diagonal with three equal numbers. The same is true for a cube. Can you distinguish spheres and cubes by their rotation Challenge 501 s behaviour? ∗∗ Challenge 502 s Is there a motion in nature whose symmetry is perfect? ∗∗ Can you show that in two dimensions, finite objects can have only rotation and reflec- tion symmetry, in contrast to infinite objects, which can have also translation and glide- reflection symmetry? Can you prove that for finite objects in two dimensions, if no ro- tation symmetry is present, there is only one reflection symmetry? And that all possible 286 9 motion and symmetry rotations are always about the same centre? Can you deduce from this that at least one Challenge 503 e point is unchanged in all symmetrical finite two-dimensional objects? ∗∗ Challenge 504 s Which object of everyday life, common in the 20th century, had sevenfold symmetry? ∗∗ Here is little puzzle about the lack of symmetry. A general acute triangle is defined as a triangle whose angles differ from a right angle and from each other by at least 15 degrees. Challenge 505 e Show that there is only one such general triangle and find its angles. ∗∗ Can you show that in three dimensions, finite objects can have only rotation, reflection, inversion and rotatory inversion symmetry, in contrast to infinite objects, which can have also translation, glide-reflection, and screw rotation symmetry? Can you prove that for finite objects in three dimensions, if no rotation symmetry is present, there is only one Motion Mountain – The Adventure of Physics reflection plane? And that for all inversions or rotatory inversions the centre must lie on a rotation axis or on a reflection plane? Can you deduce from this that at least one point Challenge 506 e is unchanged in all symmetrical finite three-dimensional objects? Summary on symmetry Symmetry is invariance to change. The simplest symmetries are geometrical: the point symmetries of flowers or the translation symmetries of infinite crystals are examples. All the possible changes that leave a system invariant – i.e., all possible symmetry transform- ations of a system – form a mathematical group. Apart from the geometrical symmetry copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net groups, several additional symmetry groups appear for motion itself. Motion is universal. Any universality statement implies a symmetry. The reproducib- ility and predictability of nature implies a number of fundamental continuous symmet- ries: since we can talk about nature we can deduce that above all, nature is symmetrical under time and space translations and rotations. Space-time symmetries form a group. More precisely, they form a continuous symmetry group. From nature’s continuous symmetries, using Noether’s theorem, we can deduce con- served ‘charges’. These are energy, linear momentum and angular momentum. They are described by real numbers. In other words, the definition of mass, space and time, to- gether with their symmetry properties, is equivalent to the conservation of energy and momenta. Conservation and symmetry are two ways to express the same property of nature. To put it simply, our ability to talk about nature means that energy, linear mo- mentum and angular momentum are conserved and described by numbers. Additionally, there are two fundamental discrete symmetries about motion: first, everyday observations are found to be mirror symmetric; secondly, many simple motion examples are found to be symmetric under motion reversal. Finally, the isolability of systems from their surroundings implies that interactions must have no effect at large distances. The full list of nature’s symmetries also includes gauge symmetry, particle exchange symmetry and certain vacuum symmetries. All aspects of motion, like all components of a symmetric system, can be classified 9 motion and symmetry 287 by their symmetry behaviour, i.e., by the multiplet or the representation to which they belong. As a result, observables are either scalars, vectors, spinors or tensors. An fruitful way to formulate the patterns and ‘laws’ of nature – i.e., the Lagrangian of a physical system – has been the search for the complete set of nature’s symmetries first. For example, this is helpful for oscillations, waves, relativity, quantum physics and quantum electrodynamics. We will use the method throughout our walk; in the last part of our adventure we will discover some symmetries which are even more mind-boggling than those of relativity and those of interactions. In the next section, though, we will move on to the next approach for a global description of motion. Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net C h a p t e r 10 SI M PL E MOT ION S OF E X T E N DE D B ODI E S – O S C I L L AT ION S A N D WAV E S T he observation of change is a fundamental aspect of nature. Among all hese observations, periodic change is frequent around us. Indeed, hroughout everyday life be observe oscillations and waves: Talking, singing, hear- ing and seeing would be impossible without them. Exploring oscillations and waves, the next global approach to motion in our adventure, is both useful and beautiful. Motion Mountain – The Adventure of Physics Oscillations Page 248 Oscillations are recurring changes, i.e., cyclic or periodic changes. Above, we defined ac- tion, and thus change, as the integral of the Lagrangian, and we defined the Lagrangian as the difference between kinetic and potential energy. All oscillating systems periodic- ally exchange one kind of energy with the other. One of the simplest oscillating systems in nature is a mass 𝑚 attached to a (linear) spring. The Lagrangian for the mass position 𝑥 is given by 𝐿 = 12 𝑚𝑣2 − 12 𝑘𝑥2 , (93) copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net where 𝑘 is a quantity characterizing the spring, the so-called spring constant. The Lag- rangian is due to Robert Hooke, in the seventeenth century. Can you confirm the expres- Challenge 507 e sion? The motion that results from this Lagrangian is periodic, and illustrated in Figure 203. The Lagrangian (93) thus describes the oscillation of the spring length over time. The motion is exactly the same as that of a long pendulum at small amplitude. The motion is called harmonic motion, because an object vibrating rapidly in this way produces a completely pure – or harmonic – musical sound. (The musical instrument producing the purest harmonic waves is the transverse flute. This instrument thus gives the best idea of how harmonic motion ‘sounds’.) The graph of this harmonic oscillation, also called linear oscillation, shown in Fig- ure 203, is called a sine curve; it can be seen as the basic building block of all oscillations. All other, anharmonic oscillations in nature can be composed from harmonic ones, i.e., Page 292 from sine curves, as we shall see shortly. Any quantity 𝑥(𝑡) that oscillates harmonically is described by its amplitude 𝐴, its angular frequency 𝜔 and its phase 𝜑: 𝑥(𝑡) = 𝐴 sin(𝜔𝑡 + 𝜑) . (94) The amplitude and the phase depend on the way the oscillation is started. In contrast, oscillations and waves 289 A harmonically its position over time the corresponding oscillating phase object position period T amplitude A phase φ time period T An anharmonically oscillating object position period T amplitude A Motion Mountain – The Adventure of Physics time F I G U R E 203 Above: the simplest oscillation, the linear or harmonic oscillation: how position changes over time, and how it is related to rotation. Below: an example of anharmonic oscillation. the angular frequency 𝜔 is an intrinsic property of the system. Can you show that for the copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net Challenge 508 s mass attached to the spring, we have 𝜔 = 2π𝑓 = 2π/𝑇 = √𝑘/𝑚 ? Every harmonic oscillation is thus described by just three quantities: the amplitude, the period (the inverse of the frequency) and the phase. The phase, illustrated in Fig- ure 205, distinguishes oscillations of the same amplitude and period 𝑇; the phase defines at what time the oscillation starts. Some observed oscillation frequencies are listed in Table 37. Figure 203 shows how a harmonic oscillation is related to an imaginary rotation. As a result, the phase is best described by an angle value between 0 and 2π. Damping Every oscillating motion continuously transforms kinetic energy into potential energy and vice versa. This is the case for the tides, the pendulum, or any radio receiver. But many oscillations also diminish in time: they are damped. Systems with large damping, such as the shock absorbers in cars, are used to avoid oscillations. Systems with small damping are useful for making precise and long-running clocks. The simplest meas- ure of damping is the number of oscillations a system takes to reduce its amplitude to 1/𝑒 ≈ 1/2.718 times the original value. This characteristic number is the so-called Q- factor, named after the abbreviation of ‘quality factor’. A poor Q-factor is 1 or less, and an extremely good one is 100 000 or more. (Can you write down a simple Lagrangian for Challenge 509 ny a damped oscillation with a given Q-factor?) In nature, damped oscillations do not usu- 290 10 simple motions of extended bodies TA B L E 37 Some sound frequency values found in nature. O b s e r va t i o n Frequency Sound frequencies in gas, emitted by black holes c. 1 fHz Precision in measured vibration frequencies of the Sun down to 2 nHz Vibration frequencies of the Sun down to c. 300 nHz Vibration frequencies that disturb gravitational radiation down to 3 μHz detection Lowest vibration frequency of the Earth Ref. 218 309 μHz Resonance frequency of stomach and internal organs 1 to 10 Hz (giving the ‘sound in the belly’ experience) Common music tempo 2 Hz Frequency used for communication by farting fish c. 10 Hz Sound produced by loudspeaker sets (horn, electro- c. 18 Hz to over 150 kHz magetic, piezoelectric, electret, plasma, laser) Sound audible to young humans 20 Hz to 20 kHz Motion Mountain – The Adventure of Physics Hum of electrical appliances, depending on country 50 or 60 Hz Fundamental voice frequency of speaking adult human 85 Hz to 180 Hz male Fundamental voice frequency of speaking adult human 165 Hz to 255 Hz female Official value, or standard pitch, of musical note ‘A’ or 440 Hz ‘la’, following ISO 16 (and of the telephone line signal in many countries) Common values of musical note ‘A’ or ‘la’ used by or- 442 to 451 Hz copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net chestras Wing beat of tiniest flying insects c. 1000 Hz Fundamental sound frequency produced by the feathers 1 to 1.5 kHz of the club-winged manakin, Machaeropterus deliciosus Fundamental sound frequency of crickets 2 kHz to 9 kHz Quartz oscillator frequencies 20 kHz up to 350 MHz Sonar used by bats up to over 100 kHz Sonar used by dolphins up to 150 kHz Sound frequency used in ultrasound imaging 2 to 20 MHz Phonon (sound) frequencies measured in single crystals up to 20 THz and more oscillations and waves 291 F I G U R E 204 The interior of a commercial quartz oscillator, a few millimetres in size, driven at high amplitude. (QuickTime ﬁlm © Microcrystal) ally keep constant frequency; however, for the simple pendulum this remains the case to a high degree of accuracy. The reason is that for a pendulum, the frequency does not Motion Mountain – The Adventure of Physics depend significantly on the amplitude (as long as the amplitude is smaller than about 20°). This is one reason why pendulums are used as oscillators in mechanical clocks. Obviously, for a good clock, the driving oscillation must not only show small damp- ing, but must also be independent of temperature and be insensitive to other external influences. An important development of the twentieth century was the introduction of quartz crystals as oscillators. Technical quartzes are crystals of the size of a few grains of sand; they can be made to oscillate by applying an electric signal. They have little temper- ature dependence and a large Q-factor, and therefore low energy consumption, so that precise clocks can now run on small batteries. The inside of a quartz oscillator is shown in Figure 204. copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net R esonance In most physical systems that are brought to oscillate by an external source, the resulting amplitude depends on the frequency. The selected frequencies for which the amplitude is maximal are called resonance frequencies or simply resonances. For example, the quartz oscillator of Figure 204, or the usual vibration frequencies of guitar strings or bells – shown in Figure 206 – are resonance frequencies. Usually, the oscillations at which a system will oscillate when triggered by a short hit will occur at resonance frequencies. Most musical instruments are examples. Almost all systems have several resonance frequencies; flutes, strings and bells are well-known examples. In contrast to music or electronics, in many other situations resonance needs to be avoided. In buildings, earthquakes can trigger resonances; in bridges, the wind can trig- ger resonant oscillations; similarly, in many machines, resonances need to be dampened or blocked in order to avoid the large amplitude of a resonance destroying the system. In modern high-quality cars, the resonances of each part and of each structure are cal- culated and, if necessary, adjusted in such a way that no annoying vibrations disturb the driver or the passenger. All systems that oscillate also emit waves. In fact, resonance only occurs because all os- cillations are in fact localized waves. Indeed, oscillations only appear in extended systems; 292 10 simple motions of extended bodies A harmonic wave displacement crest or peak wavelength λ or maximum amplitude A node node node space wavelength λ trough or minimum An example of anharmonic signal Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 205 Top: the main properties of a harmonic wave, or sine wave. Bottom: A general periodic signal, or anharmonic wave – here a black square wave – can be decomposed uniquely into simplest, or harmonic waves. The ﬁrst three components (green, blue and red) and also their intermediate sum (black dotted line) are shown. This is called a Fourier decomposition and the general method to do this Fourier analysis. (© Wikimedia) Not shown: the unique decomposition into harmonic waves is even possible for non-periodic signals. Hum Prime Tierce F I G U R E 206 The measured fundamental vibration patterns of a bell. Bells – like every other source of oscillations, be it an atom, a molecule, a music instrument or the human voice – show that all oscillations in nature are due to waves. (© H. Spiess & al.). oscillations and waves 293 Motion Mountain – The Adventure of Physics F I G U R E 207 The centre of the grooves in an old vinyl record show the amplitude of the sound pressure, averaged over the two stereo channels (scanning electron microscope image by © Chris Supranowitz/University of Rochester). copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net and oscillations are only simplified descriptions of the repetitive motion of an extended system. The complete and general repetitive motion of an extended system is the wave. Waves: general and harmonic Waves are travelling imbalances, or, equivalently, travelling oscillations of a substrate. Waves travel, though the substrate does not. Every wave can be seen as a superposition of harmonic waves. Every sound effect can be thought of as being composed of harmonic waves. Harmonic waves, also called sine waves or linear waves, are the building blocks of which all internal motions of an extended body are constructed, as shown in Figure 205. Can you describe the difference in wave shape between a pure harmonic tone, a musical Challenge 510 e sound, a noise and an explosion? Every harmonic wave is characterized by an oscillation frequency 𝑓, a propagation or phase velocity 𝑐, a wavelength 𝜆, an amplitude 𝐴 and a phase 𝜑, as can be deduced from Figure 205. Low-amplitude water waves are examples of harmonic waves – in contrast to water waves of large amplitude. In a harmonic wave, every position by itself performs a harmonic oscillation. The phase of a wave specifies the position of the wave (or a crest) at a given time. It is an angle between 0 and 2π. The phase velocity 𝑐 is the speed with which a wave maximum moves. A few examples 294 10 simple motions of extended bodies TA B L E 38 Some wave velocities. Wa v e Ve l o c i t y Tsunami around 0.2 km/s Sound in most gases 0.3 ± 0.1 km/s Sound in air at 273 K 0.331 km/s Sound in air at 293 K 0.343 km/s Sound in helium at 293 K 0.983 km/s Sound in most liquids 1.2 ± 0.2 km/s Seismic waves 1 to 14 km/s Sound in water at 273 K 1.402 km/s Sound in water at 293 K 1.482 km/s Sound in sea water at 298 K 1.531 km/s Sound in gold 4.5 km/s Sound in steel 5.8 to 5.960 km/s Motion Mountain – The Adventure of Physics Sound in granite 5.8 km/s Sound in glass (longitudinal) 4 to 5.9 km/s Sound in beryllium (longitudinal) 12.8 km/s Sound in boron up to 15 km/s Sound in diamond up to 18 km/s Sound in fullerene (C60 ) up to 26 km/s Plasma wave velocity in InGaAs 600 km/s Light in vacuum 2.998 ⋅ 108 m/s copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net are listed in Table 38. Can you show that frequency and wavelength in a wave are related Challenge 511 e by 𝑓𝜆 = 𝑐? Waves appear inside all extended bodies, be they solids, liquids, gases or plasmas. In- side fluid bodies, waves are longitudinal, meaning that the wave motion is in the same direction as the wave oscillation. Sound in air is an example of a longitudinal wave. Inside solid bodies, waves can also be transverse; in that case the wave oscillation is perpendic- ular to the travelling direction. Waves appear also on interfaces between bodies: water–air interfaces are a well-known case. Even a saltwater–freshwater interface, so-called dead water, shows waves: they can appear even if the upper surface of the water is immobile. Any flight in an aero- plane provides an opportunity to study the regular cloud arrangements on the interface between warm and cold air layers in the atmosphere. Seismic waves travelling along the boundary between the sea floor and the sea water are also well-known. Low-amplitude water waves are transverse; however, general surface waves are usually neither longitud- inal nor transverse, but of a mixed type. To get a first idea about waves, we have a look at water waves. oscillations and waves 295 Water surface: At a depth of half the ave w len gth, the amplitude is negligible F I G U R E 208 The formation of the shape of deep gravity waves, on and under water, from the circular motion of the water particles. Note the non-sinusoidal shape of the wave. Motion Mountain – The Adventure of Physics Water waves Water waves on water surfaces show a large range of fascinating phenomena. First of all, there are two different types of surface water waves. In the first type, the force that re- stores the plane surface is the surface tension of the wave. These so-called surface tension waves play a role on scales up to a few centimetres. In the second, larger type of waves, the restoring force is gravity, and one speaks of gravity waves.* The difference is easily Page 293 noted by watching them: surface tension waves have a sinusoidal shape, whereas gravity copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net waves have a shape with sharper maxima and broader troughs. This occurs because of the special way the water moves in such a wave. As shown in Figure 208, the surface wa- ter for a (short) gravity water wave moves in circles; this leads to the typical wave shape with short sharp crests and long shallow troughs: the waves are not up–down symmetric. Under the crests, the water particles move in the direction of the wave motion; under the troughs, the water particles move against the wave motion. As long as there is no wind and the floor below the water is horizontal, gravity waves are symmetric under front-to- back reflection. If the amplitude is very high, or if the wind is too strong, waves break, because a cusp angle of more than 120° is not possible. Such waves have no front-to-back symmetry. In addition, water waves need to be distinguished according to the depth of the water, when compared to their wavelength. One speaks of short or deep water waves, when the depth of the water is so high that the floor below plays no role. In the opposite case one speaks of long or shallow water waves. The transitional region between the two cases are waves whose wavelength is between twice and twenty times the water depth. It turns out that all deep water waves and all ripples are dispersive, i.e., their speed depends on their frequency; only shallow gravity water waves are non-dispersive. Water waves can be generated by wind and storms, by earthquakes, by the Sun and * Meteorologists also know of a third type of water wave: there are large wavelength waves whose restoring force is the Coriolis force. 296 10 simple motions of extended bodies Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net F I G U R E 209 Three of the four main types of water waves. Top: a shallow water gravity wave, non-sinusoidal. Bottom left: a deep water ripple – a sinusoidal surface tension wave. The not-shown shallow water ripples look the same. Bottom right: a deep water gravity wave, here a boat wake, again non-sinusoidal. (© Eric Willis, Wikimedia, allyhook) Moon, and by any other effect that displaces water. The spectrum of water waves reaches from periods shorter than 100 ms to periods longer than 24 h. An overview is given in Table 39. The table also includes the lesser known infra-gravity waves, ultra-gravity waves,, tides and trans-tidal waves. The classification of periodic water waves according to their restoring force and to the influence of the floor give four limit cases; they are shown in Figure 210. Each of the four limit cases is interesting. Experiments and theory show that the phase speed of gravity waves, the lower two cases in Figure 210, depends on the wavelength 𝜆 and on the depth of the water 𝑑 in the following way: 𝑔𝜆 2π𝑑 𝑐 = √ tanh , (95) 2π 𝜆 where 𝑔 is the acceleration due to gravity (and an amplitude much smaller than the oscillations and waves 297 TA B L E 39 Spectrum of water waves. Type Period Propertie s Genera- tion Surface tension < 0.1 s wavelength below a few cm wind with more waves/capillary than 1 m/s, other waves/ripples disturbances, temperature Ultra-gravity waves 0.1 to 1 s restoring forces are surface wind, other dis- tension and gravity turbances Ordinary gravity 1 to 30 s amplitude up to many wind waves meters, restoring force is gravity Infra-gravity 30 s to 5 min amplitude up to 30 cm, wind, gravity waves/surf beat related to seiches, restoring waves force is gravity Long-period waves 5 min to 12 h amplitude typically below storms, earth- Motion Mountain – The Adventure of Physics 10 cm in deep water, up quakes, air to 40 m in shallow water, pressure changes restoring force is gravity and Coriolis effect Ordinary tides 12 h to 24 h amplitude depends on loca- Moon, Sun tion, restoring force is gravity and Coriolis effect Trans-tidal waves above 24 h amplitude depends on loca- Moon, Sun, tion, restoring force is gravity storms, seasons, copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net and Coriolis effect climate change wavelength is assumed*). The formula shows two limiting regimes. First, so-called deep water or short gravity waves appear when the water depth is larger than about half the wavelength. The usual sea wave is a deep water gravity wave, and so are the wakes generated by ships. Deep water gravity waves generated by wind are called sea they are generated by local winds, and swell if they are generated by distant winds. The typical phase speed of a gravity wave is of the order of the wind speed that generates it. For deep water waves, the phase velocity is related to the wavelength by 𝑐 ≈ √𝑔𝜆/2π ; the phase velocity is thus wavelength-dependent. In fact, all deep waves Page 301 are dispersive. Shorter deep gravity waves are thus slower. The group velocity is half the phase velocity. Therefore, as surfers know, waves on a shore that are due to a distant storm arrive separately: first the long period waves, then the short period waves. The general effects of dispersion on wave groups are shown in Figure 211. The typical wake generated by a ship is made of waves that have the phase velocity of the ship. These waves form a wave group, and it travels with half that speed. Therefore, from a ship’s point of view, the wake trails the ship. Wakes are behind the ship because * The expression for the phase velocity can be derived by solving for the motion of the liquid in the linear regime, but this leads us too far from our walk. 298 10 simple motions of extended bodies Water wave capillary waves or surface tension waves or ripples dispersion sinusoidal, of small amplitude, dispersive GAM ripples in shallow water ripples in deep water 2 4 𝜔 = 𝛾𝑑𝑘 /𝜌 CC 𝜔2 = 𝛾𝑘3 /𝜌 104 102 shallow water 𝜔2 = (𝑔𝑘 + 𝛾𝑘3 /𝜌) tanh 𝑘𝑑 deep water or long waves 𝑑𝑘 or short waves 10-4 10-2 1 102 104 dispersive 2 2 2 𝜔 = 𝑘 (𝑔𝑑 + 𝛾𝑑𝑘 /𝜌) 10-2 𝜔2 = 𝑔𝑘 + 𝛾𝑘3 /𝜌 10-4 𝜔2 = 𝑔𝑑𝑘2 𝜔2 = 𝑔𝑘 tanh 𝑘𝑑 𝜔2 = 𝑔𝑘 Motion Mountain – The Adventure of Physics tides, tsunamis wakes, sea, swell, etc. non-dispersive gravity waves non-sinusoidal, amplitude large, but still much smaller than wavelength (thus without cnoidal waves, Stokes waves or solitons) Water wave wavelength dispersion (m) tide copyright © Christoph Schiller June 1990–March 2023 free pdf ﬁle available at www.motionmountain.net 106 tsunami 𝑑𝑘 = 1 104 storm waves storm waves on open sea 2 at shore 10 depth (m