Seven Tales of the Pendulum 9780191004841, 9780199589517

Citation preview

SEVEN TALES OF THE PENDULUM

This page intentionally left blank

SEVEN TALES OF THE PENDULUM Gregory L. Baker Bryn Athyn College, Bryn Athyn, Pennsylvania

1

1

Great Clarendon Street, Oxford ox 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Gregory L. Baker 2011 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2011 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by SPI Publisher Services, Pondicherry, India Printed in Great Britain on acid-free paper by CPI Antony Rowe, Chippenham, Wiltshire ISBN 978–0–19–958951–7 1 3 5 7 9 10 8 6 4 2

Contents Preface Acknowledgements List of Figures Part I: Introduction

ix xi xiii 1

What is a Pendulum? I.1 Why the pendulum? I.2 The motion of the pendulum I.3 Picturing the motion

3 3 8 14

Part II: Seven Tales of the Pendulum

21

Tale 1: Pendulums Measure the Earth 1.1 Shape 1.2 Density 1.3 Rotation – the Foucault pendulum 1.4 Frames of reference 1.5 Foucault pendulums are tricky 1.5.1 Starting the pendulum 1.5.2 Finite swing size 1.5.3 Asymmetry 1.5.4 Energy dissipation 1.5.5 Some modern solutions 1.6 Pendulums go public

23 23 28 33 42 44 45 45 46 51 53 56

Tale 2: Pendulums Keep Time 2.1 Mechanical clocks pre-pendulum 2.2 The development of the pendulum clock 2.2.1 Galileo (1564–1642) 2.2.2 Huygens (1629–1695)

62 62 66 66 67

vi

Contents 2.3 2.4 2.5 2.6

Time as a standard of length? Escapements Temperature compensation Pendulum clocks into the twentieth century

74 76 79 82

Tale 3: Pendulums Measure Fundamental Forces 3.1 The historical torsion pendulum 3.1.1 Coulomb and the electrostatic force 3.1.2 Cavendish and the gravitational force 3.2 Modern torsion pendulums 3.2.1 The universal gravitational constant 3.2.2 The principle of equivalence (revisited)

87 91 91 97 101 101 103

Tale 4: Pendulums Swing High 4.1 A religious pendulum 4.2 Yes, the period does depend on the amplitude 4.3 Phase space revisited 4.4 Parametric forcing 4.5 Pendulum of horror

107 107 111 118 123 128

Tale 5: Pendulums Go Chaotic 5.1 Introduction and history 5.2 The chaotic pendulum 5.3 The geometry of chaos 5.3.1 The state diagram 5.3.2 Phase planes, Poincaré sections, time series 5.3.3 Fourier analysis 5.3.4 Bifurcation diagrams 5.4 Fractals 5.5 Information and prediction in chaos

131 131 136 139 139 142 147 150 152 156

Tale 6: Pendulums Stick Together 6.1 Synchronized pendulums 6.2 Secure communication 6.3 Control of the chaotic pendulum 6.4 Large-scale synchronization

160 160 166 169 171

Contents

vii

Tale 7: Pendulums in the Small 7.1 Potential energy 7.2 A new physics 7.3 Two simplified quantum systems 7.3.1 The linearized quantum pendulum 7.3.2 The free quantum rotor 7.4 The nonlinear quantum pendulum 7.5 Microscopic pendulums? 7.5.1 The Stark effect 7.5.2 Naturally librating molecules – ethane 7.6 Does quantum physics partially wash out chaos?

175 175 178 180 181 187 188 191 191 194 198

Part III: Conclusion

201

Odds and the End C.1 The double pendulum C.2 The cradle pendulum C.3 The relativistic pendulum C.4 The Long Now clock C.5 Some final thoughts

203 203 206 210 213 216

Further Reading Glossary of Terms Index

219 221 225

This page intentionally left blank

Preface In 2005 Oxford University Press published a unique physics textbook, The Pendulum: A Case Study In Physics, by Gregory Baker (this author) and James Blackburn. This new book, Seven Tales of the Pendulum, grew from that work. The subject of the original work was a single physical system – the pendulum – which exhibits remarkably varied and complex behavior under many different conditions. We used this single system to build a comprehensive description of the pendulum’s classical, chaotic, and quantum behavior into a single book. In the course of our research it became apparent to us that, beyond the physics, the pendulum has importance for the historical development of applied science, and as such, has left a significant imprint on human thought and culture. Many of these nonscientific, but culturally enriching, connections found their way into the text. The combination of these factors resulted in that publication being devoted to the science, history, and culture of the pendulum. The Pendulum was written as a text for students and researchers of mathematics, physics, and other quantitative sciences, and therefore utilized advanced mathematics. Yet, the diversity of the science of the pendulum and its importance in the history of human intellectual development are exciting stories that needed to be presented to a wider audience. Therefore, one of us (GLB) decided to write an almost nontechnical version, the book you have in hand. Seven Tales of the Pendulum is intended for the general reader with an interest in science and its relation to other human endeavors. It is an accessible description of the science of the pendulum, liberally sprinkled with related historical and cultural matters. What about the mathematical prerequisite? Almost all of the mathematical formulas from the old work have disappeared. Instead, I describe quantitative relationships with words, pictures, and graphs. Occasionally, I use “scientific” notation for numbers that are very big or very small. For example, the approximate size of an atom is about 0.0000000002 meters or, in scientific notation, 2 × 10−10 meters. The United States’ national debt is, at this writing, about 13 trillion dollars or, in scientific notation, 13×1012 dollars. Occasionally, we will use a letter or symbol to represent a quantity. For example, we use the Greek letter theta h to signify the

x

Preface

angle of a pendulum away from the downward vertical position. And very occasionally, a simple algebraic formula will be used to reinforce the meaning of the graphs and words. (I do sneak in the sine curve from trigonometry, but pictures should make its meaning clear.) Readers may rest easy knowing that I am mindful of the warning made famous by Stephen Hawking, that every formula reduces the readership by a factor of two. While this book relies heavily on The Pendulum: A Case Study in Physics, it is structured somewhat differently, with a clearer distinction between small-amplitude and large-amplitude motion. The material has been rearranged, completely edited, condensed in many places, and expanded in others. Furthermore, certain sections of the 2005 production that may be either too esoteric or too mathematical for the general reader are not included. For example, beyond passing mention, I do not describe the mathematical analogy of the damped driven pendulum with a superconducting Josephson junction, a topic that previously occupied a whole chapter. Similarly, the chapter on synchronization is now heavily edited to omit difficult technicalities. Yet many surprising features are still described in sufficient detail to give the reader a sense of that growing field of study. As behooves a popular work, detailed references are not included. Instead, there is a short section of “further reading” that points to generally accessible additional material. Those wishing references must consult the over two hundred sources in the 2005 book. Finally, there is a short glossary of terms. The passage of several years has only strengthened the conviction that the story of the pendulum is both interesting and important. It is part of our cultural and scientific history, providing roots of that history as well as pointing toward a future for the pendulum that is yet to be charted. It is most gratifying to be offering this story to a wider audience. Gregory L. Baker Huntingdon Valley Pennsylvania 2010

Acknowledgements Because this book rests heavily on The Pendulum: A Case Study in Physics, much credit must go to those who aided in that production. However, their contributions are recognized in that book and I refer the reader to the acknowledgments there. In this book, I would like to recognize individuals who contributed specifically to this new work. These include some who helped with the earlier work as well as new people. I am grateful to Jonathan Betts, senior curator of horology at the National Maritime Museum, Greenwich, for pointing me to certain documents on how early clocks were calibrated. Further information on clocks was kindly provided by Bob Holmstrom. I would like to thank many people who freely allowed me to again use diagrams or figures from the earlier work or that are new to this book. These include Dan Allen, John Byrd, Beryl Clotfelter, Richard Crane, Jens Gundlach, J. Carter Harris, John Lindner, Juan Sanmartin, Bill Underwood, Margaret Walker, and James Yorke. I would like to thank Steve Fazzio for allowing me to choose from among his lovely photographs of O Botafumeiro. Jason Gallas produced beautiful diagrams using his scientific expertise and many hours of computer time, for which I am very grateful. I also appreciate the help of organizations that freely allowed me to use new diagrams or those from the previous work. This group includes the American Association of Physics Teachers, the American Physical Society, the American Chemical Society, Cambridge University Press, the Long Now Foundation, the National Watch and Clock Museum (USA), Oxford University Press, the Rare Books and Special Collections section of Princeton University Library, and the Society of Exploration Geophysicists. I also acknowledge permissions for figure use from the National Maritime Museum, l’ Observatoire de Paris, and the Science Museum of London. The nature of this book provides a strong incentive for the insertion of many figures, taken from a variety of sources. I have made every effort to determine original sources and obtain permissions for the use of these illustrations. Credit is found in the figure captions. Figures that were generated either by James Blackburn or myself for The Pendulum: A Case Study in Physics are typically not given specific credit. New figures generated by me for this book are again not given specific credit.

xii

Acknowledgements

All reasonable effort has been made to contact the holders of copyright in materials reproduced in this book. Any omissions will be rectified in future printings if notice is given to the publisher. I appreciate help from David Perry of the Swedenborg Library at Bryn Athyn College in acquiring important materials. Various people read preliminary drafts of certain chapters and provided valuable comments. These include Thane Glenn, Robert Hilborn, Robert Johns, and Julian Clint Sprott. I feel a special sense of gratitude to Sonke Adlung, the senior physical science editor at Oxford University Press. His patience and encouragement fortified my resolve to search for, and hopefully find, a style and format that would be suitable for a popular audience. I also appreciate the help of April Warman as I worked through the publication process. I valued her clear and timely responses to my many questions. My coauthor with the previous book, colleague with other projects, and friend, James Blackburn, deserves special thanks. Many of the diagrams that he created for The Pendulum: A Case Study in Physics populate this new work. He also very kindly created new or improved artwork for this book. His aid, encouragement, and wise counsel have been a strong support for me as I took on this new challenge. I value his professional acumen and his abiding friendship. Last, and most deserving of my gratitude, is my wife, Margaret. She read every submission and every draft of the book, catching my grammatical slips, but more importantly, acting as my archetypical reader. Her persistence in asking that I explain things more clearly or more fully has been immeasurably important to creating an understandable and readable book. Without her gentle and determined insistence on continual improvement, this book would not have happened. And during the process, she has continued to be the wonderful partner that she has been for more than four decades. I am profoundly grateful for our life together.

List of Figures I.1 I.2 I.3 I.4 I.5 I.6 I.7 I.8 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16

An early primitive Chinese seismometer A portrait of Galileo, from his book on sunspots The cathedral at Pisa An idealized picture of the simple pendulum The dependence of the restoring force on the pendulum Time series of the angular displacement of the small-angle pendulum Phase-plane and phase-space representations of the pendulum’s motion A plot of probability versus angle The Kater reversing pendulum The little “bump” on the earth’s surface represents a whole mountain range A schematic of the earth with a mine shaft The Vening Meinesz pendulum A schematic of the crime scene in Aird’s novel A Foucault pendulum (similar to the original) hanging in the Pantheon, Paris Leon Foucault, 1819–68, inventor of the Foucault pendulum, at age 47 Public display of the Foucault pendulum Computer simulation of a few oscillations of the pendulum bob An accumulation of the pendulum bob’s path as it would appear from above H. Kamerlingh Onnes The support for the pendulum and a drawing depicting the complete apparatus An engraving of the electrical impulse mechanism used by Foucault The surviving remnant of Foucault’s original electromagnetic driver Two members of the south pole group with their Foucault pendulum bob An engraving from L’Illustration, Paris, 1851

5 6 7 9 11 15 16 18 26 29 31 33 35 36 37 39 40 47 48 49 52 53 54 57

xiv 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 4.1

List of Figures The Astronomical Clock in the town square of Prague The verge-and-foliot mechanism The clock from Dover Castle Galileo’s design for a pendulum clock A portrait of Christiaan Huygens The basic arrangement for Huygens’ pendulum clock escapement Huygens’ clock design, as illustrated in the Horologium Oscillatorium (1673) A pendulum whose suspension cord is constrained between two cycloidal cheeks An anchor escapement consisting of an escape wheel and pallet The dead-beat escapement The principle of the gridiron temperature compensator The arrangement of metal rods in a gridiron pendulum The Number 16 Shortt clock, at the National Maritime Museum, Greenwich A Shortt clock, showing both master and slave mechanisms A photograph showing the mechanisms of both master and slave units A typical torsion pendulum A view looking down on a torsion pendulum twisted through an angle h The arrangement of the extra fixed masses, M, for the time-of-swing method A schematic of Coulomb’s torsion balance Coulomb’s torsion balance (1785) Three data points from Coulomb’s original experiment, and a fitted inverse-square law relationship A portrait of Henry Cavendish A schematic arrangement of a Cavendish torsion balance The original apparatus used by Henry Cavendish in 1798 The apparatus used by Gundlach and Merkowitz to determine Newton’s gravitational constant, G The 1990 torsion pendulum The cathedral of Santiago de Compostela in northern Spain

63 64 65 68 69 70 71 73 77 79 80 81 83 84 85 88 90 91 95 96 97 98 99 100 102 105 108

List of Figures 4.2 O Botafumeiro, the giant censer that hangs in the transept of the cathedral of Santiago de Compostela 4.3 The support structure for the rope holding O Botafumeiro 4.4 The period of oscillation for a rigid pendulum versus amplitude of swing 4.5 A spectrum of the amplitudes of the simple sine-wave components in the motion of a nonlinear pendulum 4.6 The response of the pendulum to forcing at its small-angle resonant frequency 4.7 A graph of the period versus the energy of the pendulum 4.8 A phase diagram showing possible orbits for a pendulum 4.9 A phase-space diagram for the damped, undriven pendulum 4.10 The relative frequency of occurrence of angle for period-2 and period-15 motion 4.11 A schematic diagram of the path of O Botafumeiro (or a playground swing) during a cycle 4.12 The size of the angular amplitude versus the number of pumps 4.13 The oscillation amplitude for each pumping cycle 4.14 A depiction of the victim in Poe’s story “The Pit and the Pendulum” 5.1 A photograph of the Blackburn–Smith pendulum 5.2 A comparison of data from the experimental pendulum with simulated data 5.3 Corresponding state diagrams for a pendulum with fixed damping Q, but variable forcing amplitude A and variable forcing frequency xD 5.4 Three examples of complex, yet periodic, motions of the driven pendulum 5.5 Time series for the pendulum’s angular velocity 5.6 Three geometric representations of the chaotic pendulum 5.7 The Fourier spectrum for a chaotic pendulum 5.8 The frequency distribution for the angle of the chaotic pendulum 5.9 A bifurcation diagram

xv 109 110 112 114 116 117 119 120 122 124 126 127 129 137 138

141 143 144 146 148 150 151

xvi

List of Figures

5.10 The Cantor set is produced by removing the middle third from each segment an infinite number of times 5.11 Increasing magnifications of the Poincaré section of the chaotic pendulum 5.12 Two angular-velocity time series for pendulums with a difference between their initial angles of one part in one hundred million 6.1 Two sets of synchronized pendulum clocks 6.2 Magnetically coupled damped, driven pendulums 6.3 The lower curve shows that after a few cycles, the two pendulums come into synchronization 6.4 Time series of the message, its transmission with a chaotic carrier, and its reception for a strongly coupled transmitter and receiver 6.5 Time series of the message, its transmission with a chaotic carrier, and its reception for a weakly coupled transmitter and receiver 6.6 Time series of the pendulum’s angular velocity 6.7 A close-up view of torsionally coupled pendulums 6.8 Experimental spatiotemporal angular velocity plots 7.1 The restoring force that acts upon a simple gravity pendulum 7.2 The potential energy of both a linearized pendulum and a nonlinear pendulum 7.3 The probability density functions for the lowest four energy states of the linearized pendulum 7.4 Classical and quantum-mechanical probability density functions for the ground state of the linearized pendulum 7.5 Classical and quantum probability density functions for the fourth state 7.6 A plot of energy levels versus q 7.7 Energy levels and probabilities for the nonlinear pendulum 7.8 The molecular structure of ethane in the “eclipsed” form 7.9 The molecular structure of ethane in “staggered” form 7.10 The fraction (F) of ethane molecules above the potential well

154 155

158 162 164 165

167

168 170 172 173 176 177 183

185 186 190 193 194 195 198

List of Figures C.1 A compound pendulum with an arbitrary distribution of mass C.2 A hand-cranked double pendulum C.3 A schematic diagram of the mathematical model of the double pendulum C.4 The chaotic time series of the angular velocity of the lower mass of the double pendulum C.5 The typical arrangement of Newton’s cradle C.6 The curve shows Hertz’s law for the interaction of colliding spheres C.7 Velocity time series for the nonrelativistic and relativistic pendulums C.8 An overall view of the Long Now clock C.9 A close-up view of the triple mass torsion pendulum that is the heart of the Long Now clock

xvii 204 205 205 207 208 209 212 214 215

This page intentionally left blank

PART I

INTRODUCTION “That was when I saw the pendulum. The sphere, hanging from a long wire set in the ceiling of the choir, swayed back and forth with isochronal majesty. . . . I knew the earth was rotating, and I with it, and Saint-Martin-des-Champs and all Paris with me, and that together we were rotating beneath the Pendulum, whose own plane never changed direction, because up there, along the infinite extrapolation of its wire beyond the choir ceiling, up toward the most distant galaxies lay the Only Fixed Point in the universe eternally unmoving.” Umberto Eco, Foucault’s Pendulum (1988)

This page intentionally left blank

What is a Pendulum? I.1 Why the pendulum? In an age in which microcircuits in computers and satellite communication around the world are the norm, a discussion about the pendulum might seem to be just an exercise in nostalgia. Inexpensive modern timepieces use a quartz crystal as their precision oscillator, while very expensive modern clocks sustain extreme precision using resonant states of atoms. Furthermore, much of our information about the earth now comes from satellite motion, and even the complex mechanical earth-measuring pendulums of the early twentieth century are now museum pieces. Except for backyard swings, the pendulum is hardly a common household object. Yet, closer examination shows that the pendulum, or often its mathematical idealization, is still worthy of our scientific interest. The pendulum is one of the few examples of the new science of nonlinear dynamics and chaos that is amenable to both theoretical and experimental study. The pendulum is also an accessible system for the study of synchronization in large and small networks. Furthermore, at the microscopic scale, the pendulum can be treated as aquantum-mechanical system. As such, it continues to live as a model for molecular motions. (And although it was felt to be outside the scope of this general work, under appropriate conditions a superconducting device, known as a Josephson junction, is effectively modeled by the nonlinear pendulum.) Beyond the issue of scientific relevance lie questions about the pendulum’s cultural and historical importance. In this book, we will see examples of the pendulum as a literary device, especially as it portends inexorable regularity and possible doom. The simple regularity of pendulums may convey a sense of fatalism, a bounded human existence. Even today, the pendulum is used by some practitioners of the occult as a diagnostic tool for personality analysis or even as means of foretelling the future.

4

Introduction

The importance of the pendulum in the history of science is well established. Galileo used it as one of his primary experimental tools. Its role in the exploration of the properties of the earth, the demonstration of the rotation of the earth, and its primacy as a clock regulator all point to its historical importance in science. In a more general sense, these scientific advancements also signaled or contributed to economic and social progress. One obvious example is that increasing precision in timekeeping has brought steady technological advancement. And while early pendulum measurements of the density of the earth or its gravitational field may have been crude, their more sophisticated successors allow us to explore intelligently for oil and to understand patterns of flow in the oceans. Today, the pendulum is still a somewhat familiar object. Its most common appearance is in old-fashioned clocks that, even in this era of quartz timepieces and atomic clocks, remain quite popular. Much of the pendulum’s fascination comes from the well-known regularity of its swing. Older students of music are very familiar with the adjustable (and sometimes annoying) regularity of that inverted ticking pendulum known as a metronome. And, of course, there is the ubiquitous playground swing. The pendulum’s influence has extended even to the arts. A sampling of works includes the novel by Umberto Eco, Foucault’s Pendulum, an award-winning Belgian film, Le pendule de Madame Foucault, a piece of music entitled Pendulum Music, by the minimalist composer Steve Reich, that uses feedback from swinging microphones, and finally, Edgar Allen Poe’s famous story of terror, The Pit and the Pendulum. Probably no one knows when pendulums first impinged upon human consciousness. One of the earliest known, slightly practical, scientific manifestations of the pendulum goes back about two millennia, to the Han Dynasty in China (202 B.C. to A.D. 220). In A.D. 132, a renowned government mathematician and scientist, Chang Heng (A.D. 78–139), constructed a crude seismometer to meet the need for faster and somewhat directional reporting of earthquakes (see Figure I.1). At the time, people believed that earthquakes had a supernatural origin, as punishment inflicted by angry gods. Being more practical, Chang Heng felt that he could provide a scientific explanation through careful categorization of seismic events. To implement his plan, he needed an instrument that would give him rapid and better information about the occurrence and location of earthquakes.

What is a Pendulum?

5

Figure I.1 An early primitive Chinese seismometer.

Although Heng’s original device no longer exists, it is surmised that inside the main container was a pendulum, possibly upside down (inverted), that was sensitive to earthquake motion. Figure I.1 is a photo of a plausible reproduction of Heng’s seismometer. The size of the container, about 1.8 meters, was a serendipitous choice. It was about the right size to resonate with, and therefore detect, sub-audio signals from a distant quake. Through a series of levers, the pendulum’s motion would cause a ball to drop out of one of the dragon heads on the container and into the mouth of one of the eight metal toads that circumscribed the container. Each toad was positioned at one of eight points of the compass. The position of the chosen toad indicated the approximate direction of the earthquake. Heng would receive his directional data in a timely manner and aid to the victims could be dispatched in the general direction of the quake. We often associate the first modern scientific observations of the pendulum with Galileo Galilei (1564–1642) (Figure I.2). The famous Galileo pendulum story (perhaps apocryphal) has Galileo sitting in the cathedral at Pisa. He observed a lamplighter

6

Introduction

Figure I.2 A portrait of Galileo, from his book on sunspots.

pushing one of the swaying pendular chandeliers of the type shown in Figure I.3. His earliest biographer, Vincenzo Viviani, suggests that Galileo then timed the swings using his pulse and concluded that, even as the size or amplitude of the swings diminished, the time or period of each swing was the same. This constancy of period with changing amplitude is called isochronism. According to Viviani, Galileo’s observations of approximate isochronism were made in 1583. A prominent modern Galileo scholar, Stillman Drake (1910–93), wrote that guides at the cathedral refer visitors to a certain lamp, which they describe as “Galileo’s lamp”. Unfortunately, this suggestion does not seem plausible as that particular lamp was not installed until late in 1587. Nevertheless, earlier swaying lamps undoubtedly did exist.

What is a Pendulum?

7

Figure I.3 The cathedral at Pisa. The thin vertical wire indicates a hanging chandelier.

Drake also surmised that Galileo actually first came to the insight about fixed vibrations, and therefore isochronism, in connection with his father’s musical instruments. He saw that a musical instrument could hold a fixed pitch despite variations in volume of air, and thought that a similar effect should occur with the pendulum. Pendulums should, he thought, keep the same period even as the swing decreased in size. Then later, perhaps in 1588, he associated isochronism with his earlier 1583 pendulum observations in the cathedral. However, Galileo did make systematic observations of pendulums in 1602. These observations did not really confirm his earlier conclusion about swings of differing amplitude. But despite the lack of firm experimental confirmation, Galileo clung to his earlier conclusion of isochronism. Aside from his earlier preconception, he believed that the universe had been ordered so that motion should be simple. In his view, there was “no reason” for the

8

Introduction

longer path to take a longer time than the shorter path. While this famous conclusion by Galileo about the pendulum has only partial legitimacy, the importance of Galileo’s work stems (a) from its being the first known scientific measurement of the pendulum’s behavior, (b) from the measurement leading to a conclusion (isochronism), and (c) from the fact that Galileo’s work on pendulums makes up part of the history of early research in physical science. In this way, the pendulum begins its history as a significant model in physical science. With Galileo’s work, we see the initial stages of the transition from wanting to know “why” to wanting to know “how” something works: from a philosophic view to a mechanical view. In the rest of this Introduction, we describe the pendulum in more detail and establish some properties that will be needed for our seven tales. (Further characteristics will be provided in individual tales as needed.) If the concepts or types of presentation in this section are new to the reader, then a small investment of careful thought may be required. But the effort will be rewarded by making what follows in the seven tales easily understood. Once mastered, the general properties of the pendulum provide a basis for understanding each of the following tales independently. Of course, the story is richer when taken in its entirety. Let us begin our consideration of the science of the pendulum by looking at various ways in which we can view the motion of the pendulum, mostly through pictorial representations.

I.2 The motion of the pendulum The simple pendulum shown in Figure I.4 is an idealization of a real pendulum, such as any chandelier that Galileo might have observed. In Figure I.4, the round ball at the bottom is called the “bob” of the pendulum. For the simplest model, we can think of all the mass of the pendulum (denoted m) as concentrated at a single point, the center of the bob. The suspension point is at the top of the pendulum and is referred to as the pivot point, or simply as the “pivot”. The pivot is sometimes idealized as a frictionless mechanism, but in reality, this can be a source of friction and can help to gradually decrease the angular displacement and the motion of the pendulum. In most of our discussions, the pendulum is idealized as rigid, with a fixed length shown as l

What is a Pendulum?

9

l θ

m Frestoring Fgravity

Figure I.4 An idealized picture of the simple pendulum.

in the figure. For simplicity, the material of the suspending length is considered to have a mass that is negligible compared to that of the bob. Of course, most pendulums are more complicated. (These are often called compound pendulums; the mass is not concentrated at a point, or the length is not fixed or rigid.) But most of our examples can be treated, at least approximately, as simple pendulums of the type described. The arrows in the figure show the direction of the forces acting on the bob. Gravity pulls the bob downward. At equilibrium, the pendulum just hangs straight down, motionless. But when the pendulum is pulled to the side and released, the bob is constrained to move along a circular path. If the bob has some speed, it will overshoot the downward position and continue upward again, thereby causing the motion to repeat. Gravity is only able to “restore” the pendulum to its equilibrium position by forcing the pendulum along the circle. So Frestoring is only that amount of the gravitational force Fgravity that is directed along the circular path. The size of the gravitational restoring force then varies along the circular path. It is very small near the bottom of the path when the pendulum is nearly vertical, but equal to the full force of gravity when the pendulum is horizontal, or 90 degrees from equilibrium. The angle by which the pendulum is displaced from the downward position is denoted in the figure by h. The relationship between this angle h and the restoring force is a very important feature of the pendulum. For small displacements, the restoring force varies approximately with the angle. The restoring force is then roughly proportional to the angular displacement; we say it is more or less linear in the displacement. Linear systems are more

10

Introduction

easily analyzed and therefore have a long history in science. Here is an early example; Robert Hooke (1635–1703) showed that the amount of stretch of a perfect spring increased proportionately, or linearly, with the amount of force causing the stretch. One unit of force provided one unit of stretch, two units of force provided two units of stretch, and so forth. This strict proportionality is true of any linear system. Another way to describe linearity is to say that the rate of growth of a system’s response to stress is constant. The linear approximation for the pendulum is used in most cases where the pendulum’s displacement from the vertical is small. For large displacements, of more than perhaps 10 degrees from the vertical, the relationship between angle and restoring force must be treated in a more complicated way. We call this more complex relationship nonlinear, and it leads to many unique features of the pendulum’s behavior. The distinction between linearity and nonlinearity is important. A system that is nonlinear does not respond in a strictly proportional manner and typically requires a more complex analysis. We all know about the straw that breaks the camel’s back. The camel may sustain an increasing burden with a proportionate or linear bending of its legs. But at the breaking point, it takes only a single piece of straw to cause a disproportionate bending, indeed buckling, of the legs as the camel falls down. This is a nonlinear response. There are many less abrupt nonlinear responses, and we could simply say, in contrast with linear systems, that the rate of change of a nonlinear system is nonconstant. For the pendulum, nonlinearity becomes increasingly important for increasing angles of displacement. These ideas are illustrated in Figure I.5 for the restoring force on a pendulum bob. The graph gives the strength of the restoring force for angle – large or small. In the middle of the graph, where the angular displacement of the pendulum is zero, the pendulum is hanging down and the restoring force is zero. As the pendulum moves upward, the force increases in magnitude (positive or negative depending on direction.) It reaches a maximum when the pendulum is at 90 degrees, or perpendicular to the vertical. Finally, if the pendulum continues upward to the vertical or inverted position where the angle is ±180 degrees, the force decreases back to zero. We note that this position is unstable, as a slight displacement on either side will cause the pendulum to experience a force again, and fall away. Even if the pendulum goes through several revolutions, the force still only depends upon the

What is a Pendulum?

11

Restoring Force

1.0

Linear Force

0.5

Horizontal

Bottom of Swing 0.0

–0.5

–1.0

Nonlinear Force Horizontal

–90

0 Angle

90

Figure I.5 The dependence of the restoring force on the pendulum is illustrated by the Nonlinear curve. Linear behavior is shown for comparison. The angle is measured in degrees from the vertical. The force units are arbitrary.

angular displacement from the downward position. This behavior gives the force the property of being periodic. That is, it is the same force curve from revolution to revolution. This force curve is a mathematical sine curve, and is applicable to all gravity pendulums. The formula for this nonlinear force restoring force is Frestoring = Fgravity sin h,

where the sine curve, or sin h formula is shown as the nonlinear force in Figure I.5. We noted that for small displacements, perhaps less than 10 degrees, the force may be reasonably modeled by the straight line (linear). But why do we bother with the inexact linear model? Why not go right to the nonlinear model? The linear model has the significant advantage that it is much easier to work with mathematically. Furthermore, there are several important applications where this simpler model tells most of the story.1 Thus we begin with a linear approximation and slowly make the transition to the full nonlinear treatment. Once we 1 For example, complex electromagnetic forces govern the motion of atoms in molecules. But because the vibrations of atoms are often relatively small, the relevant forces may be modeled as approximately linear.

12

Introduction

understand the linear approximation, the truly nonlinear pendulum will not seem so formidable. Yet, in initially defining the pendulum, nonlinearity has a role. Since the nonlinear sinusoidal restoring force is applicable to most pendulums, a mathematical definition of a pendulum might simply be any device that has that particular mathematical form of nonlinear restoring force. We will invoke this definition, for example, in our tale of “microscopic pendulums”. On the other hand, there is a pendulum configuration for which the restoring force is indeed simply proportional to the angular displacement alone. This straight line or linear behavior is characteristic of the torsion or twisting pendulum, provided that the suspension fiber or wire is not twisted too much. The torsion pendulum was very important in both the discovery of Charles Coulomb’s law of electrostatics and Henry Cavendish’s study of the density of the earth. In fact, such pendulums continue to be used in gravitational research today. As linear devices, torsion pendulums share many characteristics with Hooke’s simple spring. The term “pendulum” is sometimes used synonymously and perhaps loosely with the term “oscillator”. Certainly the pendulum oscillates, but not always as simply as a spring or simple harmonic oscillator, which is subject to a linear restoring force. As we shall see, oscillations of the nonlinear pendulum can have multiple harmonics – higher frequency motions – embedded within a basic periodic motion. Musical instruments have the same property. The violinist plays a particular note such that the string vibrates primarily at a single vibration frequency. But the richness of the tone is due to the presence of higher frequency vibrations of the string and the sound box of the violin that are harmonics or multiples of the basic frequency. Therefore the vibrating violin has the nonlinear property. In this Introduction, we begin with the small-amplitude, linearized pendulum. We model it as having a restoring force proportional to the angle of displacement, h. Within this approximation, a mathematician or physicist can use Newton’s laws of motion to show that the time for a complete swing, the period T, is proportional to the square root of the pendulum length, l, and inversely proportional to the square root of the acceleration due to gravity, g. In other words, the period increases with increasing length and decreases with increasing strength of gravity. Expressed as a mathematical formula, the period is

What is a Pendulum? T = 2p

13

l . g

An interesting and important consequence of this formula is that no other physical quantities affect the period. This means, first of all, that in this linear approximation, Galileo was correct in saying that the period did not depend on the amplitude. Of even more importance is that somehow the mass, m, of the pendulum bob seems to have no influence on the period. This might seem a bit surprising. But it is the result of two factors. The first is that the restoring force is provided by a component of the gravitational force on the bob, and the gravitational force works on the pendulum’s mass. If some force other than gravity were active, then the pendulum’s mass would indeed be involved in the period. The second reason for the noninvolvement of the mass is something called the principle of equivalence. The principle of equivalence starts from the notion that there may be two types of mass: a mass that responds to gravitational attraction, or gravitational mass, and a kind of mass associated with the resistance of an object to change in its motion or inertial mass. If inertial and gravitational mass were different, they would both appear in the formula for the period. But equivalence postulates that these two types of mass are equal. We typically think of this question as belonging to the twentieth century, and encapsulated in the equivalence principle as articulated by Einstein in 1911. And the many sophisticated experiments that have been done to test equivalence go back less than a century.2 But the equivalence principle has been implicit in mechanics for several centuries. Galileo demonstrated equivalence by rolling differently massed balls down inclined planes. Again, the mass did not have an appreciable effect. In 1685, using equal-length pendulums, Isaac Newton carefully experimented with bobs of different masses and materials: gold, silver, lead, glass, sand, table salt, water, wood, and wheat. In every case, the number of oscillations in a fixed time interval remained constant. As Galileo had

2 We also note the less sophisticated 1971 demonstration by astronaut David Scott who, on the surface of the moon, dropped a feather and a hammer. Because there is no atmosphere, the two objects could fall unhindered by friction. Both objects landed at the same time, thereby showing that the time of descent was independent of mass. After more than two millennia, Aristotle, who would have argued that large masses fall more quickly than small masses, was proved wrong in an obvious and dramatic fashion.

14

Introduction

suspected, for small amplitudes the period of a pendulum was independent of the mass of the bob. So, where have the two types of mass gone from the formula for the period? The answer is that the inertial mass and gravitational mass have cancelled each other because, according to equivalence, these two types of mass are equal. Further illustrations of equivalence and its connection to the pendulum appear in our tale of the twisting pendulum. Equivalence is one example of how the pendulum connects to broad issues in physical science.

I.3 Picturing the motion Pictorial representations of motion are an important aid in understanding physical systems. These representations are not like a video of the motion in the space of our real world, but are pictures – or even videos – in various mathematical spaces. As we progress to more complex pendulum motions, these pictorial or geometric representations in mathematical spaces become increasingly helpful descriptors of the pendulum motions. You may find this section a bit challenging, but your understanding of the following tales will be enhanced by making a good effort on this material. One of the simplest graphical representations for anything that changes in time is a time series: a series of data points in time. The dayby-day graphical representation of a stock market index is a typical time series. For the pendulum, one example of a time series is the graph of angular displacement versus time. Figure I.6 shows such a graph for the small-swing pendulum, where the angular displacement h on the left side is shown to depend on time t as a periodic wave. That is, in time, the angle of the swing moves in a repeated pattern. The repetition time, or period, of this wave is given by the previous formula involving gravity and pendulum length. This particular wave has, again, the mathematical form of a sine curve.3 Its shape is the simplest form of periodic shape and, in this case, its mathematical expression is 3

It is only a coincidence that the mathematical expressions for the general restoring force on the pendulum and for the time series for the linearized pendulum both use sine expressions. Note that the restoring force in Figure I.5 varies with spatial configuration (angular displacement h) whereas the time series in Figure I.6 is the variation of the pendulum’s angle in time t.

What is a Pendulum?

15

1.0

Angle

0.5

0.0

–0.5

–1.0

0

5

10

15

Time

Figure I.6 Time series of the angular displacement of the small-angle pendulum. The units for time and angle are arbitrary.

h(t ) = h0 sin

2 pt 4 , T

where h0 is the maximum angular displacement. The variable of the sine expression includes the time, t, and the period, T, in such a way that the shape repeats whenever the time interval is equal to the period. Linearized pendulums always have this special simple form for a time series – just one sine curve. A similarly shaped time series would also represent the velocity of the pendulum. The velocity of the pendulum is also periodic. Of course, the two curves will usually have different heights, because the maximum velocity is going to be numerically different from the maximum angle. Another difference is that the curves will be displaced from each other in time. For example, when the pendulum is vertically downward, its angle is zero, but its velocity will be greatest. Generally, when one curve is at zero, the other must be at maximum height. But the period for both angle and velocity curves is the same, and both curves have single but different sine components.

4

The sine function operates on angle. For every angle, there is a specific value of the sine of the angle. Formulas using the sine function are often more compactly expressed if the unit of angular measure is the radian rather than the degree. A radian is equal to about 57 degrees. There are 2p radians in a circle.

16

Introduction

Another pictorial representation is called a phase space. For a simple pendulum, a useful phase space might be a two-dimensional plane or graph with a horizontal axis that measures the angle of displacement, h, and a vertical axis that measures the angular speed of the pendulum, m. This representation is then a graph of angle versus speed. Figure I.7(a) shows such a graph for a small-swing pendulum. The elliptical shape in the plane occurs because, as said earlier, (a) the speed and the angle are out of phase with each other, and (b) the maximum angle and maximum speed are not equal numbers. We also noted that when the angular displacement is zero, the pendulum is moving rapidly, and when the pendulum is at its maximum displacement its speed is momentarily zero. The phase diagram indicates both (a) 0.6

Angular Velocity

0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

Angle

Angular Velocity

(b)

Tim

e

le

Ang

Figure I.7 (a) A phase-plane representation of the pendulum’s (frictionless) motion in two dimensions. As time increases, the pendulum travels (mathematically) around the ellipse. (b) A phase-space representation in three dimensions. The third dimension is time.

What is a Pendulum?

17

positive and negative speeds and displacements as the pendulum goes back and forth in its motion. We can think of the phase diagram as drawn from two time series: one for position and one for velocity. The time series for angle oscillates back and forth along the horizontal axis, whereas the time series for velocity oscillates along the vertical axis. At each point in time, their respective positions and velocities provide the coordinates for the phase orbit shown. A phase point, defined by the velocity and the angle of the moment, travels around the phase orbit.5 Depending on the complexity of the system, higher dimensions can be added to a phase-space representation. For the pendulum, it may also be helpful to add a third dimension proportional to time. As shown in Figure I.7(b), the pendulum’s previous mathematical motion around an ellipse now appears as a spiral. The ellipse spreads out in the time direction. Conversely, the phase-plane diagram (Figure I.7(a)) is just a flattening of the spiral down to two dimensions. Both phase diagrams contribute to our understanding of the small-angle pendulum. Another pictorial tool for understanding dynamics is a graph that provides a sense of the amount of time a system spends in a particular location or state. We can also interpret such figures as showing the relative frequency with which the system visits a particular location. For the pendulum, the diagram shows the amount of time that it spends near a particular angular displacement. For small oscillations, it seems obvious that the pendulum spends more time at the extremes of its swing, where the bob momentarily comes to rest, than it does at the bottom of its swing, where it is moving with maximum speed. The frequency distribution is then larger at the extremes of the motion than at the bottom of the swing, as illustrated in Figure I.8. This mathematical diagram is a confirmation of our intuition in this simple case. For nonlinear or micro-pendulums, this diagram – like time series and phase diagrams – can also be quite complicated. The pendulum’s motion can appear to be random, or may even actually be random. In these more complex situations, relative frequency diagrams now give the probability that the pendulum is located near a certain angle. The figures become probability distributions for the pendulum’s location. For simplicity we will refer to all such diagrams as probability distributions.

5 From this hint about the process, some readers might recognize the phase-plane diagram as an example of a Lissajous figure, named for Jules Antoine Lissajous (1822–80).

18

Introduction

Relative Probability

0.35

0.30

0.25

0.20

0.15

0.10 –10

–5

0

5

10

Angle

Figure I.8 A plot of probability versus angle, showing the relative amount of time that a pendulum with a swing of 10 degrees spends near a given angle.

We have now introduced three pictorial representations for the pendulum: time series, phase diagrams, and probability diagrams. The three pictures each emphasize various facets of the pendulum’s behavior. As the motion of the pendulum becomes more complex, deviating from small-angle displacements and the linear approximation, these pictorial representations will be more complex and more important to our understanding of the pendulum’s motion. We are now ready to get on with the main part of the story. Each of the seven tales focuses on a specific aspect of the pendulum – each is a tale in itself. Our first tale describes how the pendulum was used to measure the shape, density, and motion of the earth. Next, we give some of the history and science of the pendulum as a clock regulator. Both of these tales focus primarily, although not exclusively, on the linearized pendulum approximation. The third tale shows how the torsion pendulum – a truly linear pendulum – was important in the development of the classical laws of gravity and electrostatics. For the last four tales, the pendulum is considered to be fully nonlinear. In the fourth tale, we look at the nonlinear pendulum with a large, but still periodic, swing. Yet new complexity is introduced into the periodic motion. In the fifth tale, we let the pendulum be energized to the point of being chaotic. It now exhibits motion that seems random and nonperiodic. This new regime of motion introduces exciting and discretely new pictorial

What is a Pendulum?

19

richness to our time series, phase, and probability diagrams. The complexity is such that fractal geometry now appears. The sixth tale continues this trend by looking at how both periodic and chaotic pendulums couple together. Such coupling leads to remarkable results for fields as diverse as controlling chaotic systems and encryption of sensitive data. The seventh tale considers the idea of a very small pendulum. This may seem strange until we realize that a system that is mathematically equivalent to a pendulum – that has a corresponding sameness of restoring force – can still be a pendulum in some sense. Such equivalency may be realized at the molecular level. It is at this point that quantum physics comes into play. Therefore the quantum pendulum is the subject of the seventh tale. Finally, we conclude the story with brief discussions of a few miscellaneous pendulum configurations that did not fit easily into one of the more extended tales. While the science and technology of the pendulum is primary, we meet along the way bits of literary and human pendulum trivia; examples include the (fictional) murder committed by the earth’s rotation, the woman who preached perpetual motion, and the victim of the inquisition’s pendulum. We also learn some obscure facts, such as how the period of the pendulum almost became a standard not of time, but of length. Let us now begin a story that spans the last four centuries.

This page intentionally left blank

PART II

SEVEN TALES OF THE PENDULUM “And God blessed the seventh day, and sanctified it . . .” Genesis ii:3 “Wisdom hath builded her house, she hath hewn out her seven pillars . . .” Prov. ix:1 “Seven cardinal virtues: prudence, courage, temperance, justice, faith, hope, and charity:” Corpus Mundi (about a.d. 1300) “That ‘seven’ signifies an entire period . . . thus what is [a] full [state] is evident . . .” 9228, Arcana Coelestia (about a.d. 1750)

This page intentionally left blank

Tale 1: Pendulums Measure the Earth From ancient times, thinkers have speculated about, theorized upon, calculated, and measured the physical properties of the earth. Early Greek and Chinese astronomers estimated the circumference of the earth using “instruments” that were readily at hand. Think about shadows from the sun and positions of stars. Man-made instruments, such as the pendulum, arrived on the scene much later. In this first tale of the pendulum, we discuss two ways in which the pendulum has contributed to our understanding of the earth. Beginning in the seventeenth century and carrying right into the twentieth century, the pendulum became the primary tool for measuring the earth’s density and shape. Today, more sophisticated instruments carry the burden, especially measurements from satellites. But, as we describe in the first part of our tale, a general understanding of these fundamental geological properties can be traced to the lowly pendulum. The second and larger part of this tale describes the dramatic demonstration that the earth rotates. We refer to the well-known Foucault pendulum, hung for the public in 1851, in the Pantheon, which is located in Paris, France. This may have been the most public scientific experiment of all time. Its success as a demonstration was undoubtedly enhanced by the fact that it was really a validation of an already generally held belief. People of the time typically believed that the earth rotated, but to see a man-made structure affected by this rotation must have been a powerful confirmation. In our story, we will learn more about the public reception of – and even public misunderstanding about – the Foucault pendulum.

1.1 Shape By the latter part of the seventeenth century, the daily rotation of the earth was an accepted fact. In 1673, the Dutch physicist and astronomer

24

Seven Tales of the Pendulum

Christiaan Huygens suggested a picture of how this rotation might affect the earth’s shape. He proposed the idea of centrifugal force, a “center-fleeing” force. Seated inside an automobile as it turns a corner, we feel a tendency to fly outward – a centrifugal tendency. The same effect happens for the earth, especially near the equator. We can think of the earth being somewhat elastic, and therefore the centrifugal effect causes a slight bulge at the equator. The bulge makes the effective radius of the earth at the equator slightly larger than, for example, at the poles. And this means that the gravitational field is slighter smaller at the equator. (The centrifugal effect is not just limited to bulging of the earth. It decreases the effective gravitational field on any object on the surface of the rotating earth.) In this way, Huygens reasoned that the earth is fatter at the equator. In 1687, Newton published his universal law of gravity in his famous Principia, giving theoretical foundation to these thoughts.1 What emerges is a picture of the earth as an oblate ellipsoid, fatter at the equator than at the poles. But the picture required much experimental verification, and for more than two centuries, the pendulum typically provided this verification. Another important date in this story is Christmas Day 1657, when Huygens presented the world with the pendulum clock. The clock and other specially adapted pendulums were to become dominant tools in geological measurements. It is the relationship among its . . . , its length, and the strength of the gravitational field that makes the pendulum a useful tool for measuring the local gravitational field. As we noted in the Introduction both through words and formula, the stronger the field, the shorter is the period. Or, in a weaker field, a pendulum must be shortened to maintain the same period. The first recorded use of the pendulum to measure local gravity is usually attributed to the measurements of the French astronomer Jean Richer (1630–96). He made the measurements in 1672. Richer found that a pendulum clock beating out seconds in Paris at latitude 49 degrees north lost about 2½ minutes per day when placed near the equator in Cayenne, now the capital of French Guiana, 1 Newton coined the term “centripetal force” in parallel with the Huygens centrifugal force. Centripetal force is that force which keeps a moving object in circular motion. Otherwise, the object would fly off tangentially to the circle.

Pendulums Measure the Earth

25

at 5 degrees north.2 From this difference, he concluded that Cayenne was further from the center of the earth than was Paris. Newton, on accidentally hearing of this result at a meeting of the Royal Society ten years later, used it to refine his theory of the earth’s oblateness. Richer’s result was to be only one of many pendulum measurements to contribute to detailed knowledge of the earth’s shape. Richer’s swinging of pendulums had another unforeseen consequence. His demonstration confirming that pendulums are sensitive to gravity meant that pendulum clocks kept different time in different parts of the world. It then followed that, as the clock traveled the world, it would not be a reliable timekeeper of a standard time such as Greenwich time. This feature precluded the use of the pendulum clock for the measurement of longitude. Accurate measurement of longitude would be possible only with precision spring-driven clocks that were unaffected by local gravitational variations.3 Therefore, Richer’s result, while helping to determine the shape of the earth, also led to the eventual rejection of the idea of using a pendulum clock as a reliable timing standard for the measurement of longitude. While Richer may have been the first to actually make some pendulum measurements, other famous – or almost famous – scientists also proposed or used the pendulum to measure the gravitational field. Robert Hooke (1635–1703), well known for the linear law of elasticity, for his invention of the microscope, a host of other inventions, and his controversies with Newton, was one of the first to suggest, in 1666, that the pendulum could be used to measure the gravitational field. Edmond Halley (1656–1742), the Astronomer Royal (of Halley’s Comet fame), also appears in this story. In 1676 Halley sailed to St. Helena’s island, located in the South Atlantic, in order to make a star catalog for the southern hemisphere. As a friend of Hooke, he was aware of Hooke’s suggested use of the pendulum to measure gravity, and he did make 2 Independent checks on mechanical clocks typically utilized astronomical measurements. For example, by 1676 John Flamsteed, the first Astronomer Royal, was calibrating his pendulum clocks using a fixed telescope aimed at the star Sirius to measure the sidereal day (23 hours, 86 minutes, 4.1 seconds). This is the time taken for the star to reappear at a precise position in the field of the telescope. Similarly, Richer could have measured the sidereal day in Paris and near the equator with a telescope. 3 For the story of the initial determination of longitude, see D. Sobel, Longitude: The True Story of a Lone Genius who Solved the Greatest Scientific Problem of His Time (Penguin, New York, 1996).

26

Seven Tales of the Pendulum

such measurements while on St. Helena. (Certainly Halley is famous for having his name applied to the comet, yet he probably rendered a significantly more important service to mankind by encouraging and supporting Newton in the publication of the latter’s famous Principia.) In principle then, a comprehensive series of pendulum measurements taken at a variety of locations should yield a map of the strength of the gravitational field over the earth’s surface. Yet without special effort the results tend to be inaccurate. For example, it is often difficult to accurately determine the effective length of the pendulum. There can be ambiguity in the measurement at the position of the pivot or at the bob. In 1817, at the suggestion of the German astronomer F. W. Bessel (1784–1847), Henry Kater (1777–1835), a captain in the British Army, invented a reversible pendulum that significantly increased the accuracy of the measurement of the gravitational field. Kater’s pendulum is shown schematically in Figure 1.1. It consists of a rod with two sharp pivot points (P), whose positions (h1 and h2) along the rod are adjustable. The period of the pendulum is determined with the pendulum suspended from one pivot point. Then the pendulum is reversed to pivot from the other point and a further measurement of the period is made. In principle, the determination of the gravitational field g is made by adjusting the pivot points until the periods of small oscillation about both positions are equal. In practice, it is difficult to precisely adjust the pivot points, which are usually knife

p1 h1 h h2 p2

mg

Figure 1.1 The Kater reversing pendulum.

Pendulums Measure the Earth

27

edges. Instead, counterweights are attached to the rod and are easily positioned along the rod until the periods are equal. In this way, the pivot positions are defined by fixed knife edges that provide the possibility of accurate measurement. Once the periods are found to be equal and measured (T ), the acceleration due to gravity g is readily calculated using a slight variation on the formula given in the Introduction for the period of a simple pendulum. That is, T = 2p

h1 + h2 . g

Most importantly, the sum h1+h2 is just the easily measurable distance between the knife-edge pivot points. By gradually moving the counterweights, the convergence toward equal periods from both pivots creates a significantly more accurate process than that from a single pendulum. As late as 1936, the U.S. National Bureau of Standards used a modified version of the Kater pendulum to determine the acceleration due to gravity at Washington, D.C. as g = 980.080 ± 0.003 cm/s². After its invention, many of the pendulum gravity experiments were carried out using the Kater “reversing” pendulum. One of the original pendulums, number 10, constructed by a certain Thomas Jones, rests in the Science Museum in London. The display card reads as follows: “This pendulum was taken together with No. 11, which was identical . . . on a voyage lasting from 1828 to 1831. During this time Captain Henry Foster swung it at twelve locations on the coasts and islands of the South Atlantic. Subsequently it was used in the Euphrates Expedition, of 1835–6, then taken to the Antarctic by James Ross in 1840.” Meanwhile, some form of the pendulum continued to travel the world measuring local gravitational fields. Sir Edward Sabine (1788–1883), an astronomer with Sir William Parry in the search for the northwest passage (through the Arctic Ocean across the north of Canada) spent the years from 1821 to 1825 determining measurements of the gravitational field with the pendulum along the coasts of North America and Africa, and, of course, in the Arctic. The American philosopher Charles S. Pierce (1839–1914) makes a surprising appearance in this historical context. Although known for his contributions to logic and philosophy, Peirce rarely held academic positions in these branches of learning. Instead, he made his living with the U.S. Coast

28

Seven Tales of the Pendulum

and Geodetic Survey. Between 1873 and 1886, Pierce conducted pendulum experiments at a score of stations in Europe and North America in order to improve the determination the earth’s elliptical shape. However, his relationship with the Survey administration was fractious, and he resigned in 1891.4 Thanks to these and many other measurements, we now know that the earth is an oblate ellipsoid, fatter at the equator than at the poles. Modern measurements find that the equatorial axis (the diameter through the earth at the equator) is about 21 kilometers, or 0.3 percent, longer than the polar axis.

1.2 Density Pendulums were also instrumental in determining the earth’s density. The average density of the earth is much greater than that of materials that are typically found on the earth’s surface. But direct sampling of the earth’s interior would have been an impossible task. Remarkably, Pierre Bougeur (1698–1758), a French professor of hydrography and mathematics, used a clever bit of theory and pendulum measurements to estimate the earth’s density, r, without using a direct measurement. In 1735, the French Academy of Sciences sent Bouguer to Peru to measure, among other things, the length of a degree of a meridian arc, near the equator.5 (A variety of such measurements at different latitudes, along with pendulum measurements, would help to determine the oblateness, or “bulginess”, of the earth.) While in Peru, Bougeur made measurements of the oscillations of a pendulum. His pendulum had beaten out seconds in Paris, but in Quito, Peru (latitude 0.25 degrees south) the period was different. From his original memoir, it is a little confusing to determine whether he maintained a constant-length pendulum or whether, as his data suggest, he modified the length of the pendulum to maintain a period of a second. At any rate, he used the pendulum to measure the gravitational field. Now, for the density calculation, Bouguer had to measure the gravitational field at two different elevations. He measured it close to sea level, with a resulting gravitational acceleration of gsea, and then on top 4 Pierce’s problems stemmed from a certain indifference to bureaucratic procedures and complications in his marital situation. 5 A meridian arc is a distance along a north–south line on the earth’s surface.

Pendulums Measure the Earth

29

of the Cordilleras mountain range, with a gravitational acceleration of gmount. Bouguer also measured the average density of material on the surface of the earth on the mountain range. He then developed a mathematical model for the data, using standard gravitational theory and some simplifying assumptions about the terrain. It is remarkable that using this minimal data and the mathematical model, he could actually calculate the average density inside the earth. Even without following the mathematical details, we can gain some appreciation for the cleverness of Bouguer’s method. Figure 1.2 shows a part of the earth’s surface, with the mountain range drawn as a simple “bump” of constant height on top of the earth. The letters on the diagram correspond to various physical quantities. For example, a is the earth’s radius, rearth is the average density of the earth, rsurface is the average density of the mountain range found by sampling the local soil, h is the average height of the mountain range, and A is an estimate of the average area of the mountain range. All of these ingredients, combine, at least in theory, to give the ratio of the gravitational acceleration on top of the mountain range to that near sea level, gmount/ gsea. The mathematics is complicated, but the clever result is a formula that connects all these known quantities and, with a little algebra, provides an estimate of the earth’s unknown average density rearth: gmount 2h 3h rsurface =1− + . gsea a 2a rearth A

h

rsurface rearth

a

Figure 1.2 The little “bump” on the earth’s surface represents a whole mountain range.

30

Seven Tales of the Pendulum

If the two accelerations were the same, then the ratio would just be one. This formula accounts for the differences through two correction terms. In the first (negative) term after the numeral one, the gravitational field on the mountains is reduced because the measuring point is further from the center of the earth, and this correction is called the free air term; it involves the mountain height h and the earth’s radius a. In the second (positive) term, the field is enhanced by the gravitational pull of the mountain range and this correction is called the Bouguer term. It combines the two previous quantities as well as the density of the mountain range and the average density of the earth. Every quantity could be estimated or measured, or was known at the time, and therefore it was now possible to calculate the ratio of the density of the mountainous material to that of the rest of the earth. Bouguer’s pendulum measurements convinced him that the earth’s mean density was about 4 times that of the mountains, a ratio not too different from a modern value of 4.7. In Bouguer’s own words, he had exploded two current theories: “Thus it is necessary to admit that the earth is much more compact below than above, and in the interior than at the surface. . . . Those physicists who imagined a great void in the middle of the earth, and who would have us walk on a kind of very thin crust, can think so no longer. We can make nearly the same objections to Woodward’s theory6 of great masses of water in the interior.”

Bouguer’s experiment was the first of many of this type. A common variant on the mountain range experiment compared the gravitational accelerations at the top and at the bottom of a mine shaft. In this case, we can think of the extra structure as not just a mountain range, but a whole spherical shell above a slightly smaller earth. The radius of this new “earth” is less than that of the ordinary earth by the depth of the mine shaft. Figure 1.3 shows the various quantities that go into the calculation. We see that h is the depth of the mine shaft,

6 The person referred to is probably John Woodward (1665–1728) a prominent English naturalist and geologist. Bouguer’s result showed that the earth’s interior is much denser than water.

Pendulums Measure the Earth

31

and a is the radius as measured from the center of the earth to the bottom of the shaft. Again the formula has two correction terms, although the Bouguer term differs by a factor of two from the expression used in the mountain-top experiment: gtop 2h 3hrsurface . =1− + gbottom a arearth As in the previous case, a little algebra leads to a value of the earth’s density, r earth. Coal mines were widely available in England, and the seventh Astronomer Royal and Lucasian professor of mathematics at Cambridge, George Airy (1801–92), was one of many to attempt this type of experiment. Unfortunately, his early efforts in 1826 and 1828 in Cornwall were frustrated by floods and fire. But much later, in 1854, he successfully applied his techniques at a Harton coal pit in Sunderland and obtained a value for the earth’s density of r earth = 6.6 gm/cm³, which may be compared with the presently accepted value of 5.5 gm/cm3. Pendulum experiments continued to be improved. The Austrian geographer Robert Von Sterneck (1839–1910) explored gravitational fields at various depths inside silver mines in Bohemia and in 1887 invented a four-pendulum device. The older devices were based upon the 1 second

h

a

rsurface

rearth

Figure 1.3 A schematic of the earth with a mine shaft.

32

Seven Tales of the Pendulum

pendulum, with a length of about 1 meter.7 Von Sterneck felt that little would be sacrificed and much could be gained by going to 1/2 second pendulums, which were only about 1/4 meter in length. These fitted much better into a partially evacuated container and yielded a definite improvement in accuracy. He placed two pairs of 1/2 second pendulums at right angles. Each pendulum, in a given pair, oscillated out of phase with its partner. This arrangement reduced the flexure in the support structure that ordinarily contributed a surprising amount of error to the measurements. The two mutually perpendicular pairs also provided a check on each other. Von Sterneck’s values for the mean density of the earth ranged from 5.0 to 6.3 gm/cm³. The swing of the pendulums in a pair is compared with a calibrated 1/2 second pendulum clock by means of an arrangement of lights and mirrors, as observed through a telescope. Because they have very slightly different periods, the gravity pendulum and the clock pendulum will only come into synchronization once in a while. The number of counts between such “coincidences” is observed and used in determining a precise value of the gravity pendulum period. Accuracies as high as 2 × 10–7 seconds were claimed for the apparatus. Finally, in the twentieth century, we note the work of Felix Andries Vening Meinesz (1887–1966), a Dutch geophysicist who, as part of his Ph.D. (1915) dissertation, devised a complicated pendulum apparatus. Somewhat like Von Sterneck’s device, it used the concept of pairs of perpendicularly oriented pendulums swinging out of phase with each other. As shown in Figure 1.4, the four pendulums are distinguished by the bright brass pendulum bobs: one each at the ends of the long dimension of the apparatus and a pair located in the middle on the short axis. It is clear that the device was a far cry from the original simple pendulum. With this complex device, Vening Meinesz eliminated a horizontal acceleration term due to the vibration of peaty subsoil that seemed to occur in many places where gravity was measured. Vening Meinesz’ apparatus was also fitted for measurements on or under water and contained machinery that compensated for the motion of sea. Aside from the interruption caused by World War II, some version of this device was used on submarines from 1923 until the late 1950s.8 7 Mathematically inclined readers will note that the formula in the Introduction would give a period of about 2 seconds for a 1 meter pendulum. However, in early pendulum terminology, the period was taken as only one-half of a full oscillation cycle. 8 A famous, but different, approach to measuring the earth’s density is described in the tale of the twisting pendulum.

Pendulums Measure the Earth

33

Figure 1.4 The Vening Meinesz pendulum. Four pendulums are arranged in mutually perpendicular pairs. The four bobs are the bright fixtures near the bottom of the apparatus. (Courtesy of the Society of Exploration Geophysicists Geoscience center. Photo © 2004 by Bill Underwood.)

1.3 Rotation – the Foucault pendulum The Foucault pendulum demonstrates the rotation of the earth. It is one of the most famous and fascinating experiments of all time. This fascination extends the pendulum’s reach even into literature: “At 11:30 PM in the bell tower of the old Saxon church at Randall’s bridge, a huge statue toppled and smashed. Heavy blocks of broken marble now lay up against the door barring entry or exit. The solitary window was too narrow for a man to pass through; the belfry high above led only to the steep roof which rose beyond the reach of any ladder. When Detective Inspector C. D. Sloan put his shoulder to the door (blocked by the broken statue) it opened only a crack. Through it he could clearly see the room was empty – except for the bells, the debris of shattered marble . . . and

34

Seven Tales of the Pendulum the protruding arm of a dead man. How could a murderer have escaped from this sealed tower? Sloan’s only clues: a spent match, a black thread (burned at the end), a pendulum hanging in the center of the tower. And an emerald earring.”

In 1981, novelist Catherine Aird wrote a mystery entitled His Burial Too that uses the Foucault pendulum as a murder weapon. Aird’s story opens with the murder of a certain Richard Tindall, owner of an engineering company and father of an attractive daughter – requisite characters for detective stories of a certain vintage. Various suspects are developed who have stronger or weaker motives for promoting Tindall’s death and thus a range of possibilities is created. Yet alibis seem airtight and certain aspects of the execution of the crime remain a mystery for much of the story. Later in the course of the investigation, the fog of ignorance begins to dissipate. “From Sloan’s superior: “What’s all this business about pendulums, then?” Sloan shook his head. “I don’t think the pendulum would do the trick, Sir. The sculpture was off-line from the window.” Cold trickled down Sloan’s spine. Then his head came up. . . . There was such a thing as a pendulum that turned. He’d seen it. In a museum. . . . It was coming back. What the pendulum needed was a smooth start, and half an hour later it was off course. . . . The smooth start had been important. He remembered watching while a man from the museum started it by burning the anchoring string in two with a match.” And so the investigation proceeds to its now-inevitable conclusion. The pendulum oscillation was initiated some 5 hours before the actual murder, giving the murderer an opportunity to develop an alibi for his whereabouts at the time of the killing. During the intervening 5 hours the plane of oscillation rotated about 60 degrees, until it hit a massive statue that was delicately balanced to fall on the previously tied up Mr. Tindall. When the statue fell, it killed the victim and sealed the door to the chamber. The other clues are explainable and point to a suspect once the basic physics of the Foucault pendulum is understood. The general positioning of the tools of the murder is illustrated in Figure 1.5. (At least two features of the story seem slightly implausible. First, it is very unlikely that this particular pendulum could have sustained its oscillations for 5 hours without some sort of driving force. The second difficulty lies with the fact that the trigger for the destabilization of the statue was the very small change in the plane of oscillation of the

Pendulums Measure the Earth

35

DOOR

STATUE

BOB STRING

TOP SUPPORT

Figure 1.5 A schematic of the crime scene in Aird’s novel. (Drawing by Richard Crane. Reprinted with permission from The Physics Teacher, May 1990, p. 265. © 1990, American Association of Physics Teachers.)

pendulum. The murderous statue must have been balanced very precariously indeed!) Let us get back to reality. What is a Foucault pendulum, what is its history, and what physical phenomenon does it demonstrate? In theory, any earth-based pendulum is a Foucault pendulum. Every pendulum oscillates in a plane, and a realistic Foucault pendulum is a one that is specially suspended so that its plane of oscillation is free to rotate with the rotation of the earth. Another way to conceptualize the Foucault pendulum is to think of the pendulum as having a suspension mechanism that is fixed relative to the stars.9 Then the earth rotates under the pendulum’s oscillations. From the earth’s point of view, the pendulum’s plane of oscillations seems to rotate, but seen from a distant star, it is really the earth that rotates, not the pendulum’s oscillations. Figure 1.6 shows a replica of the Foucault pendulum hanging in the Pantheon, where it made its public debut.

9

This is a bit of a simplification.

36

Seven Tales of the Pendulum

The inventor, for whom this special pendulum is named, was Jean Bernard Léon Foucault, the son of a Parisian publisher (see Figure 1.7). He was born September 18, 1819 and died February 11, 1868. Foucault received his secondary education at the prestigious College Stanislas and later enrolled in the Paris medical school. Yet sometime around 1844 he abandoned medicine to pursue his strong interests in physical science. Many of his early experiments were in optics. These included efforts to improve Daguerre’s10 new photographic processes. He also studied the intensity of sunlight, the phenomenon of the interference of light that is observed by combining special beams of light, and the effect of crystals on light. Foucault showed that light traveled faster in air than in water and also determined the absolute speed of light in vacuum. Perhaps as result of his pendulum work,

Figure 1.6 A Foucault pendulum (similar to the original) hanging in the Pantheon, Paris, where the original was first publicly displayed. (Photo by GLB.) 10 Louis Daguerre (1787–1851) is known for his invention of a precursor to the photograph labeled, not surprisingly, as the daguerreotype.

Pendulums Measure the Earth

37

Figure 1.7 Leon Foucault, 1819–68, inventor of the Foucault pendulum, at age 47. Foucault made many contributions to science and technology. (Image from Wikipedia.)

Foucault invented the precision gyroscope and demonstrated its use as a compass. In 1855, Foucault was appointed physicist at the Paris observatory. There, he invented techniques that would allow for the manufacture of large mirrors for reflecting telescopes, thereby significantly increasing the power and utility of the telescope. Remarkably, the electric currents induced by magnets that we now refer to as eddy currents were, during the nineteenth century, sometimes known as Foucault currents, since Foucault studied them experimentally. While not a brilliant theoretician, Foucault was an extremely creative experimentalist and this brief description hardly does justice to his many contributions. In early 1851, Foucault had an insight that led to his famous discovery. This bit of serendipity occurred during construction of a pendulum clock to regulate the drive of a telescope. Foucault had secured a rod in the chuck of a lathe. Perhaps accidentally, the rod was disturbed so that it vibrated. Foucault noticed that the plane of oscillation of the vibrating

38

Seven Tales of the Pendulum

rod remained fixed in orientation even as he slowly turned the chuck. This observation inspired him with the notion that the vibrating motion of a freely suspended pendulum might somehow be related to the earth’s rotation. For a pendulum suspended above the north pole, the plane of vibration would appear to complete a full circle in 24 hours, whereas at the equator, the plane of vibration would not rotate; the period of rotation then being infinite. So, depending on the latitude, the period of rotation is between 24 hours and infinity. For the latitude of Paris – Foucault’s city – the plane of a pendulum’s vibration rotates about 0.19 degrees per minute, completing the entire rotation in about 32 hours. The mathematical relation between the period for a complete rotation of the plane of vibration and geographical latitude is given by the formula: Period of Rotation =

24 hours 11 . sin(latitude )

In January 1851, Foucault made his initial observations with a pendulum suspended in his mother’s basement. It had a length of 2 meters and a bob mass of 5 kilograms. While he was able to observe the anticipated effect with the basement pendulum, he quickly understood the benefits of an even longer pendulum. For example, he realized that the slight change in the point of maximum displacement of the pendulum after each oscillation would be more noticeable. Longer pendulums would also diminish the effect of other spurious motions that mask the Foucault effect. Moreover, a long pendulum with a massive bob would have more inertia, and therefore likely oscillate longer before needing another push. A very long pendulum would certainly provide a more compelling demonstration. His next pendulum was 11 meters long. In February, it was hung in the Paris Observatory. Meanwhile the prince-president of France, LouisNapoléon Bonaparte, became interested in Foucault’s work. With princely backing, Foucault was able to construct a mammoth 67 meter, 28 kilogram pendulum under the dome of the Pantheon in Paris, in March 1851 (see Figure 1.8). For Foucault’s large Pantheon pendulum, the period of oscillation was 16.4 seconds. Foucault had a wooden circle with a 6 meter diameter set up, with its axis concentric with the vertical wire of the pendulum. The circumference of the circle was divided into fractions of 11

The ubiquitous sine curve appears once again.

Pendulums Measure the Earth

39

Figure 1.8 Public display of the Foucault pendulum. (Engraving from L’Illustration, Paris, 1851.)

a degree. During one of the slow oscillations, the plane of vibration would turn by about 0.051 degrees, and the incremental change in position of the pendulum bob at maximum displacement above the floor was found to be about 2.5 millimeters for a single oscillation. After a few swings, the cumulative change was obvious. In order to display the effect, a spike was fixed downward at the bottom of the bob so that it would cut a groove in a layer of fine sand spread below. Figure 1.9 (four consecutive bob paths from a computer simulation) shows a schematic (and much exaggerated) diagram of the pendulum’s motion. Over the next 150 years, the pendulum’s history would be connected to that of the Pantheon. But, in order to understand this history, we look briefly at the story of the Pantheon itself. A good starting point is the year a.d. 451. The fearsome Attila the Hun was marching toward the then-island community of Paris, causing much consternation to the residents. Genevieve, a young shepherd girl, had predicted the invasion, but also encouraged the inhabitants to prepare a defense rather than flee. At the last moment, Attila decided not to attack the city and this seemingly miraculous event led to Genevieve being considered a patroness of the city, and later canonized. In 512 she died and was buried in the church of the Holy Apostles, which became known as the church of St. Genevieve. Eventually, that church decayed to ruins. Much later, in 1744, King Louis XV, in gratitude for his recovery

Seven Tales of the Pendulum

Y

40

X

Figure 1.9 Computer simulation of a few oscillations of the pendulum bob, as projected onto the floor. The earth’s rotation is artificially speeded up in order to clearly demonstrate the Foucault effect.

from a serious illness, ordered a new church to be built on the original site. But progress was slow and the building was not completed until 1789, around the beginning of the French Revolution. Instead of keeping to Louis’ plan, in 1791 the new Republican government ordered that the building be used as a public secular mausoleum – a pantheon – for the interment of famous Frenchmen.12 In the next few decades, the building was to alternate several times between secular and sacred usage. Napoleon Bonaparte returned the building to the Church in 1806, although the crypt continued to be the burial place for famous Frenchmen. In 1830, at the end of the Bourbon dynasty, it once again became the national Pantheon under the new king Louis Philippe. Somewhat later, it reverted to the Church. In 1848, a revolution against Louis Philippe created the Second Republic, which restored the building as the secular Pantheon. Such was its status at the time of Foucault’s public demonstration. 12

1934.

The remains of Marie Curie were placed there in 1995, about 60 years after her death in

Pendulums Measure the Earth

41

The involvement of Louis-Napoléon with Foucault and his pendulum is curious, in that there is a certain parallelism between the two men. Both of them spent much of their lives as outsiders. Foucault was not part of the group of theoretical physicists and mathematicians who dominated the French Academy of Sciences. Similarly, Louis-Napoléon was not part of the French government until 1848 and, for many years, was not even allowed to set foot on French soil. Louis-Napoléon (1808–73) was a nephew of the famous Napoleon Bonaparte. Because of deaths in the family, he eventually became next in line to be emperor, had the Bonapartes stayed in power. However, the battle of Waterloo changed all that, and Louis-Napoléon grew and spent most of his first 40 years living abroad. Yet, his desire to return to France as leader was strong. He attempted two futile coups d’état and spent six years imprisoned in a French military fortress. During that time, he developed a strong interest in science – especially physics – an interest that in later life led to his support of Foucault’s experiment. Like Foucault, he was self-taught in science. In 1846, Louis-Napoléon made a daring escape from his prison and returned triumphantly to his supporters in London. He used the next year or so to build political capital in preparation for his eventual return to France. This was to happen shortly after the 1848 revolution. His political fortunes improved dramatically. In quick succession, he was twice elected to the National Assembly and then elected as president of the Assembly. In 1851, he heard of Foucault’s work and, as president, gave orders to proceed with the hanging of a large pendulum in the Pantheon. With this construction, both outsiders shared a moment of triumph. Unfortunately, the display of the Foucault pendulum in the Pantheon was short-lived as political events intervened. On December 2, 1851, Louis-Napoléon staged a coup d’état, taking over the government. On December 6 he gave the Pantheon back to the Church, as a way of gaining clerical support. Of course, the pendulum had to go. One year later, Louis-Napoléon consolidated his position by having the people elect him as emperor.13 However, not long after the original Pantheon demonstration, a more permanent exhibit was established in another part of Paris, in the Musée des Arts et Métiers in 1855. At this writing, two of the 1851 pendulums are on display in that location. 13 Louis-Napoléon, or Napoleon III as he was to become, continued to take a personal interest in Foucault’s work and career.

42

Seven Tales of the Pendulum

While Foucault pendulums were hung in important locations14 elsewhere in France, it was 50 years before another pendulum would hang in the Pantheon. At the beginning of the twentieth century, the Astronomical Society of France, presided over by the noted astronomer and mathematician Henri Poincaré, supported having a pendulum mounted in the Pantheon as it was once more a secular, public building. This was done and the exhibit officially opened on October 22, 1902. The opening was attended by many notables, including the composer Camille Saint-Saëns and the sculptors Auguste Rodin and Frédéric Bartholdi (who created the Statue of Liberty). But a year or so later the pendulum was again taken down, and it was not until the fall of 1995 that the original was brought out of storage and again hung from the dome of the Pantheon for about a year. The pendulum that is currently in the Pantheon was commissioned in 1996 and designed by Jacques Foiret. The bob is quite massive at 47 kilograms and has been gilded by electrolysis with 24 carat gold.

1.4 Frames of reference In making scientific measurements, we often choose a particular frame of reference. The Foucault pendulum demonstrates this point most dramatically. We ordinarily speak of a frame of reference as a point of view; perhaps the perspective of a specific political philosophy or set of religious beliefs. In the physical sciences, a frame of reference is the platform on which an observer is located and from which that observer makes measurements. Experience suggests that observers who are in different frames of reference may see the same event differently. A frame in rotation relative to a stationary frame is actually accelerated relative to the stationary frame. Objects in that frame are constantly changing direction; hence the acceleration. An observer in the accelerated frame thinks that new forces occur.15 For example, a passenger in the frame of reference of a car turning a corner experiences a new force called centrifugal force, or “center-fleeing” force. This car-bound observer believes that a force is pulling him out of the car, away from the direction 14

One such location was inside Reims Cathedral, where a 40 meter pendulum was hung in May 1851. This seems somewhat ironic, given the swift removal of the Pantheon pendulum as that building was again converted to a church. 15 This is another example of the equivalence principle.

Pendulums Measure the Earth

43

of the turn. Yet an observer standing on the ground only sees the passenger accelerate toward the center of the turn, in going around the corner. This latter acceleration is called centripetal,16 or “center-seeking”, acceleration. Relative to the stars, we all exist on a grand rotating frame of reference: the earth. Therefore we view things differently from an observer located on a distant star. On earth, we experience (but hardly feel) a slight centrifugal force that diminishes our weight – especially at the equator – whereas the distant observer sees us accelerate, with centripetal acceleration, toward the center of the earth. To further complicate the issue, if the observer in the rotating frame is himself moving relative to that frame, he experiences another, less obvious, force. This new force, called Coriolis force, is essential for our discussion of the Foucault pendulum. In 1837, the French mathematician Siméon Denis Poisson (1781–1840) published a paper, based upon the calculations of a one-time student, that described this force. It turned out that Poisson’s student was Gaspard Gustave Coriolis (1792– 1843), a physicist who specialized in various problems of theoretical mechanics and published his theory of rotating frames in the context of their effects upon machine operation. Although Poisson did not publicly acknowledge Coriolis’ work, justice eventually triumphed and we now rightly identify this further effect as Coriolis force. Here is a typical example of Coriolis force behavior. An observer on a rotating platform (such as a merry-go-round), who attempts to throw a ball to her friend on the same platform, would observe the ball to follow a curved trajectory rather than a straight line. During the ball’s time of flight, the platform has rotated and the friend, who is supposed to catch the ball, is no longer in the same place. It turns out that the Coriolis force – the force observed in the rotating frame – is proportional to both the ball’s velocity and the rotation rate of the platform. Expressed mathematically, the Coriolis force is the simple product of several factors. That is, F = 2mv┴X, where m is the mass of the ball, v┴ is the amount of the ball’s velocity that is at right angles to the platform’s axis of rotation, and X is the platform’s rate of rotation. 16 We recall that the terms “centrifugal” and “centripetal” were coined by Huygens and Newton respectively.

44

Seven Tales of the Pendulum

In the frame of the rotating earth, the Coriolis force on a moving object is proportional to whatever part of the object’s velocity is perpendicular to the axis of the earth’s rotation. Depending on their directions, motions in the plane of the earth’s surface may be affected by the daily rotation of the earth. At the north and south poles, the earthbound observer will see deviation to the right or left, respectively. At the equator, only that component of surface motion that is parallel to the equator will be subject to Coriolis acceleration. Even then, the acceleration will be perpendicular to the earth’s surface and perhaps less noticeable than at the poles. A dramatic example of Coriolis “force” is the deviation of artillery shells from their targets – to the right in northern latitudes and to the left in southern latitudes. This phenomenon was powerfully demonstrated in World War I. During a naval battle near the Falkland Islands, located near the southern tip of South America, the British gunners initially compensated incorrectly for Coriolis force by using adjustments that were valid for the northern hemisphere. However, once they changed the sign (plus or minus) of the numerical correction to that required for the southern hemisphere, their fire became accurate. Modern guns are less susceptible to Coriolis effects because, while their muzzle velocities are much higher and thus the Coriolis force is stronger, the projectile’s time of flight is much shorter and therefore the time during which the force acts – or the earth rotates – is relatively short. How does Coriolis force affect pendulums? In his 1837 paper, Poisson actually mentioned the possibility of an earthbound effect upon a pendulum’s plane of oscillation. But since the rotation of the earth is slow and the velocities of typical pendulums are small, Poisson concluded that the effect would be barely observable. Certainly this is more or less true for a single oscillation of the pendulum. Foucault, on the other hand, noted in his 1851 paper that the Coriolis effect would be cumulative and therefore the additive result of many oscillations would be quite observable under the right conditions.

1.5 Foucault pendulums are tricky Demonstration of the rotation of the earth is an exciting prospect! But unless you happen to have a very long rope and a very heavy bob, the creation of a decent Foucault pendulum is not a simple Saturday morning project. Certainly, every freely suspended earthbound pendulum is

Pendulums Measure the Earth

45

a Foucault pendulum, but not every one of them will manifestly display the desired effect. Many things get in the way; some are annoying, difficult to analyze, and difficult to remove. Here are a few of these problems.

1.5.1 Starting the pendulum It is often observed that the pendulum swing deviates from a desired straight line. One reason for this is that the person who starts the pendulum by displacing it from the downward vertical inadvertently gives a small bit of push at right angles to the direction of the swing. Even the tiniest push can cause problems. Therefore one early method for starting the motion removes the human interaction as follows. First, the pendulum bob is displaced from equilibrium and the displacement secured with a string. Then a flame is applied to the string. Once the flame has burnt through the string, the pendulum naturally moves exactly in the direction of its previous displacement. (Remember the murder mystery.) But – and this can be problematic for small pendulums – at the instant the string burns through, the pendulum is stationary relative to the rotating earth, and therefore, relative to the stars, the pendulum starts with a small amount of motion in a direction at right angles to the plane of oscillation – as measured in the “stellar” frame of reference.

1.5.2 Finite swing size Up to this point, we have assumed that the swing of the pendulum is small and that the period is independent of the size of the swing. But strictly speaking, even for small angles, the period of the pendulum does increase slightly with swing size. For something as delicate as the Foucault effect, even the small variation in period with swing is a problem. The combination of (a) the variation in the period of oscillation, (b) the very small (Foucault) rotation of the plane of oscillation with each swing, and (c) a possible bad start, results in the formation of a narrow elliptical orbit for the bob. With this elliptical motion, the pendulum effectively oscillates in two perpendicular directions, with two differently sized swings. And because of the differences in the sizes of the swings, there are two slightly different periods for these perpendicular oscillations. Looking down at the path of the bob, we find that these two perpendicular oscillations trace out a narrow ellipse, somewhat like that of the phase diagram described in

46

Seven Tales of the Pendulum

the Introduction. We can think of this effect as slightly analogous to two slightly out-of-tune strings on a musical instrument. One hears a pulsation or “beat” at a frequency that is the difference between the frequencies of the strings. It is similar for the pendulum. The elliptical motion of the pendulum bob is a precession17 of the plane of oscillation that can be confused with the Foucault effect. It turns out that the precession rate is proportional to the area of the ellipse, and adds or subtracts from the Foucault precession. That is, depending on how the pendulum is started, the direction of the precession can be with or against the Foucault rotation. For technical reasons, this difficulty is called the anharmonicity problem. The easiest way to minimize the problem is to make the pendulum length very long compared to the size of the swing. This is one reason why most Foucault pendulums are very long.

1.5.3 Asymmetry A very likely complication involves some sort of asymmetry in the configuration of the pendulum or the circumstances of its location. Asymmetry means that the pendulum does not oscillate equally in all directions, as required for the Foucault effect. For example, the pendulum may be suspended from a beam that has more “give” in one direction or the other. Air currents around the pendulum, such as those found in public museums, may also cause asymmetry. Long Foucault pendulums are often located in the stairwells of museums and, interestingly, currents from one-way traffic patterns of spectators going down the stairs – more than up the stairs – are sources of asymmetry. Open doors and windows can also cause problems. We can think of the situation in the following way. If the pendulum is “perfect” – totally symmetric – then the period of oscillation will be the same no matter what direction the oscillation. But if the pendulum is not “perfect” then, technically, the motion splits into two mutually perpendicular motions with two different periods. And again there is a “beat” frequency effect. But this time (unlike the anharmonicity problem), the motion is periodic with the “beat” in a different way. Again, 17 Most authors refer to the gradual rotation of the plane of oscillation (the Foucault effect) as “precession” of the plane. However, it has also been labeled as “veering” of the plane of oscillation.

Pendulums Measure the Earth

47

Y

we can picture the pendulum bob’s motion being projected onto a twodimensional plane, such as the floor of a museum. This is the motion illustrated by the phase-plane diagram of two mutually perpendicular oscillations at the slightly differing frequencies, as shown by the computer simulation of Figure 1.10. Typically, asymmetry will cause the pendulum to first display elliptical motion, then circular, then elliptical, and so forth in a repeated, time-varying, periodic pattern. For long pendulums with very low frequencies, this effect is less pronounced, since the frequency differences can be made relatively small. However, it is a serious problem for short pendulums and, typically, the asymmetry effect overshadows the Foucault effect in a relatively short time. A quarter century after Foucault’s original demonstration of the earth’s rotation, the first mathematical analysis of this asymmetry effect was carried out by Heike Kamerlingh Onnes (1853–1926), the

X

Figure 1.10 An accumulation of the pendulum bob’s path as it would appear from above the pendulum. The elliptical oscillation orbit caused by asymmetry gradually fattens to become circular. Then the orbit becomes thinner again, but in the direction perpendicular to the initial oscillation. The pattern repeats.

48

Seven Tales of the Pendulum

future discoverer of superconductivity,18 as part of his 1879 Ph.D. dissertation (see Figure 1.11). The problem was suggested to Kamerlingh Onnes by a former teacher, Gustav Kirchhoff (1824–87), whose laws of electrical circuitry are well known to undergraduate physics students. Not only did Kamerlingh Onnes do the analysis, but he also used his theoretical work as the basis for construction of a short Foucault pendulum with controllably variable asymmetry. Through a series of adjustments, he could equalize the pendulum motion in the two mutually perpendicular directions, thereby equalizing the resonant frequencies. Furthermore, through adjustment of the pivoting arrangement, there was also a capability for deliberately injecting asymmetry into the apparatus, in order to quantify its effects.

Figure 1.11 H. Kamerlingh Onnes. 18

Superconductivity occurs when electrical current travels in a material without any resistance. It requires very low temperatures, although continuing research is finding an increasing number of materials that are superconducting at less frigid temperatures. Superconductivity is used in many applications, such as in production of the very strong electromagnets found in medical MRI machines.

Pendulums Measure the Earth

49

Figure 1.12(a) shows an exploded view of the pivoting arrangement for the pendulum. Kamerlingh Onnes also placed the pendulum inside a gas-tight conical container to negate the effects of air drafts. He developed a special optical system – note the telescope in Figure 1.12(b) – to observe the pendulum displacement. The observation windows can be seen near the bottom of the container. Unlike the typical very long Foucault pendulum, this one is quite short, so that Kamerlingh Onnes could effectively study the oscillations’ changes provided by his adjustments to the pivoting arrangement, which is shown in the figure at the top of the support. Kamerlingh Onnes’ careful experimental work in this context was a precursor to his later and more famous experimental contribution to low-temperature physics. At this writing, the Kamerlingh Onnes apparatus is in storage at the Boerhaave Museum in Leiden, near the university where he did his Nobel prize-winning work in low-temperature physics – the liquefying of helium and the discovery of superconductivity.

(a)

50

Seven Tales of the Pendulum

(b) B

b ~ w c

A

k

z

Figure 1.12 (a) Several views of the support for the pendulum: adjustments could be made in the pivoting to exaggerate or minimize asymmetry. (b) A drawing depicting the arrangement of the complete apparatus: the apparatus was less than 2 meters high. (From Kamerlingh Onnes’ thesis.)

Occasionally, there are some very surprising connections between two areas of scientific endeavor. Kamerlingh Onnes’ work on superconductivity provides just such a link to one of those amazing connections; in this case, a connection between superconductivity and the pendulum. To see this connection, we need a little information about superconductivity and something called a Josephson junction. Superconductivity – discovered at Kammerlingh Onnes’ laboratory in 1911 – is a low-temperature phenomenon, in which electric current flows without resistance. It is important for both pure and applied science and, even today, is an active area of research. In 1961, Brian Josephson predicted some novel effects that might be found in a low-temperature device consisting of a junction: a thin piece of an appropriate nonconductor sandwiched

Pendulums Measure the Earth

51

between two pieces of superconducting material. He predicted the existence of (a) the spontaneous flow – without any applied voltage – of electric current across the junction (the d.c. Josephson effect), and (b) with an appropriate added constant voltage, the spontaneous creation of an alternating current across the junction, whose frequency depends only upon the size of the applied voltage (the a.c. Josephson effect). All this may seem a little technical, but these were novel and unconventional predictions with important implications. After much discussion of the theory and some experimental work, Josephson’s predictions were shown to be correct, and Brian Josephson was awarded the Nobel Prize in 1973. Today, Josephson junctions are used in a variety of high tech instruments, such as sensitive magnetometers.19 Now, here is the interesting coincidence involving pendulums. The mathematical equations that describe a typical configuration of the Josephson junction have an exact parallel with the equations of a driven pendulum that has friction (damping). The correspondence is so precise that a superconducting Josephson junction can, in a certain way, be thought of as a high-frequency pendulum – oscillating hundreds of millions of times per second. And parallel with that coincidence is the amazing fact that the man who was a pioneer in the world of superconducting research, Kamerlingh Onnes, started his own scientific career with research on the pendulum!

1.5.4 Energy dissipation Every real pendulum is subject to various frictional forces and a Foucault pendulum is no exception. A Foucault pendulum will need to be restarted periodically. However, it is more pleasing to incorporate some sort of drive mechanism into the pendulum, so that it receives a pulse of energy for each oscillation. Then the energy loss on each cycle is smoothly made up by the drive mechanism. The equipment in Figures 1.13 and 1.14 indicate that Foucault soon solved this problem – as early as 1855. In fact, his 1855 pendulum had the electromagnetic drive mechanism illustrated in Figure 1.13.

19 Magnetometers are sensitive instruments for measuring small variations in magnetic fields. They are commonly used in metal detectors, for oil and gas exploration, and in satellites.

52

Seven Tales of the Pendulum

Figure 1.13 An engraving of the electrical impulse mechanism used by Foucault to maintain the pendulum’s motion against frictional losses. (Bibliothèque de l’Observatoire de Paris.)

This remnant of this original magnetic drive, shown in Figure 1.14, is now stored at the Paris Observatory. It may be compared with the engraving in Figure 1.13. Many modern pendulums also have various mechanisms for giving the pendulum a push. Perhaps because these mechanisms are hidden from view, they sometimes – as we will learn – lead the casual observer to the wrong conclusion.

Pendulums Measure the Earth

53

Figure 1.14 The surviving remnant of Foucault’s original electromagnetic driver from the impulse mechanism in Figure 1.13. (Bibliothèque de l’Observatoire de Paris.)

1.5.5 Some modern solutions We have seen that the main problems associated with building a working Foucault pendulum can be minimized by making a long pendulum with a massive bob. Foucault’s dramatic 1851 presentation in the Pantheon worked because his pendulum was long – 67 meters – and because he used a massive bob of 28 kilograms, giving the pendulum plenty of inertia, so that it would oscillate for long enough to illustrate the desired effect. Furthermore, asymmetry in long pendulums is not a problem. But long heavy pendulums can cause a safety problem. A month or two after Foucault swung his pendulum in the Pantheon, the suspending wire broke at the point of suspension. This was very dangerous because the tension in the wire caused it to act like a dangerous whip on any spectators who might be nearby. As a result of this

54

Seven Tales of the Pendulum

Figure 1.15 Two members of the south pole group with their Foucault pendulum bob. (Courtesy of John Bird.)

accident, Foucault designed a “parachute” that would support the wire just below the suspension, in case of breakage near the pivot. The parachute is a kind of cylinder that is clamped to the wire and, in the event of breakage, falls a short distance to a supporting ring. Foucault’s original parachute, together with the suspension structure, is stored in the Paris Observatory. For short pendulums, all of the previously described effects are significant. In 1931, the asymmetry problem was partially solved by a simple ring, now known as the Charron ring. It is often placed just below

Pendulums Measure the Earth

55

the pivot point, such that it would surround the suspending wire. The diameter of the ring is made slightly less than the full swing of the wire, and therefore the wire20 contacts the ring at the extremes of its swing. This delicate impact arrests the various spurious precessional motions of the pendulum. However, as is well known, the ring succeeds only partially, in that it does introduce a faster dissipation of the pendulum’s energy. It now becomes even more important to provide a source of energy to keep the pendulum in motion. Various methods have been developed to provide energy to the pendulum. We noted that Foucault himself built an electromagnetic drive for the 1855 pendulum. A more sophisticated and modern design was used by the American physicist Richard Crane (1981). He placed a small permanent magnet in the bob of the pendulum that interacted with a combination of permanent and time-varying electromagnets fixed in the rigid structure below the pendulum. The electromagnet gave a pulse that changed orientation every quarter of an oscillation. In this way, the magnet in the bob received a push outward when the bob was in the middle and an inward pull when the bob was at its extreme displacement. With this setup, Crane was able to eliminate spurious precession and, at the same time, give a small periodic push to the pendulum bob. A different mechanism that mitigates problems and provides energy input is parametric pumping. The word parametric refers to some characteristic of the pendulum such as its length, mass, or the operative gravitational field. Rather than pushing the pendulum at the bob, the pendulum is driven by a periodic vertical movement of the suspension point. This motion periodically changes the effective strength of the gravitational field – a parameter of the system – and thereby injects energy into the motion. The British physicist Brian Pippard has shown that parametric pumping of the pendulum helps to eliminate the tendency of the Foucault pendulum toward elliptical motion, as well as keeping it in motion through the periodic infusion of energy. The pendulum installed in the London Science Museum in 1988 was built with such a mechanism, according to a design by Pippard. (Another variant of parametric pumping is the effort a child on a swing exerts to keep the swing moving. The periodic motion of the child changes the 20 In some cases the ring is actually placed below the bob, where it makes its slight contact with a sharp protrusion from the bob.

56

Seven Tales of the Pendulum

effective length – another physical parameter – of the playground swing or pendulum. Again, the motional energy is transformed into larger swings.) We noted that the Foucault effect is most obvious at one of the earth’s poles. A crude Foucault pendulum was constructed and tested at the south pole by three adventurous experimenters, Michael Town, John Bird, and Alan Baker. Their measurement is probably the closest ever made to one of the earth’s poles. The pendulum was erected in a six-story staircase of a new station that was under construction near the pole. Conditions were challenging; the altitude was about 3,300 meters (atmospheric pressure only about 65 percent that at sea level) and the temperature in the unheated staircase was about −68°C (−90°F). The pendulum had a length of 33 meters and a 25 kilogram bob. Interestingly, it is against the Antarctica Treaty to have an open flame at the south pole, and therefore the authors could not use the burnt-thread technique to start the pendulum. Instead, they became adept at dropping the bob so that it would oscillate in a plane. Installing even a crude pendulum in these conditions was challenging. But once they overcame the obstacles, these researchers did confirm a period of about 24 hours as the rotation period of the plane of oscillation (see Figure 1.15).

1.6 Pendulums go public Before television, the internet, and other modern media, scientists often took their science to the public by giving spectacular demonstrations. Following this tradition, in February 1851 Foucault boldly invited contemporary scientists to a demonstration at the Paris Observatory with the words “Vous êtes invités à venir tourner la terre . . .” (“You are invited to come to the turning of the earth . . .”). The much longer Pantheon demonstration pendulum, rapidly constructed for a March opening, produced a flood of visitors. Foucault’s pendulum had indeed gone public. Reaction to Foucault’s demonstration was widespread. A few examples will suffice. Two pieces from the Proceedings of the Royal Society of London are instructive. First, the English geologist John Philips (1800–74), a Fellow of the Royal Society (FRS), communicated work that Thomas Cooke, an optician in the city of York, had done with his own version

Pendulums Measure the Earth

57

Figure 1.16 An engraving from L’Illustration, Paris, 1851.

of the pendulum. Cooke used a 16 meter pendulum with two different bobs, one an oblate spheroid with a mass of about 18 kilograms, and the other a prolate spheroid with a mass of 44 kilograms. The pendulums performed as anticipated, except that in some experiments, there was a small but noticeable elliptical aspect to the path of the pendulum’s motion and a veering of the elliptical motion. Philips also undertook

58

Seven Tales of the Pendulum

experiments in his house with much smaller pendulums and found that the spurious effects – which we described above – were more pronounced. Second, the contemporary British Astronomer Royal, George Airy, FRS (1801–92), provided Philips and Cooke with a partial explanation of the so-called clockmaker’s formula that quantifies the rate of turning of the ellipse of a pendulum due to anharmonicity. Also, the same issue of the Proceedings had a contribution from an English experimentalist and Fellow of the Royal Society, Charles Wheatstone (1802–1875), who was a prominent British physicist in the field of instrumentation. (His most famous invention, the electric telegraph with various improvements, earned him a knighthood in 1868.) Wheatstone emphasized (as we described earlier) that Foucault had come to the idea of his pendulum’s rotation of the plane of oscillation by noting the effect of coupling one motion with another. Wheatstone went on to describe several other instances of this type of interaction and had himself built a mechanical device that coupled together vibration of a spring, rather than a pendulum, with rotation of the earth. The point that Wheatstone made with this device was that gravity was not necessary to achieve motion analogous to the Foucault effect. On the other side of the Atlantic Ocean, a veritable ‘pendulum mania’ occurred. Over 30 large Foucault pendulums were built at diverse locations along the East Coast of the United States. Scholarly discussion in American journals was also forthcoming. For example, the Journal of the Franklin Institute, in Philadelphia, records several contributions on the subject in 1851. Charles Allen provided a geometric analysis of the pendulum’s motion. In the same volume, J. S. Brown also proposed an analysis. He prefaces his communication with the provocative statement: “ . . . in all the explanations of Foucault’s pendulum . . . not one has given . . . a full exposition of its phenomena, and many have made false statements”. In a footnote, the journal’s editor takes polite but firm issue with Brown and, in reference to Allen’s contribution, he states that it “ . . . appears to us to be thoroughly correct”. Finally, on the rotation of the earth, there is the much quoted tongue-in-cheek note to the editor of a contemporary issue of Punch magazine: “Sir, – Allow me to call your serious and polite attention to the extraordinary phenomenon, demonstrating the

Pendulums Measure the Earth

59

rotation of the Earth, which I at this present moment experience, and you yourself or anybody else, I have not the slightest doubt, would be satisfied of, under similar circumstances. Some sceptical and obstinate individuals may doubt that the Earth’s motion is visible, but I say from personal observation it’s a positive fact. I don’t care about latitude or longitude, or a vibratory pendulum revolving round the sine of a tangent on a spherical surface, nor axes, nor apsides, nor anything of the sort. That is all rubbish. All I know is, I see the ceiling of this coffee-room going round. I perceive this distinctly with the naked eye – only my sight has been sharpened by a slight stimulant. I write after my sixth go of brandy-and-water, whereof witness my hand, Swiggins Goose and Gridiron, May 5th, 1851.”

The Foucault pendulum was certainly science with a high profile! Even today, the Foucault pendulum continues to provide a dramatic and obvious demonstration of the earth’s rotation; it is found in many museums, schools, and other public buildings. A representative list is given in Table 1.1. Some of the listed pendulums are very long and have massive bobs, a combination that allows them to partially sustain their motion against the inevitable loss of energy due to friction in the bearing, air resistance, and so forth. Other pendulums, often with smaller dimensions, use some sort of drive mechanism to supply energy that compensates for the losses owing to friction. Unfortunately, the indefinitely sustained oscillations provided by hidden drive mechanisms can sometimes reinforce misconceptions. Donald Ivey, a professor at the University of Toronto, once told his physics class a story regarding such a misconception and the Foucault pendulum. During a visit to New York City, Ivey joined a tour of the UN building. At the site of its Foucault pendulum, a fairly large crowd observed the slow back-and-forth motion of the pendulum, hoping to discern the veering motion of the plane of oscillation. At some point, a woman within earshot of Ivey stated emphatically that the phenomenon of the ongoing oscillations was an example of perpetual motion. Ivey began to squirm. As a physicist, he knew that perpetual motion contravenes the physical principles of both conservation of energy and the Second Law of Thermodynamics. Eventually, Ivey could not contain

60

Seven Tales of the Pendulum

himself. He turned to the woman and quietly explained that there was no such thing as perpetual motion, and that the pendulum had a hidden drive mechanism. She flatly and noisily disagreed with him. Ivey then, and perhaps foolishly, revealed himself to be an expert, namely a professor of physics. At this point, the woman loudly shared with the crowd her low opinion of experts in general and of university professors in particular. In the face of this verbal assault and possible public ridicule, Professor Ivey wisely decided to retreat from this field of “intellectual” combat. The degree of “disconnect” between the scientist Ivey and the unknown tourist is not unique. Figure 1.16 illustrates corresponding attitudes in the nineteenth century as the scientist tries to explain the pendulum to the girl.

Table 1.1 A partial list of the existing Foucault pendulums. Location Pantheon, Paris UN Building, New York City Franklin Institute, Philadelphia University of Guelph, Canada University of Maryland Griffith Observatory, Los Angeles Yale University Ryerson Library, Michigan Morrison Planetarium Science Museum, London Ottawa High School, Michigan

Length (m)

Mass (kg)

67 23

47 (1995 bob) 90

26

410

0.83

4.5

14 12

28 100

11 22

12 14

9 20

87 9

9

14

Pendulums Measure the Earth

61

Quoting from a translation of the original caption: “This tableau, which appeared soon after . . . (the original 1851 demonstration of the Foucault pendulum) . . . reveals some of the typically French attitudes toward science and nature. The “Philosopher” (or scientist) pays no attention to the beauty of the scene, seeing neither the vegetation nor the pretty girl (on) the swing. What he observes is abstracted to a geometrical problem, free from passion and life. The girl’s response to his mathematical analysis is ‘True, but who cares?’ To the philosopher, science is measurement; to the young girl, it is irrelevant because it omits life.”

Whatever the general perception, we see, even from this first tale, that the seemingly unremarkable pendulum produces an abundant harvest of scientific knowledge and cultural impact.

Tale 2: Pendulums Keep Time Timekeeping with a wide variety of primitive devices – water clocks, burning candles, flowing sand – has a long history. For millennia, the simplicity of individual human life and the lack of organized communal activity meant that accuracy was not a high priority. Human affairs were mainly local, and therefore exactness and standardization in the measurement of any physical quantity were not essential. Furthermore, the apparent motion of the sun as measured for many centuries by, for example, a sundial gave a quantitative idea of the passage of time that was sufficient for most purposes. Ultimately, however, more urgent needs for public timekeeping led to a progression of significant advances and inventions. The early mechanical clocks were no more reliable than sundials (when the sky was clear), but the successors to these first crude devices would eventually achieve undreamed-of accuracy.

2.1 Mechanical clocks pre-pendulum The epoch of the first European large mechanical clocks began in 1335, with a public clock erected in Milan. Soon, many towns and cities throughout Europe had installed similar clocks in cathedrals and public squares. Salisbury Cathedral has the world’s oldest working mechanical clock, dating to 1386. Another spectacular, and still-visible, example is the complex Astronomical Clock in Prague (1410), to which tourists flock in order to watch traditional religious figures make their appearance with its regular chiming (see Figure 2.1). These large timekeeping machines originally employed the so-called verge and foliot as their mechanical oscillators, as shown in Figure 2.2. The regulator function of this clock is derived from the horizontal beam (foliot), together with the combination of vertical rod and attached plates called the verge. The foliot oscillates back and forth subject to the periodic

Pendulums Keep Time

63

Figure 2.1 The Astronomical Clock in the town square of Prague. The apostles appear at the window of the upper part. The middle face is the Astronomical dial and the bottom face is the Calendar dial. (Photo by GLB.)

connection of two plates with the toothed wheel, known as the crown wheel. The weight, hanging from the spindle attached to the shaft of the crown wheel, provides the energy of the system. It pulls the crown wheel into motion and, without the periodic braking of the crown and plates by the foliot, would cause the crown to spin freely. But controlled by the plates or pallets, the crown wheel turns slowly as it repeatedly experiences push, pause, push, and so on as the drive is transferred back and forth between the upper and lower pallets. The period of the motion can be modified by increasing or decreasing the inertia of the beam through repositioning of the weights that

64

Seven Tales of the Pendulum

Figure 2.2 The verge-and-foliot mechanism. Through a series of gears, the hour hand is driven by the shaft from the crown. (The lower pallet is hidden from view behind the vertical shaft of the verge.)

are hung on the foliot, as shown in Figure 2.2. Moving these outward slows the angular spin rate and so makes the period of oscillation longer; moving the weights inward speeds up the oscillations. The overall regulation of the foliot’s frequency of oscillation – and therefore the clock’s timekeeping – is provided by the complicated interaction of the foliot and verge with the crown wheel. The clock from Dover Castle shown in Figure 2.3 exemplifies the early verge-and-foliot mechanism. The view clearly shows the foliot with its trim weights. The large crown wheel is driven via a small pinion by the rope wound around the drum. The long yoke-shaped foliot and its hanging rectangular weights are visible at the top of the apparatus. The verge and foliot does not make a very precise oscillator. Its performance is limited by the fact that there is no strong natural frequency to which it is drawn. By contrast, a good oscillator – such as the pendulum

Pendulums Keep Time

65

Figure 2.3 The clock from Dover Castle. The design is essentially fourteenth century, but the date of the clock is likely early seventeenth century. (Reproduced with permission from the Science Museum/Science and Society Picture Library, London.)

– has a well-defined frequency and possesses a sharp resonance that strongly stabilizes the central frequency.1 In these early clocks, timekeeping was no better than within about a half or quarter hour per day. There was no minute hand and the clock 1

The tightness of frequency definition is measured by the “quality” factor, or Q. Pendulums have a higher Q than the foliot. The quartz oscillator used as a regulator in typical modern clocks has very strong frequency definition and therefore a very high Q – much higher than that of a pendulum.

66

Seven Tales of the Pendulum

faces were marked only in hours. The timekeeping technology of the verge and foliot matched the rather casual needs of contemporary daily life. Yet expansion of global discovery and the consequent need to accurately measure longitude (and latitude) at sea would encourage further development.2

2.2 The development of the pendulum clock 2.2.1 Galileo (1564–1642) By 1637, Galileo had realized the possibility of using the pendulum as the central timing mechanism in a clock. Eighteen years after the fact, his young assistant Vincento Viviani recounted in a letter to Prince Leopold de Medici (dated August 20, 1659), that sometime in the summer of 1641 “ . . . it came into his [Galileo’s] mind that the pendulum could be adapted in place of the customary balance, to clocks driven by weight or spring, in the hope that the perfect natural equality of its motion would correct all the defects of the art of these clocks”.

This is the definitive moment at which the history of the pendulum clock might be said to have commenced. Galileo died in January of the next year, 1642, at the age of 78. (At the end of the same year, on Christmas Day, Isaac Newton was born.) With the help of Galileo’s son Vincenzio, a design for a pendulum clock had been begun in 1641, but the work had then been set aside and was not resumed until April 1649, well after Galileo’s death. A central idea was to employ a falling weight as a driving force to maintain the pendulum’s motion: “ . . . and thus in a certain manner was perpetually maintained the to-and-fro motion of the pendulum, until the weight had run down to the bottom”. (Letter from Viviani to Prince Leopold, August 20, 1659)

2

Longitude at sea can easily be measured if the ship carries a clock that is synchronized with some place (on land) of known longitude, such as Greenwich, England. In the seventeenth century, there was no clock that could keep sufficiently accurate time throughout a voyage of several months or even years.

Pendulums Keep Time

67

The loss of energy of the pendulum and in the gears is replaced by energy from falling weights. Yet if the falling weights were not somehow restricted, they would cause an uninterrupted rotation of the crown wheel. Instead, the pallets of the verge, alternately hitting the crown wheel, convert the rotational motion to a back and forth motion that provides the periodic impulse to the pendulum. A device that converts rotational motion to back and forth motion is called an escapement. Galileo’s design, shown in Figure 2.4, used a form of escapement involving pivoted arms, notched wheels, and so forth. It seems to have had some problems, and a working model was left uncompleted at the time of Vincenzio’s death in May 1649. As we will see, the escapement underwent considerable improvement in the search for ever more accurate clocks.

2.2.2 Huygens (1629–1695) The Dutch scientist and mathematician Christiaan Huygens (Figure 2.5) provided the next critical steps in the evolution of the pendulum clock. In 1656–57, Huygens invented what we now consider to be the first truly practical design. His achievement lay in the creation of a working escapement with a linkage to the swinging pendulum. This clock was first described in Huygens’ small book of 1658: Horologium. His major work, Horologium Oscillatorium (The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendulums as Applied to Clocks), was published 15 years later, in 1673. Part I of this latter treatise is devoted to the design of the pendulum clock, as well as its possible application to the problem of determining longitude at sea. Huygens describes a test that was carried out during the voyage of a British ship in 1664. Two of his clocks, each with a pendulum length of 9 inches and a weight of half a pound, were fitted out for sea and sent on a voyage to the west coast of Africa. According to Huygens, the results were promising, and he went on to propose modifications to improve their stability with respect to the rolling and pitching motions of a ship. Part I of the Horologium Oscillatorium (p. 32) concludes: “Instruments which have been modified in this way have yet to be taken to sea and put to the test. But they offer an almost certain hope of success because, in experiments conducted so far, they have been found to hold up under every kind of motion much better than prior clocks.”

68

Seven Tales of the Pendulum

Figure 2.4 Galileo’s design for a pendulum clock. The drawing is dated to 1659 and is attributed to Galileo’s friend Viviani. (Reproduced with permission from the Science Museum/Science and Society Picture Library, London.)

Pendulums Keep Time

69

Figure 2.5 A portrait of Christiaan Huygens. (Reproduced with permission from the Science Museum/Science and Society Picture Library, London.)

It was not that easy. The problem of determining longitude with suitable accuracy had to wait for almost another century, and the solution lay not in pendulum clocks after all. Rather, in 1759, John Harrison perfected an extremely sophisticated marine chronometer that employed a mainspring and balance wheel. But let us return to Huygens’ clock. Huygens’ essential idea for a clock was a very simple, yet revolutionary, modification of the vergeand-foliot mechanism – just attach a pendulum directly to the verge and discard the foliot beam. Figure 2.6 illustrates this arrangement schematically and again at the top left of his complete design, as presented in Horologium Oscillatorium and shown in Figure 2.7.

70

Seven Tales of the Pendulum

Part I of the illustration shows a side view of the complete clock. The pendulum mass (X) hangs at the end of a rod from point (V), above which a double cord extends to the upper clamp, as shown in the front view of Part II of the figure. Other key mechanical components, also shown in Part I, are the crown wheel (K) with 15 teeth and the horizontal verge (LM) with two pallets. A cord with a hanging weight is wrapped around the studded wheel (D) and provides gravitational drive to the crown wheel through intermediary gears and pinions (CEFGHI). As the pendulum swings back and forth, the shaped rod (S), referred to as the crutch, follows the motion and causes the verge rod to turn

Figure 2.6 The basic arrangement for Huygens’ pendulum clock escapement. The externally driven crown wheel rotates in only one direction (front edge to the right, back edge to the left), but does so in interrupted intervals paced by the pendulum-driven to-and-fro motion of the verge.

Pendulums Keep Time

71

Figure 2.7 Huygens’ clock design, as illustrated in the Horologium Oscillatorium (1673).

correspondingly clockwise and counterclockwise. The pallets alternate between engaging a tooth on the right side of the crown wheel or a tooth on the left. Each time an engaged tooth is released, the crown wheel freely advances slightly until stopped when a tooth on the other side of the wheel hits the other pallet. The “tick” and “tock” are the clicks as crown wheel teeth alternately hit the pallets. Hence the operation of the escapement amounts to repeatedly blocking and releasing

72

Seven Tales of the Pendulum

the crown wheel. Each time the forward rotation of the crown wheel is arrested, there is a slight recoil of the pallet. Huygens describes the effect in general terms: “For the small rod S, which is moved very slightly by the force of the wheels, not only follows the pendulum which moves it, but also helps its motion for a short time during each swing of the pendulum. It thus perpetuates the motion of the pendulum which otherwise would gradually slow down and come to a stop on its own, or more properly because of air resistance.”

Huygens clearly understood the dual requirements of a clock escapement: (1) counting beats and (2) making up for energy losses. Huygens’ design incorporated one additional innovation. Part II of Figure 2.7 depicts two thin curved plates called “cheeks”. As the pendulum swings, the double suspension cord wraps and unwraps along the inside faces of these cheeks. Huygens describes his reasoning in the following quote from Horologium Oscillatorium: “We must now describe the form of the thin plates between which the pendulum is suspended, and whose most important function is to guarantee the regular motion of the clock. For without them the simple oscillations of the pendulum will not be equal over a long period of time (even though some have thought otherwise), but rather oscillations through smaller arcs will occur in a shorter time.”

Remarkably, Huygens realized that the simple pendulum was not, as Galileo had supposed, isochronous. The period of an oscillation did indeed increase with the size of the arc of the swing. Even for a clock whose pendulum swung 10 degrees, this effect amounted to a loss of almost 3 minutes per day compared to a clock with negligible amplitude. And the early clocks often had swings that were as much as 100 degrees. It turns out that this problem of the period increasing with increasing amplitude of swing could be eliminated if there were some means to gradually shorten the effective length of the pendulum for larger swings. By shortening the length, the period could be made to return to that of a pendulum with negligible swing amplitude. This shortening is accomplished by

Pendulums Keep Time

73

constraining the flexible suspension cord against some curved “cheeks”, as shown in Figure 2.8. As the pendulum swings higher, the effective vertex of the pendulum is moved down the suspension cord. The curve of these cheeks is a special curve (not quite a circular arc) called a cycloid, as shown in Figure 2.8. (Of course, if the amplitude always stayed constant, then no length compensation would be needed.) The mathematical proof that the cycloid is the correct shape for the cheeks is usually done using calculus, but at the time of the Horologium Oscillatorium, Isaac Newton’s Principia was not to appear for another 14 years (in 1687). Huygens’ methods of investigation were mainly an insightful combination of logical reasoning and geometrical constructions. He arrived at the following conclusion: “For it is clear that a pendulum which is suspended and set in motion between two plates curved in the shape of a semi-cycloid describes an arc of a cycloid by its motion, and thus oscillations, whatever their magnitude, are completed in equal times.”

Figure 2.8 A pendulum whose suspension cord is constrained between two cycloidal cheeks (shaded). By wrapping around the upper portion of a cheek, the cord is effectively shortened slightly for larger displacement angles, and the pendulum bob traces a cycloidal, not circular, path.

74

Seven Tales of the Pendulum

By constraining the suspension of the pendulum between cycloidal cheeks, the period can be made independent of the amplitude of the swing. The cheeks are shown in Huygens’ design in Figure II of Figure 2.7 and again schematically in Figure 2.8. We therefore see that Huygens combined mathematical insight with experimental ingenuity in the design of the clock, to go beyond the sort of first approximation one might have expected from a first working model. Despite the beautiful ingenuity of this method, we shall see that later developments in the pendulum clock mechanisms ended the need for the cycloidal cheeks. But, before we learn more, let us take an intermission from the technical aspects of the clock in order to see how the period of the pendulum almost became part of the definition of a scientific standard for length. It seems amazing that a measurement of time almost became a standard of distance.

2.3 Time as a standard of length? An ideal simple pendulum at sea level and latitude 45 degrees, executing very small angle oscillations without any frictional losses, will take exactly 1 second to swing from left to right or right to left, provided that its length is precisely 99.357 centimeters (39.12 inches). Such a pendulum has historically been referred to as a “seconds pendulum”.3 This coincidental relationship between the half-period of the pendulum and its length played an important, but short-lived, role in the definition of the standard meter. That is, the length of such a pendulum almost became the standard for a meter. Let us see how this could happen. We now take for granted the existence of worldwide standards of measurement such as the precisely defined meter, second, and kilogram. Yet in the late 1700s and early 1800s, the quantities used for measurement were anything but standardized. For example, in France prior to the Revolution, a whole array of different length measurements was in common usage. These included the “toise”, which was defined as six royal feet, or about 2 meters. Or there was the “parish” measuring rod, which consisted of the sum of the lengths of the feet of 16 randomly 3 Recall that in modern parlance, the period denotes the time for a complete dynamical cycle, so the period of a seconds pendulum is actually 2 seconds.

Pendulums Keep Time

75

sized men, measured as they left the parish church. Aside from the plethora of imprecisely defined units, the same unit could also mean different measures in different places or different circumstances. For example, a pint of wine in Paris was relatively small compared to a pint of wine in Précy-sous-Thil, where it was more than three times the Parisian pint. The French system also had a feudal component, in that the landowner would always want his tenants to pay him in the largest size of a particular unit of measurement, whereas he would want to sell produce in the smallest size of the same unit. In England, the situation was somewhat less confusing, and there were even laws that mandated a national standard for weights and measures. Nevertheless, standard measures of length and volume contained inconsistencies. For example, a gallon typically contained less, by a half pint, than four quarts. Contemporary with these domestic issues, scientists were swinging pendulums in various parts of the world to determine the shape of the earth, as noted in our first tale. Momentum on both sides of the Channel built for using the length of the “seconds” pendulum as the standard meter. Christopher Wren (1632–1723), the noted English architect, is believed to have been the first to suggest this standard, but it also received early support from Christiaan Huygens and Jean Picard. Of course, because of the deviation of the earth’s shape from sphericity, and its variable topography and density, the standard pendulum would have to be swung in a specified location. After the French Revolution, the National Assembly turned to the question of a single standard. A Committee of Weights and Measures was appointed. In 1790, Talleyrand, French statesman and diplomat during the turbulent years before and after the Revolution, suggested that the unit of length be that of a seconds pendulum at 45 degrees latitude. And in 1792, new measurements were made on the seconds pendulum at the Paris Observatory. But even by 1791 events had moved away from a pendulum standard. The National Academy, upon which the Assembly relied in matters of science, had another agenda, namely their longstanding project of measuring the meridian through Paris. Therefore the Assembly voted to define the meter as one ten-millionth of the distance from the equator to the north pole, measured along the meridian of longitude passing through Paris. Monies were appropriated, new surveys along the meridian were funded, and in 1799 these

76

Seven Tales of the Pendulum

measurements led to a brass meter bar as the contemporary standard meter. In England, the pendulum seemed to fare somewhat better. For example, Henry Kater, of Kater’s pendulum fame, was appointed a member of the 1819 royal commission on weights and measures. Yet, there was still the matter of where to swing the pendulum, and as pendulum measurements became increasingly more precise, this issue became increasingly urgent. In 1824, the pendulum was given a temporary legal niche in the measurement of a meter in a rather unique way. The Weights and Measures Act of 1824 actually mandated an Imperial standard yard, based upon a length previously constructed in 1760. However, the act stipulated that in the event that this physical yard was destroyed or damaged, a new standard length would be created using a pendulum swung in a place with the same latitude as that of London. (This lack of precision did not help the pendulum’s cause.) Remarkably, on October 16, 1834, a fire in Westminster burnt the House of Commons and damaged the standard yard beyond repair. It would seem that the pendulum was about to create a new standard of length. But the promise was not to be realized. Because of delays, lack of agreement as to where the pendulum should be swung, and advances in other measurement techniques, the British pendulum standard for length was abandoned in 1851. Thus the pendulum’s legal celebrity as a standard of length was relatively short-lived. The definition of the meter has changed significantly. Today, it is the distance traveled by light in a vacuum within a time of 1/299,792,458 of a second. The length definition is derived from the universality of the speed of light, and the definition of the second (which is now based on a resonant frequency of the cesium atom: 9,192,631,770 Hertz). Let us now return to some more details of the development of the pendulum clock.

2.4 Escapements The escapement of a pendulum clock is at the heart of its operation. Energy is typically provided by raised weights and the escapement acts to control the flow of this energy, in small installments, to the pendulum and also to the rest of the clock mechanism. In his original clock, Huygens used the verge and crown wheel (now more generally referred to as an escape wheel, or as the scape wheel).

Pendulums Keep Time

77

In this original version, the verge has to rock back and forth through a considerable angle – perhaps 20 degrees or so. Consequently, the pendulum must swing with a rather large amplitude for the verge to work properly. The large swings introduce errors, but these, as we noted, could in turn be rectified with cycloidal cheeks, at least theoretically. Within a short time, significant technical advances took place in escapement design. In about 1671, the anchor escapement appeared. The originator of this innovation is not known for certain, but it is variously attributed to the brilliant scientist Robert Hooke or to the clockmaker William Clement. A drawing of this escapement is shown in Figure 2.9. Note the vague resemblance to a ship’s anchor. One notable change is that whereas in the verge escapement the shaft of the verge and the shaft of the crown wheel are perpendicular to one another (see Figure 2.6), in the anchor escapement the shaft of the anchor is parallel to the shaft of the escape wheel. That is, for the anchor escapement, all rotating components (pendulum, escape wheel, and anchor) move in parallel planes. This means that a thin clock mechanism can be produced. More importantly, in contrast to the verge and crown wheel, the anchor escapement can accommodate quite small pendulum oscillations. This improvement in performance immediately made possible long pendulum clocks and reduced the potential usefulness of cycloidal cheeks. In fact, cycloidal cheeks brought their own set of problems and are now no longer used in high-performance pendulum clocks.

THE ANCHOR ESCAPEMENT

Figure 2.9 An anchor escapement consisting of an escape wheel and pallet.

78

Seven Tales of the Pendulum

The year 1675 seems to have been important one for clock development. On March 2, Charles II signed the order creating John Flamsteed (1646–1719) the first Astronomer Royal and founding the Royal Observatory at Greenwich. Flamsteed was determined to have accurate clocks for his astronomical work and energetically pursued improvements to existing technology. Richard Towneley (1629–1707), a member of a group of scientists from northern England, often corresponded with Flamsteed on this matter. He is credited, in the same year, with producing a design for a new escapement that was supposed to eliminate the recoil problem of the anchor escapement. In the old anchor movement, just after a tooth of the escape wheel is intercepted by one of the pallets, there is a slight recoil as the pallet pushes further in and forces the escape wheel slightly backward. In clocks that employ this form of escapement, there is a slight observable shiver in the seconds hand at each tick. Towneley’s new design was implemented by the artisan Thomas Tompian; and by July 1676, Flamsteed had Tompion pendulum clocks that were supposed to run for a year without resetting the weights. As experience was gained with those clocks, Tompion improved on Towneley’s design. Flamsteed himself seems to have also contributed modifications in design. However, it was not until 1715 that the clockmaker George Graham really solved the recoil problem and created a reliable dead-beat escapement. The dead-beat design, shown in Figure 2.10, significantly reduces recoil through clever shaping of the contact surfaces of the two pallets. Each plate (pallet) has a relatively large stopping face and a smaller, angled impulse face. The particular shape and angle of each face is such that when contact is made with the escape wheel, there is a bit of push provided from the wheel to the pallets and indirectly to the pendulum. These pushes from the escapement act to make good the frictional losses and keep the pendulum running. The important improvement here is that no recoil is generated as a wheel tooth slides along a stopping face because the stopping faces are arcs, with the pallet axis as their center. There are many designs for escapements, of which could be mentioned the pin-wheel, the Brocot pin pallet, the Mudge gravity, the Grimthorpe three-legged gravity, the Riefler, and the Bloxham. They can be quite complex and ingenious in their attempts to refine the art and enable more and more accuracy in clocks. As well as art, there is also some science behind the escapement, and in 1872, George Airy, the

Pendulums Keep Time

pallet A stopping face impulse face

79

pallet B stopping face impulse face

Figure 2.10 The dead-beat escapement.

seventh Astronomer Royal, published a mathematical analysis of the mechanics of escapements in pendulum clocks.

2.5 Temperature compensation At this point, we are very aware that the length of a pendulum is allimportant to its role as a clock regulator. Heat and cold will change this length and there needs to be some means of minimizing resultant changes. According to the usual laws of thermal expansion at everyday temperatures, the change in the length of a rod is proportional to the original length of the rod and the size of the temperature difference from some standard temperature. For example, a 1 meter rod made from steel, undergoing a temperature rise of, say five centigrade degrees, would increase in length by 0.06 millimeters (a relative change of six parts in 100,000). The lengthened steel rod would result in a period change of three parts in 100,000, enough to produce an error of more than a second per day. We might not mind this small change in a household clock, but such a change would be significant for the longitude toward the end of a three-year sailing voyage. In 1729, John Harrison (whose spring-driven marine clock eventually solved the longitude problem) invented the so-called gridiron pendulum. This is one method that cleverly and beautifully compensated for

80

Seven Tales of the Pendulum

temperature variations. The underlying principle of the gridiron compensator is illustrated in Figure 2.11 by a three-segment construction. In essence, pieces of two different metals are alternately attached together such that, with heating, one metal expands downward while the other metal expands upward, each doing its job by just the right amount so that the overall length of the structure remains the same. The pairs of complementary metals are chosen according to their thermal expansion properties and the number of segments to be in the gridiron. For example, for a five-segment gridiron, the combination of red brass and carbon steel is a suitable choice. The relationship of thermal expansion properties to the number of segments in the gridiron leads to one of those strange numerical coincidences that occasionally turns up in science. For example, for a threesegment gridiron, the ratio of thermal expansion of the pairs of metals must be close to 2/1. Yet the coincidence is easy to understand. Looking at the three-segment gridiron, we see that the inside rod must, by itself, compensate for the two outside pieces. Similarly, for a five-segment gridiron, the two inside pieces must compensate for the other three pieces, and therefore the ratio of the thermal expansions must be 3/2. For the seven- and nine-segment gridirons, the respective ratios are 4/3 and 5/4. It seems quite remarkable that the numerical factors in the fraction actually add up to equal the required number of segments in the gridiron! Finally, we see that, because of the decreasing ratios as the number of segments increases, the thermal properties of the two

L1 L2 L3 D1 D2 D3

Figure 2.11 The principle of the gridiron temperature compensator. The middle rod is of a different metal than that of the outer rods. The top segment shows the unexpanded version, whereas the bottom segment shows the effect of heating. Yet the overall length stays the same.

Pendulums Keep Time

81

metals may be increasingly similar. Therefore gridirons with a large number of segments are more likely. A full nine-segment gridiron of an early nineteenth century Parisian clock is shown in Figure 2.12. Because of their beauty, gridirons were relatively common in quality pendulum clocks even into the early twentieth century. However, ambient temperatures in modern homes are relatively stable and while modern pendulums may appear to have gridirons, they are usually only simulated. While the gridiron solution is perhaps the most elegant, other strategies have been devised for temperature compensation. In 1721, George (a)

(b)

Figure 2.12 The arrangement of metal rods in a gridiron pendulum, in a French clock dated to circa 1815. (Photo by GLB, courtesy of the National Watch and Clock Museum, USA.)

82

Seven Tales of the Pendulum

Graham placed a container with mercury at the end of the steel pendulum rod. When warmed, the steel rod lengthens, but the mercury also rises, just as in a warmed thermometer, and this keeps the effective pendulum length constant. However, in 1895, the need for interesting temperature compensation mechanisms was greatly diminished when C. E. Guillaume produced an alloy of iron and nickel, known as Invar. The important feature of Invar is that it expands about 100 times less than normal metals – about five parts in ten million. But even though Invar expands very little, in clocks that require great precision, some compensation may be required. One method is to make the Invar rod hollow and partially fill it with mercury. As in Graham’s method, when the rod lengthens with temperature increase, expansion of the mercury compensates, thereby keeping the effective length to its original value.

2.6 Pendulum clocks into the twentieth century By the end of the nineteenth century the error in pendulum clocks, as typified by those from Siegmund Riefler of Munich, had diminished daily error to around 0.1 second. Riefler’s own design took the additional step of providing a partial vacuum for the pendulum, by enclosing the clock mechanism within a metal cylinder. Riefler regulators were the standard for timekeeping for about 30 years. But in 1921 a revolutionary masterpiece of pendulum clock design appeared – William H. Shortt’s subtle and intricate dual pendulum arrangement. In providing a very brief description of the Shortt clock, we definitely move away from the historic pendulum clock, as much more complexity is introduced. Furthermore, the Shortt clock is not just a mechanical pendulum clock; it also utilizes electricity. The idea behind the Shortt clock is that a really “free” pendulum will keep better time if it is supplied only with enough energy to keep it going, and is otherwise unencumbered with all the other parts of the clock that go into timekeeping and recording. This requires two pendulums: a master and a slave. The master pendulum – the free pendulum – subtly communicates regulation to the slave pendulum, whose tasks are all of the other things clocks do, namely counting pulses and displaying time (see Figure 2.13).

Pendulums Keep Time

83

Figure 2.13 The Number 16 Shortt clock, master on the left, on display at the National Maritime Museum, Greenwich. (Detail. © National Maritime Museum, London.)

The genesis of this arrangement was to be found in earlier systems for electrically synchronizing networks of clocks such as were located in schools and railways. Just to provide some idea of the complexity of the Shortt clock, a schematic is given in Figure 2.14. Briefly, the operation of the clock may be described as follows. The timekeeping element is a pendulum that swings freely, except that once every half-minute it receives an impulse from a gently falling lever. This lever is released by an electrical signal transmitted from the slave section of the clock. After the impulse has been sent, a synchronizing signal is transmitted back to the slave section, which ensures

84

Seven Tales of the Pendulum

SLAVE DIAL

MASTER DIAL

MASTER

PENDULUM

SLAVE

PENDULUM

THE ‘SHORTT’ CLOCK. MECHANICAL AND ELECTRICAL DETAILS OF SYSTEM

Figure 2.14 A Shortt clock, showing both master and slave mechanisms. (From The Science of Clocks and Watches, by A. W. Rawlings.)

that the impulse to the free pendulum will be released exactly a halfminute later than the previous impulse. We noted that in contrast to all of the preceding pendulum clocks, which were purely mechanical, this system is electromechanical. Only electrical wires connect master and slave, a fact apparent from Figure 2.14. Close-up photographs of the master and slave mechanisms are shown in Figure 2.15.

Pendulums Keep Time

85

Figure 2.15 A photograph showing the mechanisms of both master and slave units. (Photo by James Blackburn and used by permission of the Science Museum/Science and Society Picture Library, London.)

A comparison of Figure 2.14 and Figure 2.15 shows that the components in the schematic drawing of Shortt’s design are clearly recognizable in the two photographs. It took nearly six centuries to progress from the verge and foliot to the Shortt pendulum clock. Over this span, western Europe has moved from the Middle Ages through the Renaissance, the Enlightenment, the Industrial Revolution, and into the twentieth century. The error in the early mechanical clocks of about a half-hour per day was ultimately reduced to about one second per year for the very best pendulum-based devices. The Shortt master–slave clock was the pure essence of technical craft and ingenuity. It was an accurate, albeit delicate, timekeeping instrument. This seems like a good place to close our tale of the mechanical pendulum clock. The Shortt clock and other similarly improved clocks incorporated electrical features. Strictly mechanical pendulum clocks seemed to have reached their zenith.4 4 The Russians went on to further refinements, particularly the Fedchenko clock, which claimed an accuracy of one third of a second per year. These clocks were used as national time standards and in early Russian space flights. (Private communication from Bob Holmstrom.)

86

Seven Tales of the Pendulum

The accelerating rate of technological change caught up with the just-invented Shortt clock. Not long after its introduction in 1921, a remarkable advance in technology sealed the fate of pendulum clocks as the ultimate reference timekeepers. It was discovered that resonant quartz crystals could serve as the frequency-stabilizing elements in electronic oscillators with much sharper frequency definition than pendulums. By 1929, quartz resonators had been applied to clocks. The performance of these quartz clocks was extraordinary, bringing errors down to less than a second in a decade. Beginning in the 1970s, the microelectronics revolution created the integrated circuit. Now, a relatively inexpensive plastic watch with a small silicon chip and a tiny quartz crystal can rival the best products of the clockmaker’s art, at least insofar as accuracy is concerned. Nevertheless, pendulum clocks persist. Their mechanisms continue to fascinate and they continue to tick away in their outdated mechanical elegance. It is an almost curious fact that a pendulum whose length is comparable in scale to the size of a person oscillates in the gravitational field of the earth at a rate that is so sympathetic to human rhythms.

Tale 3: Pendulums Measure Fundamental Forces In a previous tale, we saw how the gravitational field affected the period of the pendulum, making it an important instrument for determining the shape of the earth. But a different configuration of the pendulum can allow us to probe more deeply into the nature of gravity itself. In this new configuration, the pendulum does not oscillate back and forth in a plane; rather, it oscillates in a twisting motion. This new twisting pendulum is usually called a torsion pendulum. The twisting or torsion pendulum played a significant role in the development of fundamental laws of gravity and electricity. In this tale, we explore that history and also describe some current research on gravity that still uses the torsion pendulum, albeit in very sophisticated versions. Let us start with a description of the ideal torsion pendulum and compare it with the linearized gravitational pendulum that has been the focus of the last three chapters. We remember that the linearized gravitational pendulum is one for which the restoring force (the force that brings it back to verticality) is approximately proportional to the angular displacement from verticality, but only for small angles of displacement. In contrast, an idealized torsion pendulum is linearly proportional to the amount of twist, for any (reasonable) amount of twist. This result may be tested experimentally for many materials, and it may be modeled theoretically using ideas from the theory of the elastic properties of materials. The point is that the restoring force of a torsion pendulum is then intrinsically linear. And we emphasize that this proportionality is true, at least in theory and often in practice, for any reasonable amount of twist – large or small. Figure 3.1 is a schematic diagram of a torsion pendulum where, for example, a dumbbell-like mass is suspended from a thin fiber or wire that is secured at the top end. In the figure, the curved arrows indicate the degree of twist.

88

Seven Tales of the Pendulum

Figure 3.1 A typical torsion pendulum. As the dumbbell-shaped mass turns, the suspending fiber twists. The arrows show the direction of the pendulum’s motion. The supporting stand is rigid. (Figure created by James Blackburn.)

The fiber resists any initial external twist and this intrinsic resistance acts as a restoring twisting force that leads to the desired back-andforth oscillations (clockwise, counterclockwise, clockwise, etc.). The plane in which this motion takes place is perpendicular to the direction of gravity, which means that gravity plays no role in the oscillatory motion, other than to give the pendulum weight and thereby keep the fiber under tension. We see that the torsion pendulum differs from the gravitation pendulum (1) by being intrinsically linear for all amplitudes of motion, and (2) by the fact that gravity plays no essential role in the restoring force. The restoring force will now depend on the strength and geometry of the supporting fiber. A stiffer fiber, a thicker fiber, or a shorter fiber will give a stronger restoring force, whereas a weaker, thinner, or longer fiber will give a weaker restoring force. And from our discussion of the

Pendulums Measure Fundamental Forces

89

gravity pendulum we know that a stronger restoring force leads to a shorter period – higher frequency – of oscillation. Another consequence of there being no gravity in the restoring force is that the mass (inertial mass) no longer cancels away in the determination of the period of oscillation. Mass is now important and the distribution of that mass is important. The period increases from increased inertia resulting from either an increase in the mass of the pendulum, or from a distribution of the mass further out on the arms of the pendulum, or from both effects. It turns out that inertia, together with fiber stiffness, determine the period. Thus, increased inertia increases the period, whereas increased fiber stiffness decreases the period. This information is encapsulated in the formula for the period: I T = 2p j .

The moment of inertia I is a complicated expression that includes the pendulum mass and its spatial distribution, like that of the beam shown in Figure 3.1. It quantifies the overall resistance to motion of the pendulum’s mass. The constant j is a measure of the stiffness of the fiber, and it depends on the composition and geometry of the fiber. We may compare this formula with that for the period of the small-amplitude gravity pendulum. It seems that the moment of inertia roughly takes the place of the length l in that other formula, and the stiffness constant provides the restoring force, just as gravity does for the simple pendulum. If the torsion pendulum is not principally affected by gravity, how is a torsion pendulum used to measure the forces of gravity or electricity? These forces come into play when external sources of mass or electrical charge are applied. In the case of gravity, the masses at the ends of the horizontal beam are acted upon by attractive gravitational forces from some other neighboring mass or masses. In the electrical case, electrostatic repulsion is produced by like electric charges on both the pendulum beam and on some other neighboring object(s). Figure 3.2 shows how either type of force, acting on the masses, can twist the pendulum beam. The instantaneous deflection angle h can be sensed by a small mirror, as shown in the diagram. The mirror – sometimes called an optical lever in this configuration – is attached to the apparatus and causes an incoming light beam to be deflected at twice the twist angle. The deflection is

90

Seven Tales of the Pendulum F

h mirror h h

F

Figure 3.2 A view looking down on a torsion pendulum twisted through an angle h by an external force F, either gravity or electricity. An incoming light beam is deflected by an angle 2h from its original direction, thereby showing the amount of twist.

readily detected because for a large distance between the mirror and detector, a very small angular deflection is “multiplied” into a considerably larger scale reading. Therefore one way to use the torsion pendulum is to very slowly apply some sort of force to the masses or charges on the beam and observe the amount of twist. The amount of twist gives information, either about the applied force, or about the properties of pendulum’s suspension fiber. This is a static method of measuring the force on the pendulum beam and it provides a simple way to study the fundamental forces of gravity and electricity. The pendulum can also be used in a dynamic mode with a “time-ofswing” method. The pendulum is now allowed to oscillate, first without extra masses and then with adjacent masses. Without the extra masses, the pendulum will have a certain oscillation period determined by a restoring force that is only due to the suspending fiber. Then the large masses are stationed near the small masses connected to the oscillating pendulum beam in such a way that, while they do not interfere with the path of the swing, they do certainly affect the oscillation period. Figure 3.3 shows a typical arrangement. In this configuration, both the intrinsic resistance of the fiber and the gravitational force of the two extra masses act. The pendulum’s

Pendulums Measure Fundamental Forces

91

M F m

d

Fx θ

L/2

m M

Figure 3.3 The arrangement of the extra fixed masses, M, for the time-ofswing method.

restoring force is enhanced and the period of oscillation is decreased. By measuring the differing periods of the two configurations, one can independently calculate both the stiffness of the supporting fiber and the strength of the gravitational pull.

3.1 The historical torsion pendulum It is quite remarkable that a seemingly simple device like the torsion pendulum should be involved in the study of two of the fundamental forces of nature: electromagnetism and gravity. The illustrious history of this work began with two profoundly important scientific applications that occurred late in the eighteenth century. These nearly concurrent developments were the creative products of the emerging technological cultures of France and England. Although, as is nearly always the case, many persons contributed to the final achievements, the most celebrated individuals were Charles Coulomb and Henry Cavendish.

3.1.1 Coulomb and the electrostatic force Charles Augustin Coulomb (1736–1806) was a French military engineer. He was a student at the engineering school at Mézières (now called Charleville-Mézières, a town northeast of Paris, near the border

92

Seven Tales of the Pendulum

with Belgium), an establishment mostly reserved for minor nobility, but occasionally, as in Coulomb’s case, open to other students with clear promise. After graduating in 1761 as an officer in the Corps of Engineers, he spent less than three years within France before being posted to the island of Martinique in February 1764. There, he was in charge of the design and construction of a new defensive fort at Port Royal. In spite of bouts of ill health, his stay lasted eight years. He returned to France, finally, in June 1772. Remarkably, Coulomb was able to pursue his interests in science and engineering during his protracted stay in the West Indies, so that by the spring of 1773 he was in a position to read two scientific memoirs to the Academy in Paris. For the next nine years following his return to France, Coulomb was posted to various military sites, ending in 1781 with a permanent transfer to Paris. His rank then allowed him a degree of freedom to pursue his interests in science, most particularly in physics, but also including public health and education, canals and navigation, structural engineering, and military topics. Coulomb’s accomplishments in electricity and magnetism ranged from his early (1777) memoir entitled “Investigations of the Best Method of Making Magnetic Needles”, through a famous series of seven memoirs read to the Academy from 1785 through 1791. In the First (1785) and Second (1787) of these, the torsion balance was introduced, and the fundamental force law for electrostatics was established. The Third Memoir treated charge leakage, while the remaining memoirs dealt with charge distributions on conductors of various shapes, and magnetism. Coulomb resigned from military service on April 1, 1791. He lived in Paris during the period from 1781 to July 1793, when he moved to his country house, north of the city. Thus he escaped the nightmare of the Reign of Terror led by Robespierre in 1794. Soon after, for further safety, he moved to a property near Blois, where he remained until at last he returned to Paris around 1797. He died in his home on August 23, 1806. The professional life of Coulomb spanned some of France’s most significant and turbulent years. The Bastille was stormed in July 1789, and shortly thereafter Louis XVI and the royal family were imprisoned in the palace of the Tuileries. Louis was guillotined in the Place de la Concorde on January 21, 1793 (about six months before Coulomb left the city). Napoleon rose to power in the interval 1795–99, when he became First Consul, and then in 1804, Emperor of France. War with .

Pendulums Measure Fundamental Forces

93

Britain was an ongoing and prolonged affair. Toulon was besieged in 1793 and blockaded again by Nelson at the beginning of new hostilities in 1803. Other important events included the Battle of the Nile in 1798 and the Battle of Trafalgar in 1805. War with Russia and Austria culminated at the Battle of Austerlitz in December 1805. Coulomb did not live to see Napoleon’s forced abdication in 1814, nor his second and final defeat at Waterloo in 1815. It is remarkable that such scientific achievements were possible during these upheavals and dangers (the great chemist Lavoisier was guillotined in May 1794). Coulomb built several versions of a practical torsion pendulum, one of which is shown below in Figure 3.5. This sort of instrument was used to determine the specific ways in which the period of the pendulum depended on individual factors such as the suspension wire material (iron and brass were tested), wire length and diameter, and tension. With this knowledge about the pendulum’s characteristics, Coulomb was able to move on to his most famous achievement, his use of the torsion pendulum to determine the nature of the law governing the attraction or repulsion of electrical charge. A proper perspective on this accomplishment must take account of what little was known about electrical charge in the late eighteenth century. It was to be more than 100 years before the true nature of negative charge was determined. Still ahead and unforeseen were the discoveries of Michael Faraday during the 1820s and 1830s regarding electromagnetism and the law of electrolysis. The identification of negative charge with electrons and the measurement of the charge-to-mass ratio of the electron occurred much later, with the work of J. J. Thomson in 1897. The precise determination of the charge of the electron by means of the celebrated oil-drop experiment was achieved later still, by Robert Millikan in 1909. We now know the fundamental particle carrying the basic unit of positive charge to be the proton, but this understanding emerged only with the clear picture of the atom as a small, massive, and positively charged nucleus surrounded by orbiting electrons. This nuclear model was established by Ernest Rutherford in 1910. In 1787, all of this lay ahead in the unknown future. There was, nevertheless, wide interest in the phenomenon of static electricity. Benjamin Franklin’s celebrated kite experiments in thunderstorms took place in 1752. Franklin conceived a two-fluid picture of electricity in which “positive” and “negative” charges either attracted or repelled one another. Static charge was usually produced by the rubbing

94

Seven Tales of the Pendulum

of dissimilar materials. The fortuitous discovery in about 1745 by Pieter van Musschenbroek1 and by Georg von Kleist of what became known as the Leyden jar made it possible to accumulate and store charge. The Leyden jar was the forerunner of the modern electrical capacitor. The inspiration for Coulomb’s work came from the previous century. One hundred years earlier, in the Principia of 1687, Isaac Newton had established the foundations of mechanics and had discovered the universal law of gravitation which states that the force between two masses, m1 and m2, is proportional to their product and inversely proportional to the square of their separation. In equation form, this famous law is mm F = G 12 2 , r where G is now known as the universal gravitational constant and r is the separation of the masses. The word “universal” reinforces the fact that gravity applies to all masses throughout the universe. We may think of the masses as providing the “source” of gravity, whereas the increasing separation between the masses causes a fairly rapid falling off of the force. With these ideas about gravity in mind, it would have been quite natural to wonder if the law governing electrostatic attraction or repulsion might be similar. In the electrical case, the source of electrical force would be the charges, and again there might be some similar falling away of electrical force as the separation increased. Of course, gravity can produce only mutual attraction – there are no “positive” or “negative” masses. Coulomb adapted his torsion pendulum to the electrical situation. The basic idea is illustrated in Figure 3.4. An equal charge is placed on two spheres, one fixed and the other located at the end of an arm that is suspended from a thin fiber. The electrostatic repulsion causes a twist that is then measured and from which the force could be calculated. The apparatus employed by Coulomb in 1785 is depicted in Figure 3.5. The apparatus was not especially large: overall, it was 0.305 meters (about 12 inches) in diameter by 0.914 meters (about 36 inches) in 1 Musschenbroek did his work in Leiden, Holland, from which this first electrical capacitor gets its name, the ‘Leyden’ jar.

Pendulums Measure Fundamental Forces

95

height. In Figure 3.5, the suspension fiber (silver, copper, or silk) is seen to be attached to the horizontal beam at P. The ends of the beam were occupied by a paper-disc counterweight (g), which also served as an air-damper, and a pith-ball that would hold charge (a).2 A second pith-ball was located at the lower end of an insulated rod that extended down through the top plate at m. Two angular scales were engraved on the apparatus, one around the wall of the large glass cylinder and the other as part of the fiber attachment at the top. To begin a typical measurement, the suspension head was then rotated, turning the fiber and beam until the two pith-balls were just in contact. By connecting the (temporarily joined) pair of pith-balls to a Leyden jar, equal charge became distributed on each, causing a force of repulsion and leading to a twisting away of the beam. In the light of our discussion about what little was known about electricity at this time, the amount of charge on the pithballs would, of course, be unknown. Nevertheless, it was still possible to determine the dependence of the electrical force on distance between the charges. Figure 3.6 shows three measurements of electrical repulsive force between the charged pith-balls matched up with three corresponding distances between the balls. The graph joining the points shows that the force indeed decreases in inverse proportion to the square of the

q q h

counterweight

Figure 3.4 A schematic of Coulomb’s torsion balance (seen looking down the axis of the suspending fiber). When the two spheres have like charges, the spheres repel. 2 A pith-ball (pith, as in a pith helmet) was originally dried material from the spongewood tree of Bengal (or other similar plants). Coulomb employed gilded elderwood spheres (of about 1/6 inch diameter) for this purpose.

96

Seven Tales of the Pendulum

Figure 3.5 Coulomb’s torsion balance (1785), used for determining the law of electrostatic repulsion. (Image from Wikipedia.)

separation, exactly as in Newton’s law of gravity. In this way, Coulomb verified that the electric force between charged objects follows the same mathematical dependence on separation of charge – an inversesquare law – as that followed by separated masses in Newton’s law of gravity. Modern methods and experiments have also verified, to very high precision, the inverse-square dependence in Coulomb’s law.

Pendulums Measure Fundamental Forces

97

600

500

Relative Force

400

300

200

100

0

0

10

20

30

40

50

Angular Separation (corrected)

Figure 3.6 Three data points from Coulomb’s original experiment, and a fitted inverse-square law relationship.

3.1.2 Cavendish and the gravitational force Henry Cavendish (1731–1810) was a British physicist and chemist (see Figure 3.7). He was born in Nice, France, of British parents and educated at Peterhouse College at the University of Cambridge. Cavendish was a shy, eccentric, person who shunned social exchange, especially with women. He was able to avoid speaking more than necessary, even to his servants, by having a special back staircase constructed in his house. However, Cavendish was an extremely gifted and careful experimenter. Born an aristocrat, he was able to finance his own experiments and took great pleasure in constantly perfecting his scientific apparatus. In his earliest experiments, he studied the capacities of various substances to absorb heat. In 1766, Cavendish discovered the properties of the element hydrogen and determined its specific gravity. His most celebrated work was the discovery of the composition of water; he stated that “water consists of dephlogisticated air (oxygen) with phlogiston (hydrogen)”. By what is now known as the Cavendish experiment, he

98

Seven Tales of the Pendulum

Figure 3.7 A portrait of Henry Cavendish. (Image from Wikipedia.)

determined that the density of the earth was 5.45 times as great as the density of water, a calculation very close to the 5.5268 established by modern techniques. Cavendish also determined the density of the atmosphere and made important investigations of electrical currents. Cavendish and Coulomb were almost exact contemporaries. It is a widely held notion that Cavendish experimentally verified Newton’s famous gravitational law, but as the biographical details above hint, such was not the case. In the first place, Newton himself did not write his “law” in the now familiar form, nor did he introduce the “universal” gravitational constant G. One hundred years had to pass after Cavendish before the modern idea of “G” was to appear in a paper entitled “On the Newtonian Constant of Gravitation”, by British physicist Charles Vernon Boys (1894). It is also worth keeping in mind that our familiar system of

Pendulums Measure Fundamental Forces

99

units was not established at the time of Cavendish. For example, a universally accepted unit for force was not introduced until 1873. We saw, in our first tale, how eighteenth century scientists were preoccupied with determining the relative density of the earth, and it was to this task that Cavendish initially turned his attention. Again the torsion pendulum is the basic apparatus, as indicated by the schematic of Figure 3.8. In this view down the axis of the suspension fiber, two small masses can be seen fixed to the ends of a horizontal beam. Two large masses are positioned at either pair of sites, numbered 1,1 or 2,2. Gravitational attraction will rotate the beam and twist the support fiber through a counterclockwise angle h when the large masses are in the first position, and through a clockwise angle h when they are in the second position. The experiment consists of determining the total change in angle 2h that occurs as the large mass positions are switched. The periodic switching of the large masses acts to “pump” the oscillation, just as a child on a playground swing periodically changes his body position in order to pump the swing. It is generally acknowledged that Cavendish improved upon an original concept by his friend and colleague the Reverend John Michell (1724–93).3 Figure 3.9 shows the torsion beam built by Cavendish. It h 1

2

2

1

Figure 3.8 A schematic arrangement of a Cavendish torsion balance. The view is looking down the axis of the suspension fiber. 3

John Michell was an astronomer who, in a letter to Cavendish, dated 1784, proposed the existence of stellar objects so big that they could trap light. Today we term such objects “black holes”. Laplace made a similar suggestion, probably independently, in 1796.

100

Seven Tales of the Pendulum G

p

H

m

H

M

G

M P

r

r F

K

F I

L

L E A T

s

m nb D x B

G

E g

A D

w

w

C R

R

bn x

s

T

C B G

Figure 3.9 The original apparatus used by Henry Cavendish in 1798 to measure the density of the earth. (Reprinted with permission from the American Journal of Physics, vol. 55, 1987, pp. 210–13. © 1987, American Association of Physics Teachers.)

was quite large, with a beam 73.3 inches (1.86 meters) in length and very large masses that were lead spheres 1 foot (0.305 meters) in diameter and weighing 348 pounds (a mass of 158 kilograms). The small spheres were 2 inches (0.051 meters) in diameter and the center separation distance r was 8.85 inches (0.225 meters). The copper suspension wire was 39.25 inches (0.997 meters) in length. As suggested in the drawing, the entire apparatus was housed in a room provided with illuminating lanterns and sighting telescopes, so that the experimenter remained outside and air currents were reduced. Arriving at a mathematical expression for the density of the earth required some ingenuity, particularly as, at that time, the value of the gravitational constant, G, was unknown. But it turns out that the right combination of measurable quantities does circumvent this lack. These quantities are: (a) the period of oscillation of the torsion pendulum T0; (b) (strangely) the period of oscillation of a simple gravity pendulum whose length is equal to the arm of the torsion pendulum TP; (c) the angle of twist caused by the large masses h; and (d) some readily measurable properties of the torsion pendulum

Pendulums Measure Fundamental Forces

101

itself. For the Cavendish apparatus, the first three required values were T0 = 848 seconds, TP = 1.94 seconds, and h » 0.22 degrees. It is interesting to note that the torsion pendulum took more than 14 minutes to complete one oscillation. Yet this is not surprising given the magnitude of the apparatus and the properties of the supporting wire. From these data, the mean density of the earth is calculated to be 5.48 gm/cm³. Remarkably, the presently accepted value for the density of the earth is 5.5268 gm/cm³. It is quite amazing that Cavendish was able to obtain a meaningful result for the density of the earth – a testimony to his careful experimental technique. The force of gravity is very weak compared, for example, to the electric force. By using heavy masses Cavendish hoped for enough gravitational attraction to get useful measurements. In retrospect, as the properties of pendulum suspensions were better understood, Cavendish would have achieved better experimental sensitivity by using smaller masses and thinner suspension wire or fiber. It turns out that the sensitivity of such experiments lies in the use of thin fibers. The scale of Coulomb’s apparatus was better suited to such measurements, as his thin fiber suspensions were more sensitive than Cavendish’s thick suspension, which he needed to support his heavy masses. Cavendish-type experiments have steadily improved, and typical modern designs have masses of only a few kilograms.

3.2 Modern torsion pendulums 3.2.1 The universal gravitational constant We have stressed that the particular notion of a “universal gravitational constant” G was not included in either seventeenth century (Newton) or eighteenth century (Cavendish) formalisms. However, by the nineteenth century, modern mathematical language and the idea of “fields” – electromagnetic or gravitational – had taken root. Fundamental constants of nature, such as Avogadro’s number, had assumed a certain pre-eminence. The Cavendish experiment can give a value for the universal gravitational constant from a static measurement, and from the dynamic time-of-swing method that we described previously (see Figure 3.3). We learned that the arrangement of large masses near the torsion pendulum increases its frequency of oscillation. By knowing the frequencies of oscillation with and without the extra masses, and certain

102

Seven Tales of the Pendulum

measurable properties of the pendulum, it is possible to develop an equation for G for which one can measure all the necessary quantities. Modern experiments generally use some variation on the time-ofswing method. Following in Cavendish’s tradition of careful experimentation, modern experiments take extreme care as well. For example, one such experiment was carried out inside a vacuum system situated in an isolated building on a mesa in New Mexico. Another experiment was isolated in the center of Yu-Jia Mountain in Wuhan, People’s Republic of China. With today’s technology, modern experimental apparatus is enormously complex, and we can give only a bit of the flavor of these experiments. In 2000, a new determination of G was made using a very elegant torsion pendulum, as shown in Figure 3.10. This design, by Jens Gundlach and Stephen Merkowitz of the University of Washington, employed a simple Pyrex slab in place of the usual dumbbell. External to the fully enclosed pendulum was an array of 8 kilogram steel balls. The entire pendulum was placed on a rotating turntable. As the pendulum then vacuum chamber –7 10 Torr

xi

pendulum 1.5 mm thick pyrex

torsion fiber 17 lm dia. W

attractor mass stainless 4 × 8.14 kg

Sphere seats

xa air bearing (not visible)

angle encoder 36000 lines

attractor mass turntable

one of two read heads eddy current motor 50 cm

Figure 3.10 A photograph and cutaway view of the apparatus used by Gundlach and Merkowitz to determine Newton’s gravitational constant, G. (Photograph and drawing courtesy of Jens Gundlach.)

Pendulums Measure Fundamental Forces

103

turned past the steel balls, the suspension fiber would exhibit periodic twists. A further design refinement added feedback control to the turntable, so as to accelerate and decelerate the rotation to minimize fiber twist. In the spring of 2000 they reported the best current value for G: (6.674215 ± 0.000092)×10−11m3/kg–1/s–2. After more than 200 years, the torsion pendulum remains the definitive instrument for determining G.

3.2.2 The principle of equivalence (revisited) Earlier, we introduced the distinction between gravitational mass (mass as it is gravitationally attracted to other mass) and inertial mass (that mass which is accelerated by force). The equivalence principle proposes that these two masses are the same. We saw how the formula for the period of the gravity pendulum reflected this equivalence by canceling out the two masses, since they are assumed equal. Another consequence of equivalence is that acceleration of an object is independent of the material of the object: iron and wood both undergo the same dynamics. This may be easily illustrated for a falling body accelerated by gravity. We also saw how Newton’s law of gravitation describes the mutual attraction of a pair of masses, m1 and m2. At the surface of the earth, the force of attraction of the earth (mass ME, radius RE ) acting on a test mass mG, a gravitational mass, near the earth’s surface is given by Newton’s law of gravity, F =G

mG ME . RE2

Furthermore, Newton’s second law of dynamics tells us that force is equal to inertial mass multiplied by acceleration, F = m I a,

and therefore the downward acceleration on the test mass is simply the above expression for force divided by the small test inertial mass mI. The downward acceleration is called the acceleration due to gravity and has a special symbol g.4 We now take the critical step of invoking 4 The well-known value for the acceleration g of free fall at sea level at latitude 45 degrees is 980.616 cm/s2, or 32.17 ft/s2.

104

Seven Tales of the Pendulum

the equivalence principle, that the inertial mass and the gravitational mass of the test particle are the same. That is, we assume mI = mG = m. Then, the resulting acceleration downward becomes F mM M g = = G G 2E = G 2E , mI mI R E RE independent of the mass of a falling object. Thus, as Galileo surmised 400 years ago, all objects fall at the same rate. We might dispute this by recourse to our common experience in comparing the free fall of a piece of paper and a stone. But there are other forces acting here, such as the interaction of air with the paper, that distort the reality of equivalence. When the experiment is performed in a vacuum, the prediction is confirmed. Recall the 1971 experiment by the astronaut, standing on the moon, who dropped an eagle feather and a hammer. Both reached the ground at the same time. How does the torsion pendulum enter the discussion? From time to time ingenious experiments, involving torsion pendulums, have been devised to test this hypothesis of material independence. One of the most celebrated experimenters was Baron Roland von Eötvös of Hungary. He used a torsion balance consisting of a 40 centimeter long beam suspended from a platinum–iridium wire. Weights made from different materials were placed at the ends of the beam, and the apparatus, enclosed so as to reduce air currents, was aligned in an east–west direction. Twisting of the fiber was detected by means of the usual optical lever, a light beam reflected from a mirror attached to the fiber/ beam similar to that shown in Figure 3.2. Since the apparatus was fixed to the surface of the spinning earth, the two beam masses were subject to gravitational forces from the earth and centrifugal forces generated by the slowly rotating reference frame of the laboratory. Any small difference between these two forces as they were experienced by the beam masses would have resulted in a net torque on the torsion pendulum. In experiments carried out between 1889 and 1908, no torque was observed within the sensitivity limits of the apparatus. This famous null result confirmed the material independence of inertial and gravitational forces, thereby verifying the equivalence principle. Since this early work, fresh attempts have been made to examine the question. In the early 1960s, R. H. Dicke and others at Princeton University improved on the original Eötvös torsion-pendulum design.

Pendulums Measure Fundamental Forces

105

This apparatus was placed in a 12 foot deep pit near the Princeton football stadium. In the end, this experiment improved the precision over Eötvös by a factor of 50, and once more gave a null result. A 1990s test of universal free fall, by Y. Su, J. H. Gundlach, and others, involved the torsion pendulum shown in Figure 3.11. This pendulum was located in a laboratory situated in a hillside on the campus of the University of Washington. The 20 micron5 suspension 13

14

1

12

6

10

6

2

11

9 8

3 3

7

4 3

5 6

Figure 3.11 The 1990 torsion pendulum. The principal components include the suspension fiber (1), the turntable (5), and the pendulum itself (8). The concrete block (11) is 1.23 meters high. (Reprinted with permission from Su et al., Physical Review, vol. D50, 1994, pp. 3614–36. © 1994 by the American Physical Society.)

5

A micron is a one-millionth part of a meter.

106

Seven Tales of the Pendulum

fiber was a gold-coated tungsten wire, 0.8 meters long. The free oscillation period of the pendulum was 695 seconds. As the diagram illustrates, the entire pendulum was placed on a turntable that continuously and slowly rotated about the vertical axis. The details of this experiment are quite complex, but the measurements again confirmed the equivalence principle. We have seen that the torsion pendulum has played an enormous and fundamental role in important areas of physics, gravitation, and electrostatics. Yet there are more ways in which the torsion pendulum has proved useful. For example, it has had a modern role in testing the limits of Newton’s second law of motion – force equals mass times acceleration – for extremely small forces. In 2007, Gundlach and his team at the University of Washington have, using a torsion pendulum, pushed the size of the forces for which Newton’s law is valid down by a factor of 1,000. Even today, the torsion pendulum continues to have an important place in such studies. We could also have mentioned other areas where the torsion pendulum is important. Undergraduate laboratories still use torsion pendulums to measure charge with instruments called ballistic galvanometers. In quite a different application, recent studies with a sophisticated torsion pendulum have shed light on the behavior of a state of solid helium that moves without friction, called supersolidity in analogy with superfluidity. In yet another area, research studies of viscosity and granular materials also use torsion pendulums similar to that used by Coulomb. With all these multiple contributions, it soon becomes apparent that the story of the torsion pendulum, a relatively simple piece of technology, is indeed surprisingly rich.

Tale 4: Pendulums Swing High 4.1 A religious pendulum In the northwest corner of Spain, in the province of Galicia, lies the mistshrouded town of Santiago de Compostela, the birthplace of the cult of Santiago (St. James, the major apostle), and the home of the magnificent cathedral that is presumably built upon the bones of that martyred apostle (see Figure 4.1). For a thousand years, pilgrims have sought out this cathedral as a shrine to St. James where they might worship and receive salvation. It is believed that the remains of the apostle were transported from Jerusalem to the site of the cathedral. The most famous and unique feature of the celebration of the mass at this cathedral, at least since the fourteenth century, is O Botafumeiro, a very large incense burner suspended by a heavy rope from a point 70 feet above the floor of the nave, and swung periodically through a huge arc – about 80 degrees in amplitude. The rapid motion through the air fans the hot incense coals, making copious amounts of blue smoke, and the censer itself generates a frightening swooshing sound as it passes through the bottom of its arc. Thus we dramatically introduce the large-swing pendulum! The censer, or incense burner itself, stands more than a meter high and is suspended by a thick rope whose diameter is 4.5 centimeters. (One can imagine something about the size of a backyard barbecue grill.) Over a period of 700 years, a variety of censers have been used. The original censer, which seems to have been silver, was later replaced by another silver one, donated by the French king, Louis XI. A papal bull from Pope Nicholas V in 1447 threatened excommunication to anyone who stole it. It was probably this latter censer that was destroyed when it suffered a violent fall in 1499. The censer is sporadically mentioned in records over the next couple of centuries, with yet another silver replacement being made as late as 1615. There is some evidence – but not conclusive – that French troops took a silver censer during

108

Seven Tales of the Pendulum

Figure 4.1 The cathedral of Santiago de Compostela in northern Spain. (Photo by Margaret Walker.)

Pendulums Swing High

109

Napoleon’s 1809 campaign. At some point prior to 1852, the censer was made of iron, but at that date it was replaced by a censer of silvered brass, which is the one in use today (see Figure 4.2). The current censer has a mass of 53 kilograms and is about 1.5 meters in overall height. Its center of mass is about 55 centimeters above the base. Approximately 3 meters of rope are used just in tying the heavy knot that secures the rope to the censer. High above the floor of the nave, the rope is wrapped around rollers on a frame that is supported by an iron structure. The rope then continues back down to floor level. The iron structure consists of four large struts, joined at the central frame,

Figure 4.2 O Botafumeiro, the giant censer that hangs in the transept of the cathedral of Santiago de Compostela. (Photo by Steve Fazzio.)

110

Seven Tales of the Pendulum

Figure 4.3 The support structure for the rope holding O Botafumeiro. (Reprinted with permission from the American Journal of Physics, vol. 52, 1984, p. 939. © 1984, American Association of Physics Teachers.)

that stretch to four posts near the upper part of the tower. The rollers provide the mechanism used for pumping the censer (see Figure 4.3). In the early days, the complete support structure was wooden; the iron structure was built later, in 1602, in part because the cumbersome wooden support obscured light coming into the tower. The length of rope between the rollers and the top of the knot is 19.4 meters. Therefore, the length from the rollers to the center of mass of the censer is 20.6 meters and the length to the floor is 21.8 meters. This means that the bottom of the censer comes within about half a meter of the floor of the nave as it passes rapidly through the bottom of its arc. The censer is pumped by periodically shortening and lengthening the rope as it is wound up and then down around the rollers. This kind of pumping is now known as parametric forcing, because the length – a parameter of the system – is used to provide energy to the system. The pumping action is carried out by a squad of men called tiraboleiros, or ball-swingers. They are all grouped to one side and each holds a rope that is a strand of the main rope that goes from the pendulum to the rollers and back down to near the floor. The tiraboleiros periodically pull on their respective ropes in response to orders from the chief verger of the cathedral. Unlike the child on a swing who senses the effect of her pumping, the team must simply act as a unit and follow the changing

Pendulums Swing High

111

timing prescribed by the verger, who acts as team leader. One of the more terrifying aspects of the pendulum’s motion is the fact that its large incense burner has a high velocity – about 68 km/h – as it passes through the bottom of its arc, spewing smoke and flames! How does O Botafumeiro differ from the simple small-swing gravitational pendulum described in the Introduction? For our purposes, the important feature is the large size of its swing. This has several implications, including for example, that contrary to Galileo’s conclusion, the period of the motion is not independent of the size of the swing, or amplitude. The subject of this tale is, then, some of the complications to pendulum dynamics suggested by the motion of a large-swing pendulum such as O Botafumeiro. As a postscript to this introduction, we note that the history of O Botafumeiro has not been without mishap and even violence. On July 25, 1499, an accident involved the four chains that attach the censer to the rope. The chains broke, thereby sending the censer crashing into one of the side doors. On May 23, 1622, the rope broke. This time, the censer fell approximately vertically, since the amplitude of the motion was small at the moment. In a more sinister vein, it has been suggested that the suspending rope was used in the murder of the Archbishop Suero Gomez de Toledo at the entrance to the cathedral on June 29, 1366. Two noblemen, F. Perez Churruchao and G. Gomez Gallinato, were involved. Apparently they also killed the dean of the cathedral, Pedro Alverez, who, when he discovered the fate of the archbishop, ran back into the cathedral. The killers and a third person, Pedro el Cruel, chased him around the triforium before catching him. Pedro later denied any involvement. Let us now leave the exotic world of O Botafumeiro, and return to the generic large-swing pendulum.

4.2 Yes, the period does depend on the amplitude We recall from our Introduction on the pendulum that for very small amplitudes (swings), the pendulum is subject to a restoring force that is approximately linear; the amount of gravitational force on the pendulum bob increases in approximate proportion to the size of the swing. In this approximation the period of the motion is constant, as Galileo concluded. However, once the swing becomes more than a few degrees, the

112

Seven Tales of the Pendulum Undamped Oscillations

Period (normalized)

1.2

1.1

1.0

0.9

0

10

20

30 40 50 60 Angle (degrees)

70

80

90

Figure 4.4 The period of oscillation for a rigid pendulum versus amplitude of swing up to a maximum of 90 degrees. Not shown is the fact that from 90 to 180 degrees the graph increases dramatically, until the period becomes infinite when the pendulum can just get to the upper vertical position with zero velocity. This latter position is an unstable equilibrium position. (The small-angle period is taken as 1.0.)

linear approximation of the force is no longer valid and the restoring force is manifestly nonlinear (see Figure I.5). This nonlinearity in the force causes the period of the pendulum to increase with increasing swing amplitude. Careful observation shows this to be the case. And mathematically, one can also show how the variation in period arises from the nonlinear force. The derivation of the mathematical relationship is a very challenging exercise, but the main result is T = 2 p l / g [1+positive factor that depends on h0 ],1

where h0 is the angular amplitude of the motion. That is, the period increases with the amplitude, as portrayed in Figure 4.4. The graph shows 1

For those with a mathematical bent who are curious about this new factor, the detailed formula is 2 ⎡ ⎛ 1⎞ 2 ⎛ h ⎞ ⎛ 1⋅ 3 ⎞ ⎛h ⎞ ⎤ T = 2 p l / g ⎢1 + ⎜ ⎟ sin 2 ⎜ 0 ⎟ + ⎜ sin 4 ⎜ 0 ⎟ + ⎥ , ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 2⎠ ⎦ 2 2 2 ⋅ 4 ⎣

where h0 is the angular amplitude of the motion.

Pendulums Swing High

113

a portion of the relationship between period and angular amplitude, up to a swing that carries the pendulum to a momentary horizontal position. When the amplitude is zero, the period is exactly equal to the period determined from the linearized version of the pendulum. As the amplitude approaches p radians (180 degrees), the bob approaches verticality, and period becomes infinite. Thus we see that the first important difference between small-swing and large-swing pendulums is the nonconstancy of the period. There is another important consequence of the nonlinearity of the pendulum’s equation of motion. For the linearized pendulum, the behavior of the angle in time is that simplest of waves, a single sine wave with a single period T as described in the Introduction: h(t ) = h0 sin

2 pt . T

But such is not the case for the nonlinear pendulum. While the motion of the pendulum is still periodic, its time variation is no longer that of a single sine wave. Its motion is a complex of many single sine waves. Think of a note from a musical instrument. A note with just a single wave and single period (like the small-swing pendulum) would have a very unmusical, un-rich, tone. The rich tone of a musical instrument comes from the simultaneous combination of several sine waves or harmonics, which are waves that oscillate at multiples of the basic frequency (or fractions of the basic period). It turns out that the notion of any periodic motion having many harmonics is quite general. In 1822, Jean Baptiste Joseph Fourier, a French mathematician (1768–1830), wrote a treatise on the mathematical theory of heat. In that work, he proposed that any mathematical dependency relationship (technically called a function) could be represented by an infinite sum of sine (and cosine) waves. Actually, this result is really true only for periodic relationships, but we still call such expressions Fourier series, and other mathematicians explored these relationships further to eventually develop a branch of mathematics called Fourier analysis. Therefore, the large-swing pendulum motion might be represented mathematically by a sum over many single sine-wave terms, each having the form h0,n sin

2 p(2n + 1)t , T

114

Seven Tales of the Pendulum FFT: Nonlinear Pendulum 0.0 Fundamental

Log of Relative Power

–5.0

–10.0

–15.0

–20.0

–25.0 0.0

0.1

0.2

0.3

0.4

0.5

Frequency (units of f /f0)

Figure 4.5 A spectrum of the amplitudes of the simple sine-wave components in the motion of a nonlinear pendulum. In this picture, the basic frequency is one unit. Note that the strong components are at odd harmonic frequencies of 3, 5, 7, . . . times the fundamental frequency. (Reprinted with permission from the American Journal of Physics, vol. 74, 2006, pp. 482–9. © 2006, American Association of Physics Teachers.)

where n = 0, 1, 2, 3, . . . The factors (2n + 1) are the odd numbers 1, 3, 5, and so forth, and the period T is the basic period of the large swing. Each simple component sine-wave oscillates correspondingly faster, as determined by the odd number multiplier. It is remarkable that the physical motion of the pendulum is described mathematically by a series of harmonics of the fundamental pure sinusoidal wave. We see that the largeswing motion is now significantly richer than that of the small swing, yet it is based mathematically on the small-swing motion. The strange-looking symbol h0,n is just the amplitude or size of each simple sine-wave component, and these amplitudes are approximately displayed in Figure 4.5. Note that, as with musical instruments, the contribution of each harmonic decreases with increasing frequency of the harmonic. This is generally true for large-swing periodic pendulums. Fourier’s life can be connected to some important historical events. Like other contemporary French mathematicians, Fourier made his contribution to mathematics during a turbulent period of French history. He was active in politics and, as a student during the “Terror”, was

Pendulums Swing High

115

arrested and temporarily detained. Later, when Napoleon went to Egypt, Fourier accompanied the expedition and coauthored a massive work on every possible detail of Egyptian life, Description de l’Egypt. This multivolume work included nine volumes of text and 12 volumes of illustrations. During that same campaign, one of Napoleon’s engineers uncovered the now-famous Rosetta Stone, so-named for being found near the Rosetta branch of the River Nile in 1899. The significance of this find was that it led to an understanding of ancient Egyptian hieroglyphics. The stone was inscribed with the same text in three different languages – Greek, demotic Egyptian, and hieroglyphics. Only Greek was understood, but the size and the juxtaposition of the texts allowed for the eventual understanding of hieroglyphics and the ability to learn much about ancient Egypt. In 1801, the victorious British, realizing the significance of the Rosetta Stone, took it to the British Museum in London, where it remains on display and is a popular artifact. Much later, the writings from the Rosetta Stone become the basis for translating the hieroglyphics on the Rhind Papyrus and the Golenischev Papyrus; these two papyri provide much of our knowledge of early Egyptian mathematics. The French Egyptologist Jean Champollion (1790–1832), who did seminal work in the translation of hieroglyphics, is said to have actually met Fourier when the former was only 11 years old, in 1801. Fourier had returned from Egypt with some papyri and tablets, which he showed to the boy. Fourier explained that no one could read them. Apparently, Champollion replied that he would read them when he was older – a prediction that he later fulfilled during his brilliant career of scholarship. After his Egyptian adventures, Fourier concentrated on his mathematical researches. His 1807 paper on the idea that functions could be expanded in trigonometric series was not well received by the Academy of Sciences of Paris, because his presentation was not considered sufficiently rigorous and because of some professional jealousy on the part of other Academicians. But eventually Fourier was accepted as a first-rate mathematician and, in later life, acted as a friend and mentor to a new generation of mathematicians. Besides the generation of harmonics, the fact that the period of the nonlinear pendulum increases with increasing swing size has another interesting physical effect. The dependence of natural frequency and period on amplitude leads to variation in the response of the pendulum to continued forcing at a fixed frequency. Let us suppose that a pendulum is initially forced at its low-amplitude natural frequency. At first, it will

116

Seven Tales of the Pendulum 100 80 60

Angle (degrees)

40 20 –0 –20 –40 –60 –80 –100

0

20

40

60

80

100 Time

120

140

160

180

200

Figure 4.6 The response of the pendulum to forcing at its small-angle resonant frequency. For larger amplitudes, the forcing frequency is increasingly different from the pendulum’s natural frequency, thereby causing amplitude modulation.

respond well to that forcing. However, once the amplitude is more than a few degrees, the original forcing is not as effective, because the forcing frequency is, at this point, faster than the frequency to which the largeswing pendulum responds. The original forcing goes out of step with the new slower resonant frequency suitable for higher amplitudes and becomes counterproductive. As the forcing becomes further out of synchronization with the pendulum, the pendulum’s amplitude of motion will actually diminish back to zero. In this way, the size of the swing is periodically changed or modulated. This periodic effect on the pendulum’s motion is called amplitude modulation and is shown in Figure 4.6. Therefore, if the pendulum is a typical playground swing, the rider on a swing must automatically nullify this effect by adjusting her pumping rate to match the immediate resonant frequency of the swing and thereby achieve maximum swing. In essence, the rider’s intuitive efforts to adjust her frequency of pumping form part of a feedback loop with the swing, a phenomenon known as autoresonance. Similar considerations hold true for the pumping of the largeamplitude oscillations of O Botafumeiro. The team of tiraboleiros

117

Period

Pendulums Swing High

0

1

2

3

4

5

6

7

8

9

10

Energy

Figure 4.7 A graph of the period versus the energy of the pendulum. The vertical line shows where the pendulum bob has just enough energy to reach the top.

becomes part of a feedback loop, controlled by the verger who must slow the pumping frequency as the amplitude increases. These considerations apply to energies that are sufficient to give the pendulum a large, but still less than 180 degrees, amplitude of oscillation. Let us now consider the very particular case for which the pendulum has just sufficient energy to reach an inverted vertical position. We call this energy the critical energy. The pendulum approaches verticality as the angle approaches p radians (180 degrees) and, as we noted earlier, the period becomes infinite. This is a rather special case in that it marks the boundary between oscillatory motion and rotational motion of a special kind. What happens to the period of pendulum motions if it is given a more energetic start than that required for the critical value? Now the pendulum goes over the top and has enough energy to continue some sort of rotation. Yet the rotation does not have a uniform speed. The pendulum moves quickly along the bottom of its arc and less quickly (slowed by gravity) over the top of the arc of its motion. We term the rotations as hindered because of their particular nonuniformity. The rotary motion is always in one direction and therefore the pendulum

118

Seven Tales of the Pendulum

does not have to stop and reverse its motion at the end of the swing. We might even expect that the period of the rotation would be less than that of very-large-swing pendulums. The mathematics is beyond us here. As with the pendulum at large amplitudes, the expression for the period of the rotary motion is complex. But in Figure 4.7 we graph the period for a large range of possible motions. For energies less than critical, the period increases as expected. For energies that are much larger than the critical energy, the motion is almost entirely rotary without much hindering, and the period steadily decreases as the rotary motion speeds up with further increase in energy. In summary, we see that just by looking at the variation in period with increasing energy, we can define several regimes of motion: almost linear oscillation, nonlinear oscillation, critical motion, hindered rotation, and almost-free rotation.

4.3 Phase space revisited We introduced phase space earlier as a means of visualizing the totality of the pendulum’s motion. For the small-amplitude pendulum, the phase portrait was approximately an ellipse. With the large-swing pendulum, phase space becomes more complex, corresponding to the increasingly complex dynamics. We noted that the pendulum can have the critical amplitude where the angle is 180 degrees (or p » 3.142 radians) from the downward vertical position. And even more energy will cause hindered rotation that can make the pendulum spin through many revolutions. The full range of motion is shown in Figure 4.8. In the regime of such rotations, one might think that we would need to have a very large phase space with an angle scale of many hundreds of degrees. But this is not necessary. The top position of +180 degrees can also be thought of as being at an angle of -180 degrees, as if the pendulum had traveled in the opposite direction to arrive at the top. Thus, to represent all possible angles, the range of angle need only go from −180 degrees to +180 degrees, or –p to p radians.2 Any possible physical position or angle of the pendulum bob can be represented within this confined range of angles. We call these sorts of limits on the range periodic boundary conditions. A simple way to picture this periodicity is to roll 2

Just a reminder that 180 degrees equals p (3.14159 . . .) radians.

Pendulums Swing High

119

2.0 1.5

Angular Velocity

1.0

Hindered rotor Critical Orbit

0.5 0.0 −0.5

Oscillator Saddle Point

−1.0 −1.5 −2.0 −4

−3

−2

−1

0 Angle

1

2

3

4

Figure 4.8 A phase diagram showing possible orbits for a pendulum: subcritical, critical, and supercritical in energy. (Angles are measured in radians.)

the flat phase diagram of Figure 4.8 up into a tube so that the right side wraps around to attach to the left side. (Actually, since the edges of the diagram go slightly beyond the above range, it would be necessary to chop the edges of the diagram at angles of exactly ± p radians.) Now that we have given some boundaries to the diagram, we will see that Figure 4.8 contains a lot of information about the pendulum’s motion. The space is divided by the critical orbit into two parts that correspond to two regimes of energy – oscillatory and rotary – below and above the critical energy for which the pendulum gets just to the top. For energies that are quite small compared to the critical value, the phase orbit is somewhat elliptical, because the pendulum is reasonably approximated by its small-swing, linearized version. The phase-space motion – labeled “Oscillator” – is well inside the critical orbit and is approximately elliptical (compare with Figure I.7(a)). For large swing energies that are still less than the critical energy, the ellipses become slightly squashed even as they become bigger. At the critical energy, h = ± p radians, the period of the pendulum becomes infinite. The critical phase orbit marks the separation between oscillatory and hindered rotary motion of the pendulum. By the way, except for the saddle points, none of the orbits cross. This is a consequence of the deterministic nature of the motion.

120

Seven Tales of the Pendulum x

h

–2p

–p

0

p

2p

Figure 4.9 A phase-space diagram for the damped, undriven pendulum. (Reprinted from Chaotic Dynamics: An Introduction, 2nd edn., by G. L. Baker and J. P. Gollub, 1996, p. 19, with the permission of Cambridge University Press.)

Once started in a particular way, the pendulum’s motion is determined for all time. It does not jump or even cross a different orbit. In time, the pendulum’s phase-space motion is always clockwise around the subcritical orbits. For supercritical orbits, the motion will just continue to the right as the angle grows steadily. The flow of the motion is indicated by arrows, and for the critical orbit, the flow arrows go into and come out of the points where h = ± p. Because some of the arrows are going into these points and others are pointing away, these special points are called saddle points, and the arrows indicate whether the pendulum tends to go toward or away from the saddle point along the particular orbit. Trajectories that point toward the saddle point are called stable, and those that point away are called unstable. Once the energy is greater than critical, the motion becomes that of a hindered rotor and the pendulum’s phase orbit is no longer confined to a finite region of phase space. The wavy line continues in one direction only. Finally, if the energy were to increase dramatically, the hindered rotor would become virtually a free rotor with a constant angular velocity, and the phase orbit would be just a simple horizontal line. The addition of damping or friction creates a new scenario, as shown in Figure 4.9. Now the motion of the pendulum decays: each swing gets smaller. For small-amplitude oscillations, the phase-orbit ellipse becomes a spiral toward the center as the motion dies away. Even for larger amplitudes, the motion decays and therefore the final state of any of the pendulum motions is one of rest. Decaying phase orbits always move to these centers, and therefore the centers are called attractors.

Pendulums Swing High

121

The attractors (h = –2p, 0, 2p) and the saddle points (h = –p, p) are located at what are called fixed points. The attractors are coordinates of phase space for which, as time goes on, the pendulum does not move, whereas the pendulum generally moves away from saddle points. Finally, if a small amount of periodic forcing or pushing is added to a damped pendulum, an equilibrium state will be reached in which the added energy from the forcing just compensates for the energy dissipation, and the motion will be a steady periodic oscillation, similar to – but not equivalent to – the undamped, unforced state. The phase-space orbit will again be a closed curve, although not necessarily elliptical in shape. The closed orbit is sometimes called a limit cycle. Unlike the undamped, unforced case where the phase orbit is immediately a closed figure like an ellipse, the forced, damped pendulum needs a certain transient time to arrive on its limit cycle. That is, the final steady state, limit cycle, orbit attracts the motion and is called an attractor for the motion, no matter what the initial conditions. Now let us ask what may seem a strange question. What happens to the pendulum’s motion if we push it at a frequency that is different from or non resonant with the natural frequency? It seems quite plausible that this can cause a host of complex phenomena. (Imagine the effect of pushing a child on a swing at a nonresonant frequency. The child would very quickly complain about a bumpy ride!)3 For example, moderate nonresonant forcing may cause a pendulum’s motion to take two forcing periods to complete one full oscillation. This phenomenon is called period doubling, or period-2. Such a motion can be displayed in a variety of ways, as will become evident in the next tale. Let us look at another geometric representation. With Figure I.8, we were introduced to a picture of the relative frequency of occurrence in time of the pendulum’s angular position. For that most simple motion there were two peaks, one at each extreme of the pendulum’s swing. This seems obvious, since the pendulum spends a relatively large 3

Here, we emphasize the distinction between forcing at the pendulum’s natural frequency and forcing at a nonnatural frequency. Forcing at the natural frequency, like playing a stringed or woodwind instrument, causes harmonics that are at double, triple, or higher than that of the natural frequency. In contrast, forcing at a nonnatural frequency causes subharmonics. These components are at one half, one third, or less that of the natural frequency.

122

Seven Tales of the Pendulum (a)

Angular Distribution: Period-15 0.014

Relative Frequency

0.012 0.010 0.008 0.006 0.004 0.002 0.000 –4

–3

–2

–1 0 1 Angle (radians)

2

3

4

–1 0 1 Angle (radians)

2

3

4

(b) 0.0008

Relative Frequency

0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0.0000 –4

–3

–2

Figure 4.10 The relative frequency of occurrence of angle for (a) period-2 motion and (b) period-15 motion. (Reprinted with permission from the American Journal of Physics, vol. 74, 2006, pp. 482–9. © 2006, American Association of Physics Teachers.)

amount of time at its extremes. But for a motion with period-doubling there are four peaks. Figure 4.10(a) shows one example of perioddoubled motion with the four peaks, although two of them are somewhat squashed together in this case. The squashing indicates that a momentary peak of the motion is not at the same angle on the left side as on the right side. This is a new complication. The symmetry between the two sides of the pendulum is now broken.

Pendulums Swing High

123

As the periodicity of the motion increases to become larger multiples of the forcing period, the number of peaks increases and the complexity dramatically increases. In general, the number of peaks is twice the multiple of the forcing period, although the peaks are not always all distinguishable in the relative frequency diagram. Figure 4.10(b) is a relative frequency graph for a periodic motion of the pendulum that takes 15 forcing periods to complete one cycle. We will find, in the next tale, that a host of interesting motions can be created by pushing the pendulum strongly at a nonresonant frequency. But for now we step back from the world of complex motions and their related geometries. We look at a common, but quite general, method of forcing pendulum motion at the natural frequency.

4.4 Parametric forcing Parametric forcing is any method of forcing that varies some parameter of the pendulum’s configuration – length, center of mass – in a periodic manner. The energy required for the parameter change becomes converted to motional energy for the pendulum. A common example of periodic parametric forcing is the periodic pumping of a playground swing performed by the rider in order to build and maintain the swing’s motion. In this case, the rider’s action involves movement of his or her body so that the effective length of the swing is periodically altered. A more exotic example is found with the pumping of O Botafumeiro by the team of pullers. Here, the actual length of the pendulum varies as the suspending rope is periodically wound up and wound down around the rollers high above the transept. In both cases, the energy input periodically changes a parameter – the length – of the pendulum. These changes also provide a change in the effective gravitational field on the pendulum bob due to the momentary forcing and consequent up-and-down acceleration of the bob. The energy of motion along the length converts to angular motion of the pendulum. For the playground swing, there are two common methods for parametric forcing. One method requires the rider to alternately stand and squat on the swing, such that the effective length of the swing shortens and lengthens periodically. The other common method is for the rider to remain seated but to alternately rock his body, again so that the effective length of the swing changes periodically. Variations to either of these methods make use of pulling and pushing on the ropes of the

124

Seven Tales of the Pendulum

swing to further enhance the motion. A further distinction can be made between methods of (a) initiating the motion, and (b) enhancing and maintaining the motion. Interestingly, the last three decades have seen a flurry of scholarly analysis on the physics of pumping a swing. Consider the case of a rider pumping the swing by alternately standing and squatting on the swing. The rider is squatting when the swing descends and then quickly stands as the swing passes through its lowest point. He can choose either to repeat the pattern on the return half of the cycle or simply to remain standing until the beginning of a new cycle. This activity periodically changes the center of mass of the swing and thereby its effective length in a manner very similar to that shown in Figure 4.11. Digging deeper into the technical aspects can show us how the pumping creates the motion. In going from a squatting to a standing position at the point of lowest arc, the force provided by the pumper is along the line of the pendulum, or radial in direction. Therefore there

O

L

r h

1

2

5

4 DL 3

Figure 4.11 A schematic diagram of the path of O Botafumeiro (or a playground swing) during a cycle. (Reprinted with permission from the American Journal of Physics, vol. 52, 1984, p. 940. © 1984, American Association of Physics Teachers.)

Pendulums Swing High

125

is no twisting force or torque, and the product of the mass distribution and the angular rate or angular velocity remains constant. This product is called angular momentum. Thus, even though the distribution of mass and therefore the effective length of the pendulum are changed during pumping, this product remains constant.4 Therefore the periodic shortening of the pendulum by pumping leads to an increase in angular velocity. (Think of how a figure skater increases her rate of rotation by pulling in her arms and legs.) In turn, the increased angular velocity leads to a rapid increase in the energy of motion at the bottom of the arc. Finally, increased motional energy gives rise to a new maximum height of the swing. In this way, the objective of increased amplitude of the swing is achieved by the pumping action. Figure 4.11 shows how the changes in length occur during the swing cycle. This diagram, although specific to O Botafumeiro, also roughly applies to this method of pumping the playground swing. Mathematical analysis of the energy balance for this sort of pumping can lead to a prediction of how the angular amplitude of the swing increases with the number of pumps. The result of such an analysis is shown in Figure 4.12. Figure 4.12 shows what experienced swing riders know: that the swing is slow to get going. The pumping action becomes more effective as the angular amplitude increases. The act of standing leads to an increase in speed and therefore swing amplitude. Is this amplitude increase lost when the rider squats? While some amplitude is lost, the fact that the squatting occurs at the extremes of the swing motion means that the loss in energy due to squatting is less as the angle increases. Since the loss goes down with angle, the swing amplitude goes up faster at higher angle. This is indicated toward the right side of Figure 4.12. It is also interesting to enquire how much energy is required of the rider during the action of standing. It turns out that the work done by the rider in standing increases as the swing goes higher, again a common experience. This increased work is due to the rider’s feeling greater centrifugal force at the bottom of the arc from the higher velocity of

4 Of course, gravity provides a restoring torque that causes a variation in the angular momentum. The point is that the radial action of the pumper has no effect on the angular momentum. The constancy of angular momentum is specific only to a particular angle.

126

Seven Tales of the Pendulum 90 80

Amplitude q (degrees)

70 60 50 40 30 20 10 0

0

2

4

6

8

10

12

14

16

18

Number of pumps

Figure 4.12 The size of the angular amplitude versus the number of pumps.

the larger swing. The work done, in the act of rising, requires an increasingly greater expenditure of energy by the rider. On the other hand, the graph in Figure 4.12 suggests that this extra work also produces larger increments in the swing’s amplitude. Experienced swing riders know that efficient starting of the swing’s motion requires a slightly different technique: a kind of rocking back and forth of the body in a sitting position. The rocking body acts like an asymmetric dumbbell, and again there is a slight change in the effective length of the pendulum. Putting these ideas together, we see that there is an optimum mix of pumping techniques. A rider should begin with sitting pumping, and progress to standing pumping at larger amplitudes for the most rapid accumulation of swing amplitude. A more subtle form of parametric pumping occurs when the pivot point of the pendulum is periodically varied. Imagine holding a pendulum’s pivot point between your thumb and forefinger and then periodically raising and lowering your hand. In this case, the parameter that is varied is not the pendulum’s length but, rather, the effective strength of the gravitational field acting on the pendulum bob. The periodic acceleration of your hand acts, according to the equivalence principle, like an artificial

Pendulums Swing High

127

gravitational field, sometimes positive and sometimes negative. This variation of field transfers the energy in the motion of your hand to energy that pumps the pendulum to larger swings. One prominent example of this type of drive mechanism is the modern Foucault pendulum installed in the Science Museum in London, for which the pivot actually moves. But let us return to our exotic pendulum in Santiago de Compostella. Like the self-pumped swing, O Botafumeiro is a parametric pendulum whose length is periodically shortened by pulling on the rope. The chain of causation is similar to that described above for the self-pumped playground swing. Figure 4.11 shows the approximate path of the censer for one cycle, as caused by the pumping action of the tiraboleiros. Figure 4.13 shows the change in amplitude of O Botafumeiro with each cycle of pumping. Of course, as the amplitude changes, the tiraboleiros need to slightly reduce the frequency of pulling on the rope, in order to avoid the amplitude modulation effect described earlier. Like the real pendulum, the simulation starts at about 13 degrees, where a tiraboleiro initially pushes the censer. It takes about 80 seconds to reach full amplitude in about 17 forcing cycles, in agreement with the real pendulum. We also note that the complexities of O Botafumeiro considerably stretch the validity of any mathematical model that treats 90 16

Amplitude θm (degrees)

80

18

20

80

90

14

70 12

60 10

50 8

40

6

30

4 2

20 10 0 0

10

20

30

40 50 60 t (seconds)

70

Figure 4.13 The oscillation amplitude for each pumping cycle. Both sets of points are due to mathematical models, but the circles represent a more complex model that yields an accurate representation of the motion of O Botafumeiro. (Reprinted with permission from the American Journal of Physics, vol. 52, 1984, p. 942. © 1984, American Association of Physics Teachers.)

128

Seven Tales of the Pendulum

it as a simple pendulum with its mass as concentrated at a point. One complication is that the mass of the rope is a significant factor at 15.7 kilograms. This difficulty suggests that the pendulum is not rigid and might be analyzed as a more complex construction. The thickness of the rope, at 4.5 centimeters, is also important and contributes to friction effects beyond those for the censer alone. At this point, we defer further analysis of the nonlinear pendulum until the next tale, where we discuss the chaotic pendulum. Let us end this chapter with a literary note.

4.5 Pendulum of horror In 1842, the first American author of tales of horror, Edgar Allen Poe (1809–49), wrote a short story entitled “The Pit and the Pendulum”. Poe’s stories often contained a strong element of terror, in part, because he left many of the details quite vague, just as a standard technique of psychological terror is to keep the victim in ignorance as to his ultimate fate. “The Pit and the Pendulum” does exactly that to both the reader and the protagonist. The main, and practically only, character – whose name we never know – is brought as a prisoner before the court of the Spanish Inquisition in Toledo, Spain. The trial is recalled by the prisoner during a confused dreamlike state. Subsequently, he is carried into the bowels of the earth and flung into a damp and dark dungeon. He attempts to investigate the physical condition of his cell, but exhaustion forces sleep upon him. When he awakes, he is tied to a low cot, with only one hand free with which to feed himself the spiced meat that is mysteriously laid beside the cot (Figure 4.14). He now notices a large pendulum high above the cot and observes the start of its oscillations. However, the presence of rats attempting to steal his food distracts his attention from the pendulum. Meanwhile, the pendulum continues to swish by overhead and, with each arc, the pendulum bob, now seen to be in the shape of a large sharp metal blade, comes ever closer to him. The descent of the pendulum is tortuously slow, giving the victim a chance to assess his situation and his likely fate. The strap by which he is held to the cot is a long single piece that is wound many times around his body. His first thought is that the pendulum might eventually cut the strap and allow him to free himself before he suffers his apparently inevitable fate. But, to his chagrin, he notes that the only place where the strap does not cover his body

Pendulums Swing High

129

Figure 4.14 A depiction of the victim in Poe’s story “The Pit and the Pendulum”. This drawing is actually a political cartoon drawn by Udo Keppler in the 1920s, highlighting the effect of inflation during that decade. Note “Cost of Living” on the blade/bob. (Political Cartoon Collection, Public Policy Papers, Department of Rare Books and Special Collections, Princeton University Library. Reprinted with permission.)

130

Seven Tales of the Pendulum

lies in the path of the pendulum. Therefore he needs to devise some other method of regaining freedom of movement. The story continues as the pendulum draws ever closer, with increasingly large amplitudes for its swing. Like O Botafumerio, this pendulum also makes an ominous swishing sound as it describes its increasing arc. As the pendulum nears the prisoner’s body, he estimates the total range of the pendulum’s motion to be about 30 feet. Given that the room itself is only about 40 feet high, this pendular motion is of very large amplitude, again similar to that of O Botafumerio. At this point, we refer readers to the original story in order to learn the fate of the prisoner. However, it is interesting to ask whether Poe’s tale is realistic. Does it seem likely that the Inquisition would have used something like a motorized pendulum in this context? The answer is probably no. The time frame of the story is not clear, but we may impose some limitations. The Spanish Inquisition ended in the early nineteenth century and the story itself refers to capture of the city by a certain French General La Salle. Perhaps this allusion is to the Napoleonic wars, but the time could also be much earlier. Scholars suggest that while torture was a standard and accepted means of learning truth in matters both secular and religious, it was not typically used gratuitously by the Inquisition once sentence had been passed. And, in the story, there had been a trial and a sentence had been pronounced. Furthermore, instruments of torture were fairly simple and direct. A driven pendulum that would slowly and noiselessly descend, and slowly increase its amplitude of motion, all inside and above a small dungeon, is relatively complex and hardly worth the effort even if craftsmen could be found to build such a machine. Therefore Poe’s story is somewhat unrealistic in regard to the technology. Some believe that the inspiration for Poe’s tale was actually a large swinging bell that he had observed. Nevertheless, realistic or not, the image of a sharpedged pendulum makes it frightening. It is the regularity and inexorability of the pendulum’s motion that contribute to the climate of terror found in this story. The pendulum’s periodicity, that is so important in other contexts, is here made to serve the cause of literary suspense. With Poe’s pendulum, we conclude our discussion of the science of the conventional pendulum. But the ideas developed in this chapter and earlier tales also provide a basis for our understanding of the uniquely modern features of the pendulum described in the following tales: namely, the properties of the chaotic pendulum and the quantum pendulum.

Tale 5: Pendulums Go Chaotic In earlier tales of the pendulum, we talked about what might be called the classical pendulum, the pendulum as it was prior to the twentieth century. We have learned how it can be represented geometrically, how phase diagrams, time series, and other diagrams can, with a bit of mental effort, enhance our understanding of the pendulum’s behavior. For those who like to see a bit of algebra, we have also added a few formulas as a possible further aid to understanding. Now we enter the modern period of pendulum dynamics. Computers become essential, as the equations are, in general, no longer amenable to straightforward human calculation. The geometric figures introduced earlier now take on new forms and new importance in our efforts to understand complex motions. But for now, let us gently ease into our subject of the chaotic pendulum with some historical background.

5.1 Introduction and history Much of the history of physics has been characterized by the effort to understand, in great detail, increasingly smaller pieces of nature, beginning with everyday bulk matter, and progressing down to molecules, atoms, nuclei, and elementary particles. This trend became especially pronounced in the twentieth century with the development of sophisticated experimental apparatus capable of probing deeply into nature’s innermost parts. Aside from the sense that one is closer to reality at the deeper levels of nature, it is plausible to assume that a clear understanding of the small pieces of nature will lead, in the aggregate, to a clear view of the large picture. The whole is presumed equal to the sum of its parts. This approach is labeled as reductionism. More recently, in certain areas of physics, the opposite methodology has proved fruitful. New structure and organization may become evident when there is complexity, large numbers of parts, several or

132

Seven Tales of the Pendulum

many degrees of freedom, or even just sufficient energy to make a distinct change to the system. It turns out that sometimes the whole is more than the sum of its parts. This creation of new richness of behavior often occurs in the study of processes that are pushed well beyond their equilibrium configurations. Researchers find new levels of organization, new complexity that does not seem obvious from a consideration of the individual parts of the process. For example, if reactants are forced rapidly in certain chemical reactions, the resultant products may show some kind of order in space or time. Or confined liquids with large temperature differences may exhibit new structural or dynamic organization of fluid motion. Even the motion of the humble pendulum achieves a new level of complexity if it is driven energetically at nonnatural frequencies. Why should this be? Part of the answer lies in the fact that when systems are well beyond equilibrium, they are often unstable, and instability can lead to new complex states. The entire field of chaotic dynamics is an important manifestation of a new order being achieved through instability. It is remarkable that the damped driven pendulum is an archetypical example of chaotic dynamics. The chaotic pendulum is therefore our next tale in the story of the pendulum. What do we mean by chaotic dynamics? What are its characteristics? In brief, chaotic systems are unstable deterministic systems whose trajectories are extremely sensitive to the initial condition of the system. We surmise that an unstable system is one that is easily disturbed, and that a deterministic system is one whose final state is precisely determined by its initial state and the force law that governs the system. But we need to dig deeper into the definition of a chaotic system. The French mathematician and astronomer Henri Poincaré (1854–1912) is generally credited as being the first to articulate the behavior of such systems. Here are his often-quoted words that suggest how a deterministic system might behave in what we now call a chaotic manner: “A very small cause that escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately.

Pendulums Go Chaotic

133

If that enable[s] us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. [Now here is the key statement.] But it is not always so: it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.”

There is a lot of substance in this quotation, so let us try to clarify. In this statement, Poincaré makes a distinction between two kinds of deterministic systems. He notes that we would consider a system to be deterministic if our prediction of its final state were accurate to approximately the same degree as our knowledge of its original state. This is our common experience. In science education we perform laboratory exercises in which variables are slightly uncertain. Yet we are satisfied that our calculations and observations have proved some physical law if the results give approximately the “right” numbers. That is, uncertainty in the measurements of the initial state of the system is not much amplified in the measurements of the final state of the system. On the other hand, Poincaré suggests that with certain systems the initial uncertainty increases quite rapidly, and therefore the final outcome looks as if it were the result of a random process or probability. Systems that behave in this latter manner are, within a certain range, unstable. Unstable systems have the ability to be affected by very small changes in their initial conditions. This property is referred to as sensitivity to initial conditions (SIC) and it is the hallmark of chaotic dynamics. Sensitivity to initial conditions occurs when an initial uncertainty grows exponentially in time, and the rate of growth is typically denoted by the symbol k, which is called the Lyapunov1 exponent. Mathematically, a system that has a single variable x with an initial uncertainty Dx0 exhibits SIC if it has a positive Lyapunov exponent and therefore grows according to the relation

Dx = Dx0ekt. 1

Aleksandr Mikhailovich Liapounov was a Russian mathematician who contributed to the theories of dynamics, probability, fields, and differential equations. The spelling “Liapounov” is slightly different from that typically used to describe the “Lyapunov” exponent.

134

Seven Tales of the Pendulum

A bounded system that exhibits short bursts of SIC, like a strongly (and weirdly) driven pendulum, is chaotic.2 Of course, if the initial conditions are somehow known to an infinite degree of precision, then the final state of any deterministic system is completely predictable. In practice, however, no real system is completely determined, and therefore it is the instability of the system together with the initial uncertainty that leads to the possibility of widely varying outcomes. Therefore, the deterministic, unstable, chaotic system acts as if it is probabilistic and random. Chaotic dynamics, or chaos as it is popularly named, is not to be confused with the primordial chaos suggested by various mythologies to exist at the moment of creation, or with the chaos that might ensue in the wake of a parade of small children and active dogs being turned loose in a confined space. In fact, the term “chaos” was coined by James Yorke and Tien-Yien Li of the dynamics group at the University of Maryland. In 1975, Li and Yorke published a paper with the title “Period Three Implies Chaos”, and the name stuck. While confusion of the technical meaning with the common meaning is unfortunate, “chaos” is now firmly entrenched in the scientific literature as the short form for “deterministic chaotic dynamics”. To sum up, we now define chaotic systems as deterministic systems that are unstable enough to exhibit a probability-like nature through their sensitivity to initial conditions. The history of chaos might be arbitrarily divided into two segments. The first period is from the time of Poincaré’s large work on dynamics in 1892 to Edward Lorenz’s 1963 computer simulations of unstable convective heating (such as that found in weather systems). The second period is from the early 1960s to the present. This division provides an approximate distinction between the scattered efforts of individual researchers and the later epoch in which chaos came to be an accepted field, complete with journals and conferences dedicated to the subject. The later period also roughly coincides with the dominant use of the computer in studies of chaotic dynamics. Beginning with Poincaré, here are some contributions from the first epoch. In his studies of celestial motions, Poincaré realized that analysis – loosely, equations solved with pencil and paper – had limitations 2 The pendulum’s motion is bound to a circle (or sphere) whose radius is the length of the pendulum. A well-known example of unbounded exponential growth is that of a population of fruit flies with an unlimited food supply.

Pendulums Go Chaotic

135

especially when it came to nonlinear dynamics, and he revived the use of geometric methods to study stability in such systems. The American mathematician George David Birkhoff (1884–1944) continued the geometric tradition of Poincaré and helped to develop the general aspects of the connection between unstable systems and probability. In 1932, he published the first chaotic attractor to appear in the literature. The Russian mathematician Aleksandr Mikhailovich Liapounov (1857–1918) studied the question of stability in dynamical systems and his name is, as we noted, now associated with the exponent k: the rate of exponential growth of uncertainty. In 1944, the American mathematician Norman Levinson (1912–75) suggested that a system with three degrees of freedom (independent variables) and forcing, like a driven pendulum, could produce a chaotic attractor. Other important pioneers include the English mathematician Dame Mary Cartwright (1900–98), who, in collaboration with John Littlewood (1885–1977), analyzed the nonlinear oscillator of the Dutch physicist Balthasar van der Pol (1889–1959) during World War II. The Russian mathematicians Andrey Kolmogorov (1903–87), well known for his work in probability, and Vladimir Arnol’d (1937– ), together with German/American mathematician, Jürgen Moser (1928–99), collaborated to produce the famous KAM theory of nonlinear systems. The American mathematician Stephen Smale (1930– ) produced a series of seminal theoretical papers in the 1960s that are important for advanced analysis of the chaotic pendulum. On the experimental side, we have already mentioned the electrical oscillator circuit that produced chaotic behavior, which was invented by van der Pol (circa 1927). Georg Duffing (1861–1944) created mechanical devices, specifically oscillators, with spring constants that depended nonlinearly upon position, to study nonlinear vibrations (circa 1908). Finally, we note the work of the mathematician Chihiro Hayashi (1911–87) of Kyoto University, in whose research group, in 1961, then-graduate student Yoshisuke Ueda (1936–) used an analog (as opposed to digital) computer to produce a strange attractor simulation of a nonlinear oscillator. Throughout this discussion, the word “simulation” means that the mathematical equations of the system (the differential equations) were solved using point-by-point calculations on the computer. The result is not a solution in the form of a set of new solution equations but, rather, a set of data points that fit the original equations of the dynamical model. In general, the equations of nonlinear systems are only solvable in this sense.

136

Seven Tales of the Pendulum

5.2 The chaotic pendulum From the 1960s on, chaotic dynamics became a much more universal and active field. It is quite impractical to give anything like a complete history. We just note that many of the techniques and characterizations that we will discuss for the chaotic pendulum were developed in this period. The digital computer became a primary tool, and an initial series of computer simulations of the chaotic pendulum were published by Elizabeth Gwinn and Robert Westervelt, from Harvard University, during the mid-1980s. Other researchers also saw the possibilities in a study of the pendulum. The first physical model of the chaotic pendulum was built by Canadian physicists James Blackburn and John Smith. This pendulum is driven by simple sine-wave periodic forcing at a variable frequency. The amount of energy dissipation or damping may also be varied in a calibrated way. A photograph of the actual system is shown in Figure 5.1. Through an optical encoder, data giving the instantaneous angle of the pendulum are sent to a computer, where they can be processed for further analysis. The fit of such data to a simple nonlinear dynamical model based upon Newton’s laws of motion is remarkably good (see Figure 5.2). Here are some details. The pendulum is energized by the drive motor, which provides periodic forcing through rotation of the shaft to which the bob (M) is fixed. The frequency xD and the amplitude A of the forcing are both adjustable. Friction is introduced through magnetic damping.3 The closeness of the magnet (C) – which moves with the pendulum – to the copper damping plate (B) regulates the amount of damping. Thus the strength of the damping is also adjustable with the micrometer and is characterized by the parameter Q. These three parameters – the damping strength Q, the forcing amplitude A, and the forcing frequency, xD – control the motion. Adjustment of any of the parameters can radically change the pendulum’s motion. The values of each member of this set of three parameters will determine the dynamics. Figure 5.2 compares experimental data from the pendulum shown in Figure 5.1 with computer-generated, simulated data from a mathematical model. The mathematical model connects the pendulum’s 3 If a magnet moves near a metallic conductor, it will feel a force that tries to slow the motion.

Pendulums Go Chaotic

137

F

B

A

M C

Figure 5.1 A photograph of the Blackburn–Smith pendulum. A is the optical encoder, B is the copper damping plate, C is a magnet for the drive motor, F adjusts the damping, and M is the pendulum bob.

motion to the forces that impinge on it: gravity, friction, and forcing. Both the experimental pendulum and the simulation model require the three parameters described above as inputs. We see in Figure 5.2 that the simulated data closely replicates the experimental data, and this gives us confidence in using simulated data, which are relatively easy to acquire from the computer. Therefore, most of figures in this tale are the result of such model simulations. Since that initial construction, many variations on the basic chaotic pendulum have been designed and built.4 In our fourth tale, we described many characteristics of the nonlinear, large-swing pendulum. But we also hinted at some further complexities. The key point here is that nonlinearity is essential for chaotic motion, and therefore the nonlinear pendulum is an excellent vehicle for the study of chaotic motion. Why is nonlinearity a necessary 4

In 1990, this writer, with coauthor Jerry Gollub of Haverford College and the University of Pennsylvania, gathered together many aspects of the behavior of the pendulum in an accessible undergraduate text (see the Further Reading). Many of the figures in this tale are from that book and further details of much of what we describe here may be found there.

138

Seven Tales of the Pendulum Experimental Data 20

Velocity

10 0 –10 –20

–3

–2

–1

0

1

2

3

2

3

Simulation Data 20

Velocity

10 0 –10 –20

–3

–2

–1 0 1 Angle (radians)

Figure 5.2 A comparison of data from the experimental pendulum with simulated data computed from a Newtonian mathematical model of the pendulum. The geometric representation is a slice of a phase plane and is called a Poincaré section. It is described below.

condition for chaotic motion in any physical system? It is nonlinearity that leads to instability and the possibility of an abrupt transition, a bifurcation, to a new kind of motion. A classic example of nonlinearity leading to bifurcation is the straw that breaks the camel’s back. The camel may be loaded with increasing weights with only a slight proportional buckling of its legs. Yet there may come a point where the addition of just one small thing – a piece of straw – may be sufficient to “break” the camel’s back. The camel would then be completely prostrate and unable to carry any weight at all. We observe an abrupt transition from one state – the standing camel – to a completely different state – the prostrate camel. A slightly more scientific illustration is the example of a weight suspended at the end of thin wire. As more weight is added, the wire stretches proportionately (linearly) – the behavior predicted by Hooke’s law, which we described in the Introduction. But at some point a small extra weight causes a disproportionately large

Pendulums Go Chaotic

139

stretch (nonlinearity) or even breakage (bifurcation) in the wire. Again, this behavior reflects a nonlinear response by the wire. Similarly, nonlinearity allows the pendulum to exhibit transitions to complex behavior. We have learned that, for small amplitudes of forcing, the driven pendulum exhibits the usual periodic behavior that is similar to, but not identical with, the motion of a linearized nondriven pendulum. But for the nonlinear pendulum with larger amounts of forcing at nonnatural frequencies, the picture may change completely. The motion may become chaotic.

5.3 The geometry of chaos In this section, we gather together many of the geometric representations that we have discussed, and also introduce some new ones. Earlier geometries such as time series, phase diagrams, and Fourier spectra represent a particular configuration of the pendulum, a particular combination of forcing and friction. Let us start with a new type of diagram that gives us some idea of the types of motion across whole sets of configurations.

5.3.1 The state diagram Ideally, one would like to study the pendulum’s motion for every set of its three parameter values. We might imagine a global geometrical space where every point is labeled by a set of parameter coordinates (A, Q, xD). Each point would then be a specific physical–dynamical configuration of the pendulum. The type of motion of each configuration – periodic or chaotic – might be indicated by a representative symbol placed at that point. Such a three-dimensional diagram would be very time-consuming to produce and probably confusing to interpret. However, a two-dimensional version for which one parameter is held fixed can be helpful. Figure 5.3 shows such figures, sometimes called state diagrams. In this case, the damping parameter Q is held fixed, while the other parameters, forcing strength and forcing frequency, vary. The upper figure was created by James Blackburn, of Wilfrid Laurier University, with data that he obtained from a real experimental pendulum of the type shown in Figure 5.1. The natural period of the pendulum is on the order of perhaps one-half second. Hundreds of data points are required for each point, as the pendulum takes time to relax

140

Seven Tales of the Pendulum

into each new state that is dictated by a slightly modified parameter set. Once the pendulum is settled into the new state, the nature of state – whether periodic or chaotic – may be determined. From start to finish, the experimental protocol for each data point took several minutes. Since there are more than 5,000 points, creation of the figure required weeks of experiments! The lower figure is a computer simulation from the mathematical model of the pendulum. Because a simulation is much faster than an actual experiment, the resolution in the figure is very high. Still, it required literally billions of solutions of the pendulum model’s differential equations, taking several hours even on a supercomputer. There are about six million points on the diagram and therefore it shows the fine details that the more coarse experimental diagram omits. The figure was created for the book by Jason Gallas, of the Universidade Federal do Rio Grande do Sul in Brazil. The simulation shows quite beautifully the complex relationship between chaotic (a) 2.0

Amplitude (c1)

1.5

1.0

0.5 0.2

0.4

0.6 Frequency (X)

0.8

1.0

Pendulums Go Chaotic

141

(b)

A

2.0

0.5 0.0

ωD

1.2

–0.18

0

0.34

Figure 5.3 Corresponding state diagrams for a pendulum with fixed damping Q, but variable forcing amplitude A and variable forcing frequency xD. (a) Experimental data, where the squares represent chaotic motion and dots represent periodic motion. (Courtesy of James Blackburn.) (b) Simulated data generated on a supercomputer. The shadowed portions indicate chaotic motion. The thin horizontal graph shows the corresponding Lyapunov exponent. Positive Lyapunov exponents are characteristic of chaos. (Courtesy of Jason Gallas.)

and periodic motion, and confirms the trends suggested by the experimental diagram. From either part of Figure 5.3, it is clear that the regimes of periodic motion and chaotic motion are often close neighbors, and even intertwined. This means that the character of the motion is very sensitive to the values of the three parameters. A small change in any parameter may abruptly change the type of motion. Furthermore, as we have seen in the previous tale, periodic motion itself can be quite varied. A periodic motion may be double or triple the period of the driving force, or even more. Perhaps even more symbols could have been used in these graphs to further specify particular types of periodic motion. Finally, we again emphasize that to produce these more exotic motions, the drive frequency is generally not at any natural frequency for the pendulum.

142

Seven Tales of the Pendulum

5.3.2 Phase planes, Poincaré sections, time series Aside from the state diagram, we have seen that there are a variety of other ways to represent the pendulum’s motion geometrically. In the discussion of large-swing pendulums, we briefly mentioned the idea of driving the pendulum at nonnatural frequencies, so that it would oscillate with periods that were double, quadruple, or more the natural period; that is, subharmonic motion. Let us use this nonnatural forcing to further illustrate geometric representations. In Figure 5.4, such periodic motions are illustrated by sketches of the actual paths of the pendulum bob over three different cycles, two of which are subharmonic – the top and bottom examples. Situated beside the sketch of each motion is the corresponding phase-plane diagram. The phase diagram, or phase portrait, is a graph of angular velocity versus angle. Each phase diagram is traced through many drive cycles and yet, in Figure 5.4, the motion, although somewhat complex, remains periodic and therefore the phase portrait is fairly simple. As the amount of forcing increases, complexity generally increases. For example, as noted above, the period of the motion is sometimes twice or even four times that of the forcing motion. Successive period doublings can be a mechanism that leads to chaotic motion. Thus, the understanding of chaotic motion is facilitated by using a variety of geometric representations. Poincaré introduced another geometric representation, which we now call a Poincaré section. This is of great value when phase portraits become more complex. If one imagines the motion of the phase orbit occurring in a darkened room, with only a strobe light for illumination, then the phase coordinates of the angle and the angular velocity are available only when the light flashes on. If the strobe frequency were to be made equal to the forcing frequency, the observer would see only those phase coordinates that were synchronized to the forcing. Thus, for motion whose period is the same as the forcing period as shown in Figure 5.4 (middle), there would be only one point. However, when a complete motion takes two forcing periods – period doubling – as in Figure 5.4 (top) and especially in Figure 5.4 (bottom), there would be two points. The effect of the strobe is to cut a section through the phase orbit; hence the name Poincaré section. Figure 5.2 shows more complex Poincaré sections characteristic of chaotic motion. The multitude of points indicates the existence of a myriad of periodicities in the motion.

3 2 Start

+h

Angular Velocity

End 1 –0 –1 –2 –3

–3

–2

–1

0

1

2

3

1

2

3

1

2

3

Angle

2

Angular Velocity

1

+h

0

–1

–2 –3

–2

–1

0 Angle

2 1

Start

+h

Angular Velocity

End

–0 –1 –2 –3

–3

–2

–1

0 Angle

Figure 5.4 Three examples of complex, yet periodic, motions of the driven pendulum. The figures on the left portray the pendulum motion in real space. The figures on the right are the corresponding phase-plane diagrams. (Reprinted from Chaotic Dynamics: An Introduction, 2nd edn., by G. L. Baker and J. P. Gollub, 1996, p. 22, with the permission of Cambridge University Press.)

144

Seven Tales of the Pendulum

We also introduced time series earlier. Common examples include time series for the pendulum’s angle or its angular velocity, and these provide another picture of the motion. With the same parameter sets as in Figure 5.4, corresponding time series for angular velocity are shown in Figure 5.5. While the motions are all periodic, unlike those of the linearized pendulum, the time series are not simple sine waves. In the fourth tale, we introduced the idea that any periodic function could be decomposed into its constituent sine waves using the machinery of the Fourier series. We also showed the relative contribution of each harmonic component in Figure 4.5. Clearly, the subharmonic periodic motions of Figs. 5.4 and 5.5 also contain several Fourier components. Chaotic motion appears when the pattern of periodic motion is broken through instability. The motion then becomes nonrepetitive or

2 0 –2 300

310

320

330

340

350

360

370

380

390

400

310

320

330

340

350

360

370

380

390

400

310

320

330

340

350

360

370

380

390

400

1.5 –0.0 –1.5 300 2.0 0.5 –1.0 –2.5 300

Time

Figure 5.5 Time series for the pendulum’s angular velocity corresponding to parameters used in Figure 5.4. Careful examination shows that the top series is indeed period-2, whereas the bottom series is more obviously period-2. The middle series has the same period as the forcing.

Pendulums Go Chaotic

145

nonperiodic. Another way of describing the chaos is to say that it contains all periodic frequencies, but that none of them last very long. More technically, it is said that chaotic motion contains an infinite number of unstable periodic orbits. As the phase trajectory wanders around the phase space, it comes very close to lots of periodic orbits, but is unable to maintain itself for very long on any one of these orbits. Consequently, chaotic time series seem to have no discernible pattern or periodicity, the phase plane is very congested, and even the Poincaré section becomes complex. Let us take a coordinated look at the geometry of a chaotic pendulum. Figure 5.6 gives an example of a time series, a phase-plane diagram, and a Poincaré section for a typical chaotic state. Figure 5.6(c) may be compared with Figure 5.2. The simplification caused by the strobe effect of the Poincaré section now becomes apparent. Whereas the phase portrait (b) is quite congested, the Poincaré section (c) has an appearance that lends itself to characterization. We note first that (for a system with friction) the Poincaré section is confined to a particular complex region in space. No matter what the initial conditions, the points eventually enter this region, which has a special characteristic appearance. In essence, the pendulum’s motion is attracted to the set of points that form the Poincaré section. The Poincaré section and the corresponding phase portrait shown in Figure 5.6 are then said to be strange attractors. In the fourth tale, we learned that for nonchaotic motions, the attracting phase space set for periodic motion was referred to as an attractor. The adjective “strange” is related to the particular geometry of the attractor for chaotic motion. For chaotic systems in general, and the chaotic pendulum in particular, the attractor is a geometric fractal.5 Therefore, an attractor that is a fractal is called a “strange attractor”. Obviously, chaotic motion is complex. This complexity suggests that nonlinear equations of motion are difficult, and usually impossible to solve by analytic methods. This is true for the pendulum equation. Instead, as we noted, solutions are generated by numerical methods, using computers to do repetitive calculations. It is not surprising that the bulk of the work in chaotic dynamics has occurred after the computer revolution. Edward Lorenz, of the Massachusetts Institute of Technology, is generally credited with being the first to observe chaotic time series on a computer. Lorenz recounted the story 5

We shall say more on fractals later.

146

Seven Tales of the Pendulum (a) 4 3 Angular Velocity

2 1 –0 –1 –2 –3 –4 300 310 320 330 340 350 360 370 380 390 400 Time

(b) 4 3

Angular Velocity

2 1 –0 –1 –2 –3 –4 –3

–2

–1

0

1

2

3

1

2

3

Angle

(c) 3

Angular Velocity

2

1

0

–1 –3

–2

–1

0 Angle

Figure 5.6 Three geometric representations of the chaotic pendulum: (a) time series of the angular velocity; (b) the phase-plane diagram; and (c) the Poincaré section.

Pendulums Go Chaotic

147

of this bit of serendipity for a television program on chaos. During observation of the computer solution of his model equations for convective air flow in 1963, he had occasion to restart his computer. In doing this, he failed to put in the precise numbers used to start a previous computer simulation and, to his surprise, the computer gave him quite a different result for the system trajectory. This was the first recorded instance of sensitivity to initial conditions, and led to the actualization of Poincaré’s prediction, quoted above, for the unstable behavior of a set of deterministic differential equations. The computer is now the primary tool in the theoretical study of chaotic systems. In an ideal application of the scientific method, experimental apparatus should be constructed whose behavior approximates that of the computer simulation. The experimental apparatus may then be used to test the theoretical model. But while chaotic systems are believed to be ubiquitous in nature, realistic applications are difficult to construct as controlled systems suitable for laboratory experimentation. Fortunately, the chaotic pendulum is one of the few exceptions to this chronic difficulty and, as we have seen, can be realized experimentally.6

5.3.3 Fourier analysis But let us return to geometric representations of the pendulum’s motion. What about Fourier analysis? Chaotic motion never repeats exactly; it is not periodic, and the theorem that supports Fourier decomposition into components as described in the fourth tale no longer holds. There is no fundamental frequency of motion on which to base the Fourier series expansion. However, there is a generalization of the process used to obtain Fourier series that works for nonperiodic functions. This process requires a mathematical transformation known as the Fourier transform. If the time series is not periodic, as is the case with chaotic motion, then the entire nonperiodic motion can be thought of as part of one single long wave, whose period is infinitely long. The consequence of this strange idea is that the fundamental period of a possible Fourier series now becomes infinitely long or, conversely, the fundamental frequency becomes infinitely small. Creation of the Fourier transform then becomes a mysterious limit process whereby 6 While the Blackburn–Smith pendulum seems to have been the first experimental chaotic pendulum, there are other configurations.

148

Seven Tales of the Pendulum

some quantities become infinitely long and others become infinitely short. The Fourier transform of the chaotic time series shown in Figure 5.6(a) is illustrated in Figure 5.7. It seems fairly clear that the idea for the Fourier transform stems from Fourier’s 1811 paper, although other mathematicians, including Gauss, were working on similar ideas. However, for effective implementation of the transform in chaotic dynamics, we rely on a modern approach. In 1964, James Culley and John Tukey, at the urging of Richard Garwin at the IBM research center, created a better method or “algorithm” known as the Fast Fourier Transform, or FFT. What do we learn about the chaotic pendulum from the Fourier transform? The Fourier spectra of Figure 5.7 shows that chaotic motion gives rise to an infinite number of densely packed frequency components. This observation reinforces the statement made earlier that chaotic motion contains an infinite number of unstable periodic orbits. The spectrum also shows a very strong component at the frequency of the pendulum forcing. But for the bulk of the components, the ampli-

–2

log10 (power)

–3

–4

–5

–6

0.0

0.1

0.2

0.3

0.4

0.5

Frequency

Figure 5.7 The Fourier spectrum for a chaotic pendulum. Aside from the strong component at the forcing frequency (0.106 arbitrary units), the amount of energy at each frequency roughly diminishes with increasing frequency.

Pendulums Go Chaotic

149

tude of the components generally decreases with increasing frequency. This behavior points to much of the pendulum’s motion as being in (unstable) periodic orbits of very long periods, and is consistent with the notion that one route to chaos is through increasing complexity of the subharmonic periodic orbits, which – in turn – implies increasing longer periodic orbits. By way of comparison, we note that a spectrum that is inversely proportional to frequency is also characteristic of a certain type of electrical noise, called “1/f ” noise. Such noise is in addition to that which is inherent in an electrical system due to simple thermal agitation: “1/f ” noise can be caused, for example, by fluctuations in the electrical resistance in a circuit. These fluctuations are quite prominent at low frequencies. In electronic vacuum tubes – now largely a museum artifact – current was transmitted between the cathode and anode. The electron emission from the cathode was not strictly uniform and, again, this electric current generated a noise spectrum with 1/f behavior. The analogy between 1/f noise and a chaotic spectrum is important. The processes that generate electrical 1/f noise are random, yet the pendulum is deterministic. Therefore, we once again witness a manifestation of Poincaré’s statement that deterministic processes can sometimes appear to be probabilistic; that is, according to Fourier analysis, the chaotic pendulum appears to behave randomly. A final confirmation of the sense that a chaotic system is random may be seen from observation of the relative frequency of occurrence of a given angle. For a simple linear pendulum, we saw that the relative frequency distribution given in Figure I.8 was peaked at the two extremes of its motion. For a period-doubling motion of the pendulum, the relative frequency distribution was peaked at four angles, again at its extreme angles, as shown in Figure 4.10(a), and corresponding behavior was shown for period-15 motion in Figure 4.10(b). But for the chaotic motion, as shown in Figure 5.8, the relative frequency distribution is approximately rectangular or uniform across the range of possible angular displacements, somewhat like a rectangular probability distribution. The infinite number of unstable periodic orbits tends to smooth out the distribution such that no angle is given significant preference. The pendulum is found more or less equally in any orientation. The bumps are likely due to the extra strength of the forcing frequency and its harmonics, as indicated in the Fourier spectrum of Figure 5.7. Again, there is a sense that a deterministic system can look random.

150

Seven Tales of the Pendulum Angular Distribution: chaotic pendulum

Relative Frequency

0.00025 0.00020 0.00015 0.00010 0.00005 0.00000 –4

–3

–2

–1 0 1 Angle (radians)

2

3

4

Figure 5.8 The frequency distribution for the angle of the chaotic pendulum. (Reprinted with permission from the American Journal of Physics, vol. 74, 2006, pp. 482–9. © 2006, American Association of Physics Teachers.)

5.3.4 Bifurcation diagrams Previously described geometric representations (time series, phase diagrams, and so forth) of chaotic motion all picture the development of a particular pendulum being driven with a particular forcing strength, damping, and forcing frequency. We can also provide a more global picture, in which the motion of a variety of pendulums can be represented on a single diagram. One example is the state diagram of Figure 5.3. In that case we can vary two of the three parameters and, by the shading of the specific point, see the degree of chaotic motion. Another approach is to combine a bit of the Poincaré section idea with the state diagram, rather than just indicating the type of motion by a fixed symbol. As before, we observe some aspect of the motion of the pendulum for particular values of the dynamical parameters, but then change the configuration slightly and repeat the process. This time, it is done many times for incremental changes for only one of the pendulum’s parameters (A, Q, xD). But, in exchange, we obtain a more detailed view of the dynamics. We see a slice of the Poincaré section. Here is how the graph is set up. The chosen variable parameter is the scale of the horizontal axis of a graph. The vertical scale of the graph is now reserved for some dynamical variable, such as the strobed value of angular velocity of the pendulum. Essentially, this is the vertical

Pendulums Go Chaotic

151

coordinate of the Poincaré section. We can think of this diagram as a combination of a state diagram and Poincaré sections. Where the diagram has only a few lines, the motion is periodic. Clusters of points indicate chaotic motion. The bifurcation diagram shown in Figure 5.9 is a typical example and provides a deeper view than Figure 5.3, at least with respect to the forcing parameter A. The advantage of this sort of presentation is that it provides a summary of the possible types of motion over some range of the system parameter. Points where the pendulum’s motion changes to a new kind of motion are called bifurcations, and since this diagram is characterized by many such points, it is called a bifurcation diagram. At the left side, where the forcing amplitude is small, the motion has a period equal to the forcing period. Further to the right on the diagram, the motion, which is still periodic, becomes more complex through successive period doublings. After a few such doublings, the strobed angular velocity takes on many values, and the motion is now chaotic. However, the motion does not remain uniformly chaotic, but reverts to windows of periodicity throughout the range of forcing amplitudes. If the periodic motion is period-3, then three lines will appear. If the

3

Angular velocity

2

1

0

–1

–2 0.5

0.6

0.7

0.8

0.9 1.0 1.1 1.2 Forcing amplitude (A)

1.3

1.4

1.5

Figure 5.9 A bifurcation diagram: a plot of the angular velocity, taken many times at a fixed point in the forcing cycle, versus the forcing amplitude.

152

Seven Tales of the Pendulum

motion is period-15, then 15 lines will appear. We see that there is often a similarity to the progression via several period doublings from periodic motion to chaotic motion over the range of amplitudes. By looking at increasingly smaller ranges over which the system parameter is varied, we magnify the bifurcation diagram and observe another phenomenon; namely, that the number of periodic windows inside a generally chaotic regime increases as the bifurcation diagrams are increasingly magnified. This property of increasing complexity under magnification is also found in phase portraits and Poincaré sections. Understandably, this property is called approximate self-similarity, and hints at the infinite variety and complexity of nature. Similar bifurcation diagrams may be created through variation of each of the other two parameters, the forcing frequency xD, and the friction parameter Q. A set of three such diagrams would give a complete picture of the dynamics over a range of system parameters.

5.4 Fractals The geometric representation of chaotic motion by, for example, the strange attractor of the Poincaré section has an almost artistic quality. The Poincaré section is an example of fractal geometry. As we will see, the properties of the Poincaré section fractal are intimately related to the pendulum’s dynamics. But let us first give some idea of the general properties of a fractal. The term “fractal” was coined by the mathematician Benoit Mandelbrot. It comes from the Latin adjective “fractus”, which means “broken”. If we imagine the coastline of, say, Norway, and look at that country’s map at ever-increasing magnification, we would see that no characteristic length can be applied. There is no smallest straight part. Any apparently straight lines are, upon further magnification, seen to break into smaller line segments. Hence any characteristic length is “broken”. Fractals are roughly defined as geometric sets that have no characteristic length. Fractals also exhibit a property known as selfsimilarity, which was noted above in bifurcation diagrams. Consider again the geographical example. Viewed at one level, the coast of Norway appears to be fairly jagged. If a portion of that coast is now magnified to the same size as the original view, it will have a jagged appearance similar to that of the original stretch of coastline.

Pendulums Go Chaotic

153

This sameness under observation at increasing magnifications is called self-similarity. We see, then, that fractals typically have (a) no characteristic length and (b) some degree of self-similarity. Yet long before the term “fractal” was invented, several mathematicians studied geometric configurations that later came to be known as fractals. Some of the mathematicians associated with these fractal precursors are: the German mathematicians Georg Cantor (1845–1918) and David Hilbert (1862–1943), who proposed space-filling curves; the Italian mathematician Guiseppe Peano (1858–1932); Helge von Koch, the Swedish mathematician who, in 1904, published the Koch snowflake; the Polish mathematician Wacław Sierpiński (1882–1969), who published the Sierpiński gasket fractal in 1916; the French mathematician Gaston Julia (1893–1978), who gave us the family of fractals known as Julia sets; and the German mathematician Felix Hausdorff (1868– 1942), with whom we associate the idea of dimension of a fractal. (During World War II, Hausdorff and his wife, both Jewish, committed suicide in anticipation that they were within a week of being sent to a concentration camp.) Mandelbrot (1924–2010), entered the scene much later. Mandelbrot’s uncle recommended Julia’s 1918 paper to him around 1945. For various reasons, Mandlebrot did not like the paper and found his own path to Julia’s work in about 1977. With the aid of the computer, Mandelbrot greatly expanded Julia’s work and showed the potential for creating beautiful pictures. As an example, the Cantor set (1883) is a simple but popular fractal, and its construction is indicated in Figure 5.10. The set is constructed by removing the middle third out of each line segment an infinite number of times. After an infinite number of steps, there are an infinite number of disconnected pieces. Clearly the Cantor set is self-similar – the same appearance at varying magnifications – and has no characteristic length. The Poincaré section for the pendulum also has these properties in some measure. Figure 5.11 shows the Poincaré section under increasing magnification, and again there is a sense that further structure is revealed with increasing magnification. Fractals also present us with some surprising ideas about geometric dimension. We are quite used to the idea that objects have integer dimensionality – zero, one, two, three, or even more, dimensions. A single point, or perhaps a finite collection of points, has a dimension of zero. Lines are one-dimensional and consist of an infinite set of closely spaced points. Solid planes are two-dimensional. In each case, the dimension

154

Seven Tales of the Pendulum N

n steps

d = lim ε→0 = lim n→∞ =

log 2 log 3

ε

1

1

2

1 3

4

1 9

8

1 27

2n

1 3n

log (N) log (1/ε) log (2n) log (3n)