THE PENDULUM

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The PendulumA Case Study in Physics

GREGORY L. BAKER

Bryn Athyn College of the New Church,Pennsylvania, USA

and

JAMES A. BLACKBURN

Wilfrid Laurier University,Ontario, Canada

AC

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Preface

To look at a thing is quite different from seeing a thing

(Oscar Wilde, from An Ideal Husband )

The pendulum: a case study in physics is an unusual book in several ways.Most distinctively, it is organized around a single physical system, thependulum, in contrast to conventional texts that remain conned to singleelds such as electromagnetism or classical mechanics. In other words, thependulum is the central focus, but from this main path we branch to manyimportant areas of physics, technology, and the history of science.Everyone is familiar with the basic behavior of a simple penduluma

pivoted rod with a mass attached to the free end. The grandfather clockcomes to mind. It might seem that there is not much to be said about suchan elemental system, or that its dynamical possibilities would be limited.But, in reality, this is a very complex systemmasquerading as a simple one.On closer examination, the pendulum exhibits a remarkable variety of

motions. By considering pendulum dynamics, with and without externalforcing, we are drawn to the essential ideas of linearity and nonlinearity indriven systems, including chaos. Coupled pendulums can become syn-chronized, a behavior noted by Christiaan Huygens in the seventeenthcentury. Even quantum mechanics can be brought to bear on this simpletype of oscillator. The pendulum has intriguing connections to super-conducting devices. Looking at applications of pendulums we are led tomeasurements of the gravitational constant, viscosity, the attraction ofcharged particles, the equivalence principle, and time.While the study of physics is typically motivated by the wish to under-

stand physical laws, to understand how the physical world works, and,through research, to explore the details of those laws, this science continuesto be enormously important in the human economy and polity. The pen-dulum, in its own way, is also part of this development. Not just a device ofpure physics, the pendulum is fascinating because of its intriguing historyand the range of its technical applications spanningmany elds and severalcenturies. Thus we encounter, in this book, Galileo, Cavendish, Coulomb,Foucault, Kamerlingh Onnes, Josephson, and others.We contemplated a range of possibilities for the structure and avor of

our book. The wide coverage and historical connections suggested a broadapproach suited to a fairly general audience. However, a book withoutequations would mean using words to try to convey the beauty of thetheoretical (mathematical) basis for the physics of the pendulum. Graphsand equations give physics its predictive power and preeminent place in ourunderstanding of the physical world. With this in mind, we opted instead

for a thorough technical treatment. In places we have supplied backgroundmaterial for the nonexpert reader; for example, in the chapter on thequantum pendulum, we include a short introduction to the main ideas ofquantum physics.There is another signicant difference between this book and standard

physics texts. As noted, this work focuses on a single topic, the pendulum.Yet, in conventional physics books, the pendulum usually appears only asan illustration of a particular theory or phenomenon. A classicalmechanics text might treat the pendulum within a certain context, whereasa book on chaotic dynamics might describe the pendulum with a verydifferent emphasis. In the event that a book on quantummechanics were toconsider the pendulum, it would do so from yet another point of view. Incontrast, here we have gathered together these many threads and made thependulum the unifying concept.Finally, we believe thatThe Pendulum: A Case Study in Physicsmay well

serve as a model for a new kind of course in physics, one that would take athematic approach, thereby conveying something of the interrelation ofdisciplines in the real progress of science. To gain a full measure ofunderstanding, the requisite mathematics would include calculus up toordinary differential equations. Exposure to an introductory physicscourse would also be helpful. A number of exercises are included for thosewho do wish to use this as a text. For the more casual reader, a naturalcuriosity and some ability to understand graphs are probably sufcientto gain a sense of the richness of the science associated with this complexdevice.We began this project thinking to create a book that would be something

of an encyclopedia on the topic, one volume holding all the facts aboutpendulums. But the list of potential topics proved to be astonishinglyextensive and variedtoo long, as it turned out, for this text. So frommany possibilities, we have made the choices found in these pages.The book, then, is a theme and variations. We hope the reader will nd

it a rich and satisfying discourse.

Prefacevi

Acknowledgments

We are indebted to many individuals for helping us bring this project tocompletion. In the summer of 2003, we visited several scientic museumsin order to get a rst hand look at some of the famous pendulums towhich we refer in the book. In the course of these visits various curatorsand other staff members were very generous in allowing us access to themuseum collections. We wish to acknowledge the hospitality of AdrianWhicher, Assistant Curator, Classical Physics, the Science Museum ofLondon, Jonathan Betts, Senior Curator of Horology, Matthew Read,Assistant Curator of Horology, and Janet Small, all of the NationalMaritime Museum, Greenwich, G. A. C. Veeneman, Director, HansHooijmaijers, Curator, and Robert de Bruin all of the BoerhaaveMuseum of Leyden, and Laurence Bobis, Directrice de la Bibliothe`que deLObservatoire de Paris. We wish also to thank William Tobin forhelpful information on Leon Foucault and his famous pendulum, JamesYorke for some historical information, Juan Sanmartin for furtherarticles on O Botafumeiro, Bernie Nickel for useful discussions about theFoucault pendulum at the University of Guelph, Susan Henley andWilliam Underwood of the Society of Exploration Geophysicists, andRajarshi Roy and Steven Strogatz for useful discussions. Others to whomwe owe thanks are Margaret Walker, Bob Whitaker, Philip Hannah, BobHolstrom, editor of the Horological Science Newsletter, and DannyHillis and David Munro, both associated with the Long Now clockproject. For clarifying some matters of Latin grammar, JAB thanksProfessors Joann Freed and Judy Fletcher of Wilfrid Laurier University.Finally, both of us would like to express gratitude to our colleague andfriend, John Smith, who has made signicant contributions to theexperimental work described in the chapters on the chaotic pendulumand synchronized pendulums.Library and other media resources are important for this work. We

would like to thank Rachel Longstaff, NancyMitzen, and Carroll Odhnerof the Swedenborg Library of Bryn Athyn College, AmyGillingham of theLibrary, University of Guelph, for providing copies of correspondencebetween Christiaan Huygens and his father, Nancy Shader, CharlesGreene, and the staff of the PrincetonManuscript Library. GLB wishes tothankCharles Lindsay, Dean of BrynAthynCollege for helping to arrangesabbaticals that expedited this work, Jennifer Beiswenger and CharlesEbert for computer help, and the Research committee of the Academy ofthe New Church for ongoing nancial support.Financial support for JAB was provided through a Discovery Grant

from the Natural Sciences and Engineering Research Council of Canada.

The nature of this book provided a strong incentive to use gures from awide variety of sources. We have made every effort to determine originalsources and obtain permissions for the use of these illustrations. A largenumber, especially of historical gures or pictures of experimental appa-ratus, were taken from books, scientic journals, and from museumsources. Credit for individual gures is found in the respective captions.Many researchers generously gave us permission to use gures from theirpublications. In this connection we thank G. DAnna, John Bird, BerylClotfelter, Richard Crane, Jens Gundlach, John Lindner, Gabriel Luther,Riley Neuman, Juan Sanmartin, Donald Sullivan, and James Yorke. Thebook contains a few gures created by parties whom we were unable tolocate.We thank those publishers who either waived or reduced fees for useof gures from books.It has been a pleasure working with OUP on this project and we wish to

express our special thanks to Sonke Adlung, physical science editor,Tamsin Langrishe, assistant commissioning editor, and Anita Petrie,production editor.Finally, we wish to express profound gratitude to our wives, Margaret

Baker and Helena Stone, for their support and encouragement throughthe course of this work.

viii Acknowledgments

Contents

1 Introduction 1

2 Pendulums somewhat simple 82.1 The beginning 8

2.2 The simple pendulum 9

2.3 Some analogs of the linearized pendulum 13

2.3.1 The spring 13

2.3.2 Resonant electrical circuit 15

2.3.3 The pendulum and the earth 16

2.3.4 The military pendulum 19

2.3.5 Compound pendulum 20

2.3.6 Katers pendulum 21

2.4 Some connections 23

2.5 Exercises 24

3 Pendulums less simple 273.1 O Botafumeiro 27

3.2 The linearized pendulum with complications 29

3.2.1 Energy lossfriction 29

3.2.2 Energy gainforcing 34

3.2.3 Parametric forcing 42

3.3 The nonlinearized pendulum 45

3.3.1 Amplitude dependent period 45

3.3.2 Phase space revisited 51

3.3.3 An electronic Pendulum 53

3.3.4 Parametric forcing revisited 56

3.4 A pendulum of horror 63

3.5 Exercises 64

4 The Foucault pendulum 674.1 What is a Foucault pendulum? 67

4.2 Frames of reference 71

4.3 Public physics 74

4.4 A quantitative approach 75

4.4.1 Starting the pendulum 78

4.5 A darker side 85

4.6 Toward a better Foucault pendulum 86

4.7 A final note 89

4.8 Exercises 91

5 The torsion pendulum 935.1 Elasticity of the fiber 93

5.2 Statics and dynamics 95

5.2.1 Free oscillations without external forces 96

5.2.2 Free oscillations with external forces 98

5.2.3 Damping 98

5.3 Two historical achievements 99

5.3.1 Coulomb and the electrostatic force 99

5.3.2 Cavendish and the gravitational force 104

5.3.3 Scaling the apparatus 108

5.4 Modern applications 108

5.4.1 Ballistic galvanometer 108

5.4.2 Universal gravitational constant 110

5.4.3 Universality of free fall: Equivalence of

gravitational and inertial mass 113

5.4.4 Viscosity measurements and granular media 117

5.5 Exercises 119

6 The chaotic pendulum 1216.1 Introduction and history 121

6.2 The dimensionless equation of motion 125

6.3 Geometric representations 126

6.3.1 Time series, phase portraits, and Poincare sections 127

6.3.2 Spectral analysis 130

6.3.3 Bifurcation diagrams 133

6.4 Characterization of chaos 135

6.4.1 Fractals 135

6.4.2 Lyapunov exponents 138

6.4.3 Dynamics, Lyapunov exponents, and

fractal dimension 142

6.4.4 Information and prediction 144

6.4.5 Inverting chaos 147

6.5 Exercises 150

7 Coupled pendulums 1537.1 Introduction 153

7.2 Chaotic coupled pendulums 161

7.2.1 Two-state model (all or nothing) 164

7.2.2 Other models 168

7.3 Applications 170

7.3.1 Synchronization machine 170

7.3.2 Secure communication 173

7.3.3 Control of the chaotic pendulum 176

7.3.4 A final weirdness 183

7.4 Exercises 185

Contentsx

8 The quantum pendulum 1898.1 A little knowledge might be better than none 189

8.2 The linearized quantum pendulum 192

8.3 Where is the pendulum?uncertainty 196

8.4 The nonlinear quantum pendulum 200

8.5 Mathieu equation 201

8.6 Microscopic pendulums 203

8.6.1 Ethanealmost free 204

8.6.2 Potassium hexachloroplatinatealmost never free 206

8.7 The macroscopic quantum pendulum and

phase space 208

8.8 Exercises 209

9 Superconductivity and the pendulum 2119.1 Superconductivity 211

9.2 The flux quantum 214

9.3 Tunneling 215

9.4 The Josephson effect 216

9.5 Josephson junctions and pendulums 220

9.5.1 Single junction: RSJC model 220

9.5.2 Single junction in a superconducting loop 224

9.5.3 Two junctions in a superconducting loop 226

9.5.4 Coupled josephson junctions 228

9.6 Remarks 230

9.7 Exercises 230

10 The pendulum clock 23310.1 Clocks before the pendulum 233

10.2 Development of the pendulum clock 235

10.2.1 Galileo (15641642) 235

10.2.2 Huygens (16291695) 235

10.2.3 The seconds pendulum and the meter:

An historical note 244

10.2.4 Escapements 246

10.2.5 Temperature compensation 249

10.2.6 The most accurate pendulum clock ever made 252

10.3 Reflections 255

10.4 Exercises 255

A Pendulum Q 258A.1 Free pendulum 258

A.2 Resonance 259

A.3 Some numbers from the real world 261

B The inverted pendulum 263

Contents xi

C The double pendulum 267

D The cradle pendulum 270

E The Longnow clock 273

F The Blackburn pendulum 275

Bibliography 276Index 286

Contentsxii

Introduction

The pendulum is a familiar object. Its most common appearance is inold-fashioned clocks that, even in this day of quartz timepieces and atomicclocks, remain quite popular. Much of the pendulums fascination comesfrom the well known regularity of its swing and thus its link to the fun-damental natural force of gravity. Older students of music are very familiarwith the adjustable regularity of that inverted ticking pendulum known asa metronome. The pendulums inuence has extended even to the artswhere it appears as the title of at least one work of ctionUmberto EcosFourcaults Pendulum, in the title of an award winning Belgian lmMrs. Foucaults Pendulum, and as the object of terror in Edgar Allen Poes1842 short story The Pit and the Pendulum.The history of the physics of the pendulum stretches back to the early

moments of modern science itself. We might begin with the story, perhapsapocryphal, of Galileos observation of the swinging chandeliers in thecathedral at Pisa. By using his own heart rate as a clock, Galileo pre-sumably made the quantitative observation that, for a given pendulum, thetime or period of a swing was independent of the amplitude of the pen-dulums displacement. Like many other seminal observations in science,this one was only an approximation of reality. Yet it had the main ingre-dients of the scientic enterprise; observation, analysis, and conclusion.Galileo was one of the rst of the modern scientists, and the pendulumwasamong the rst objects of scientic enquiry.Chapters 2 and 3 describe much of the basic physics of the pendulum,

introducing the pendulums equation of motion and exploring the impli-cations of its solution. We describe the concepts of period, frequency,resonance, conservation of energy as well as some basic tools in dynamics,including phase space and Fourier spectra.Much of the initial treatmentChapter 2approximates the motion of the pendulum to the case of smallamplitude oscillation; the so-called linearization of the pendulums grav-itational restoring force. Linearization allows for a simpler mathematicaltreatment and readily connects the pendulum to other simple oscillatorssuch as the idealized spring or the oscillations of certain simple electricalcircuits.For almost two centuries geoscientists used the small amplitude, lin-

earized pendulum, in many forms to determine the acceleration due to

1

gravity, g, at diverse geographical locations. More rened studies led to abetter understanding of the earths density near geological formations. Thevariations in the local gravitational eld imply, among other things, thatthe earth has a slightly nonspherical shape. As early as 1672, the Frenchastronomer Jean Richer observed that a pendulum clock at the equatorwould only keep correct time if the pendulum were shortened as comparedto its length in Paris. From this empirical fact, theDutch physicist Huygensmade some early (but incorrect) deductions about the earths shape. On theother hand, the nineteenth century Russian scientist, Sawitch timed apendulum at twelve different stations and computed the earths shapedistortion from spherical as one part in about 300a number close to thepresently accepted value. During the period from the early 1800s up intothe early twentieth century, many local measurements of the accelerationdue to gravity were made with pendulum-like devices. The challenge ofmaking these difcult measurements and drawing appropriate conclu-sions captured the interest of many workers such as Sir George Airy andOliver Heaviside, who are more often known for their scientic achieve-ments in other areas.Chapter 3 continues the discussion rst by adding the physical effects

of damping and forcing to the linearized pendulum and then by a consid-eration of the full nonlinear pendulum, which is important for largeamplitude motion. Furthermore, real pendulums do not just keep goingforever, because in this world of increasing entropy, motion is dissipated.These dissipative effects must be included as must the compensatingaddition of energy that keeps the pendulum moving in spite of dissipation.The playground swing is a common yet surprisingly interesting example.A child can pump the swing herself using either sitting or standing tech-niques. Alternatively, she can prevail upon a friend to push the swingwith aperiodic pulse. Generally the pulse resonates with the natural motion of theswing, but interesting phenomena occur when forcing is done at an off-resonant frequency. Analysis of these possibilities involves a variety ofmechanical considerations including, changing center of mass, parametricpumping, conservation of angular momentum, and so forth. Another,more exotic, example is provided by the large amplitudemotion of the hugeincense pendulum in the cathedral of Santiago de Compostela, Spain. Foralmost a thousand years, centuries before Galileos pendular experiments,pilgrims have worshiped there to the accompanying swishing sound of theincense pendulum as it traverses a path across the transept with an angularamplitude of over eighty degrees. Finally, the chapter ends with a con-sideration of the most famous literary use of the pendulum; Edgar AllanPoes nightmarish story The Pit and the Pendulum. Does Poe, the non-scientist, provide enough details for a physical analysis? Chapter 3 suggestssome answers.Chapter 4 connects the pendulum to the rotational motion of the earth.

From the early nineteenth century, it was supposed that the earths rota-tion on its axis should be amenable to observation. By that time, classicalmechanics was a developed and mature mathematical science. Mechanics

Introduction2

predicted that additional noninertial forces, centrifugal and coriolis forces,would arise in the description ofmotion as it appeared from an accelerating(rotating) frame of reference such as the earth. Coriolis forcecausing anapparent sideways displacement in the motion of an objectas seen by anearthbound observer, would be a dramatic demonstration of the earthsrotation. Yet the calculated effect was small.In 1851, Leon Foucault demonstrated Coriolis force with a very large

pendulum hung from the dome of the Pantheon in Paris (Tobin andPippard 1994). His pendulum oscillated very slowly and with each oscil-lation the plane of oscillation rotated very slightly.With the pendulum, thecoriolis force was demonstrated in a cumulative fashion. While the pen-dulum gradually ran down and needed to be restarted every 5 or 6 hours, itsplane of oscillation rotated by about 60 or 70 degrees in that time. Theplane rotated through a full circle in about 30 hours. In actuality, the planeof oscillation did not rotate; the earth rotated under the pendulum. If thependulum had been located at the North or South poles, the full rotationwould occur in 24 hours, whereas a pendulum located at the equator wouldnot appear to rotate at all. Foucaults demonstration was very dramaticand immediately captured the popular imagination. Even LouisNapoleon,the president of France, used his inuence to hasten the construction of thePantheon version. Foucaults work was also immediately and widely dis-cussed in the scientic literature (Wheatstone 1851).The large size of the original Foucault demonstration pendulummasked

some important secondary effects that became the subject of muchexperimental and theoretical work. As late as the 1990s the scientic lit-erature shows that efforts are still being made to devise apparatus thatcontrols these spurious effects (Crane 1995).Foucaults pendulum demonstrates the rotation of the earth. But more

than this, its behavior also has implications for the nature of gravity in theuniverse, and it has been suggested that a very good pendulum might pro-vide a test of Einsteins general theory of relativity (Braginsky et al. 1984).Chapter 5 focuses on the torsion pendulum, where an extended rigid

mass is suspended from a exible ber or cable that allows the mass tooscillate in a horizontal plane. The restoring force is now provided by theelastic properties of the suspending ber rather than gravity. While thetorsion pendulum is intrinsically interesting, its importance in the historyof physics lies in its repeated use in various forms to determine the universalgravitational constant, G. The torsion pendulum acquired this role whenCavendish, in 1789, measured the effect on a torsion pendulum of largemasses placed near the pendulum bob. Since that time a whole stream ofmeasurements with similar devices have provided improved estimates ofthis universal constant. In fact, the search for an accurate value of Gcontinues into the third millennium. New results were described at theAmerican Physical Society meeting in April, 2000 held in Long Beach,California, that reduce the error in G to about 0.0014%. This new resultwas obtained with apparatus based upon the torsion pendulum, not unlikethe original Cavendish device. The value of the universal gravitational

Introduction 3

constant and possible variations in that constant over time and space arefundamental to the understanding of cosmologyour global view of theuniverse.The next part of our story has its origins in a quiet revolution that occurred

in the eld of mathematics toward the end of the nineteenth century, arevolution whose implications would not be widely appreciated for another80 years. It arose from asking an apparently simple question: Is the solarsystem stable? That is, will the planets of the solar system continue to orbitthe sun in predictable, regular orbits for the calculable future? With others,the French mathematician and astronomer, Henri Poincare tried to answerthe question denitively. Prizes were offered and panels of judges pouredover the lengthy treatises (Barrow 1997). Yet the important point here is notthe answer, but that in the search for the answer, Poincare discovered a newtype of mathematics. He developed a qualitative theory of differentialequations, and found a pictorial or geometric way to view the solutions incases for which there were no analytic solutions. What makes this theoryrevolutionary is that Poincare found certain solutions or orbits for somenonlinear equations that were quite irregular. The universe was not a simpleperiodic or even quasi-periodic (several frequencies) place as had beenassumed previously. The oft-quoted words of Poincare tell the story,

it may happen that small differences in the initial conditions produce very greatones in the nal phenomena. A small error in the former will produce an enormouserror in the latter. Prediction becomes impossible, and we have the fortuitousphenomenon.1

Fortuitous or random-appearing behavior was not expected and, if itdid occur, it was typically ignored as anomalous or too complex to bemodeled. Thus was born the science that eventually came to be known aschaos, the name much later coined by Yorke and Li of the University ofMaryland.The eld of chaos would have never emerged without another, much

later revolutionthe computer revolution. The birth of a full-scale scienceof chaos coincided with the application of computers to these special typesof equations. In 1963 Edward Lorenz of the Massachusetts Institute ofTechnology was the rst to observe (Lorenz 1963) the chaotic power ofnonlinear effects in a simple model of meteorological convectionow ofan air mass due to heating from below. With the publication of Lorenzswork a ood of scientic activity in chaos ensued. Thousands of scienticarticles appeared in the existing physics and mathematics journals, andnew, often multidisciplinary, journals appeared that were especiallydevoted to nonlinear dynamics and chaos. Chaos was found to be ubi-quitous. Chaos became a new paradigm, a new world view.Many of the original and archetypical systems of equations or models

found in the literature of chaos are valued more for their mathematicalproperties than for their obvious correspondence with physically realizablesystems. However, as one of the simplest physical nonlinear systems, the

1 See (Poincare 1913, p. 397).

Introduction4

pendulum is a natural and rare candidate for practical study. It is modeledquite accurately with relatively simple equations, and a variety of actualphysical pendulums have been constructed that correspond very well totheir model equations. Therefore, the chaotic classical pendulum hasbecome an object of much interest, and quantitative analysis is feasiblewith the aid of computers. Many congurations of the chaotic pendulumhave been studied. Examples include the torsion pendulum, the invertedpendulum, and the parametric pendulum. Special electronic circuits havebeen developed whose behavior exactly mimics pendular motion.Intrinsic to the study of chaotic dynamics is the intriguing mathematical

connection with the unusual geometry of fractals. Fractal structure seemsto be ubiquitous in nature and one wonders if the underlying mechanismsare universally chaotic, in some senseunstable but nevertheless con-strained in ways that are productive of the rich complexity that we observein, for example, biology and astronomy. The pendulum is a wonderfulexample of chaotic behavior as it exhibits all the complex properties ofchaos while being itself a fully realizable physical system. Chapter 6describes many aspects of the chaotic pendulum.Chapter 7 explores the effects of coupling pendulums together. As with

the single pendulum, the origins of coupled pendulums reach back to thegolden age of physics. Three hundred years ago, Christiaan Huygensobserved the phenomenon of synchronization of two clocks attached to acommon beam. The slight coupling of their motions through the mediumof the beamwas sufcient to cause synchronization. That is, after an initialperiod in which the pendulums were randomly out of phase, they graduallyarrived at a state of perfectly matched (but opposite) motions. In anothervenue, synchronization of the ashes of swarms of certain reies has beendocumented. While that phenomenon is not explicitly physical in origin,some very interesting mathematical analysis and experiments have beendone in this context (Strogatz 1994). Similarly chaotic pendulums, bothin numerical simulation and in reality, have been shown to exhibit syn-chronization. As is true with many synchronized chaotic pairs, one pen-dulum can be made to dominate over another pendulum. Surprisingly,such a master and slave relationship can form the basis for a system ofsomewhat secure communications. Again, the pendulum is an obviouschoice for study because of its simplicity. Real pendulums can be coupledtogether with springs or magnets (Baker et al. 1998). This story continuestoday as scientists consider the fundamental notion of what it means forphysical systems to be synchronized and ask the question, How syn-chronized is synchronized?During its long history the pendulum has been an important exemplar

through several paradigm shifts in physical theory. Possibly the mostprofound of these scientic discontinuities is the quantum revolution ofPlanck, Einstein, Bohr, Schrodinger, Heisenberg, and Born in the rstquarter of the twentieth century. It led scientists to see that a whole newmechanics must be applied to the world of the very small; atoms, electrons,and so forth. Much has been written on the quantum revolution, but its

Introduction 5

effect on that simple device, the pendulum, is not perhaps widely known.Many classical mechanical systems have interesting and fascinatingly dif-ferent behaviors when considered as quantum systems. We might inquireas to what happens when a pendulum is scaled down to atomic dimensions.What are the consequences of pendulum quantization? For the pendu-lum with no damping and no forcing, the process of quantization is rela-tively straightforward and proceeds according to standard rules as shownin Chapter 7. One of the pioneering researchers in quantum mechanics,Frank Condon, produced the seminal paper on the quantum pendulum in1928 just a couple of years after the new physics was made broadlyavailable in the physics literature. We learn that the pendulum, like otherconned systems, is only allowed to exist with certain xed energies. Just asthe discovery of discrete frequency lines in atomic spectra ultimately vin-dicated the quantum mechanical prediction of discrete atomic energies, soalso does the quantum simple pendulum exhibit a similar discrete energyspectrum.Does the notion of a quantum pendulum have a basis in physical reality?

We nd it difcult to imagine that matter is composed of tiny pendulums.Yet surprisingly, there are interactions at the molecular level that have thesame mathematical form as the pendulum. One example is motion ofmolecular complexes in the form of hindered rotations. We will describethe temperature dependence of hindered rotations and show that the roomtemperature dynamics of such complexes depends heavily on the particularatomic arrangement.As a further complication, many researchers have asked if quantum

mechanics, with its inherent uncertainties, washes awaymany of the effectsof classical chaotic dynamicsdescribed in the previous chapter. Theclassical unstable orbits of chaotic systems diverge rapidly from each otheras Poincare rst predicted, and yet this kiss and run quality could besmeared out by the fact that specic orbits are not well dened in quantumphysics. In classical physics, we presume to know the locations and speedof the pendulum bob at all times. In quantum physics, our knowledge ofthe pendulums state is only probabilistic. The quantized, butmacroscopic,gravity driven pendulum provides further material for this debate.In one of natures surprising coincidences, quantum physics does present

us with one very clear analogy with the classical, forced pendulum; namely,the Josephson junction. The Josephson junction is a superconductingquantum mechanical device for which the classical pendulum is an exactmathematical analogue. The junction consists of a pair of superconductorsseparated by an extremely thin insulator (a sort of superconducting diode).Josephson junctions are very useful as ultra-fast switching devices and inhigh sensitivity magnetometers. Because of their analogy with the pen-dulum, all the work done with the pendulum in the realm of control andsynchronization of chaos can be usefully applied to the Josephson junc-tion. And so ends the ninth chapter.For the tenth chapter, we return to the sixteenth century and Galileo

to consider the role of the pendulum in time keeping. Galileo was the rst

Introduction6

to design (although never build) a working pendulum clock. He is alsoreputed to have built, in 1602, a special medical pendulum whose length(and therefore period) could be adjusted to match the heart rate of apatient. This measurement of heart rate would then aid in the diagnosticprocess. The medical practitioner would nd various diseases listed atappropriate locations along the length of the pendulum.Galileowas keenly aware of the need for an accurate chronometer for the

measurement of longitude at sea. Portugal, Spain, Holland, and Englandhad substantial investments in accurate ocean navigation. Realizing theeconomic benets of accurate navigation, governments and scienticsocieties offered nancial prizes for a workable solution. Latitude wasrelatively easy to measure, but the determination of longitude requiredeither an accurate clock or the use of very precise astronomical measure-ments and calculations. In Galileos time neither method was feasible.While the mechanical clock was invented in the early fourteenth centuryand the pendulum was conceived as a possible regulator by Leonardo daVinci and the Florentine clockmaker Lorenzo della Volpaia, these ideaswere not combined successfully. Galileos contribution to clock design wasan improved method of linking the pendulum to the clock.The earliest practicable version of a clock based upon Galileos design

was constructed by his son, but in the meantime in 1657, ChristiaanHuygens became the rst to build and patent a successful pendulum clock(Huygens 1986). Although much controversy developed over how muchHuygens knew of Galileos design, Huygens is generally credited withdeveloping the clock independently. There is also a felicitous connectionbetweenHuygens invention of amethod to keep the regulating pendulumsperiod independent of amplitude, and the mathematics of the cycloid, aconnection that we discuss analytically. The longitude problem was ulti-mately solved (Sobel 1996) by the Harrison chronometer, built with aspring regulator, but the pendulum clock survives today as a beautiful andaccurate timekeeper.Pendulum clocks exemplify important physical concepts. The clock

needs to have some method of transferring energy to the pendulum tomaintain its oscillation. There also needs to be a method whereby thependulum regulates the motion of the clock. These two requirements areencompassed in one remarkable mechanism called the escapement. Theescapement is a marvelous invention in that it makes the pendulum clockone of the rst examples of an automaton with self-regulating feedback.Chapter 10 concludes with a brief look at some of the worlds mostinteresting pendulum clocks.Finally, there are interesting congurations and applications of the

pendulum that do not t neatly into the books structure. Therefore weinclude descriptions of some of these pendulums as separate notes inAppendices AF.

Introduction 7

Pendulums somewhatsimple

There are many kinds of pendulums. In this chapter, however, we intro-duce a simpliedmodel; the small amplitude, linearized pendulum. For thepresent, we ignore friction and in doing so obviate the need for energizingthe pendulum through some forcing mechanism. Our initial discussion willtherefore assume that the pendulums swing is relatively small; and thisapproximation allows us to linearize the equations and readily determinethe motion through solution of simplied model equations. We begin witha little history.

2.1 The beginning

Probably no one knows when pendulums rst impinged upon the humanconsciousness. Undoubtedly they were objects of interest and decorationafter humankind learnt to attend routinely to more basic needs. We oftenassociate the rst scientic observations of the pendulum with GalileoGalilei (15541642; Fig. 2.1).According to the usual story (perhaps apocryphal), Galileo, in the

cathedral at Pisa, Fig. 2.2 observed a lamplighter push one of the swayingpendular chandeliers. His earliest biographer Viviani suggests that Galileothen timed the swings with his pulse and concluded that, even as theamplitude of the swings diminished, the time of each swing was constant.This is the origin of Galileos apparent discovery of the approximate iso-chronism of the pendulums motion. According to Viviani these obser-vations were made in 1583, but the Galileo scholar Stillman Drake (Drake1978) tells us that guides at the cathedral refer visitors to a certain lampwhich they describe as Galileos lamp, a lamp that was not actuallyinstalled until late in 1587. However, there were undoubtedly earlierswaying lamps. Drake surmises that Galileo actually came to the insightabout isochronism in connection with his fathers musical instruments andthen later, perhaps 1588, associated isochronism with his earlier pendulumobservations in the cathedral. However, Galileo did make systematicobservations of pendulums in 1602. These observations conrmed onlyapproximately his earlier conclusion of isochronism of swings of differingamplitude. Erlichson (1999) has argued that, despite the nontrivialempirical evidence to the contrary, Galileo clung to his earlier conclusion,

2

Fig. 2.1Portrait of Galileo. #Bettmann/Corbis/

Magma.

Fig. 2.2Cathedral at Pisa. The thin vertical wire

indicates a hanging chandelier.

in part, because he believed that the universe had been ordered so thatmotion would be simple and that there was no reason for the longer pathto take a longer time than the shorter path. While Galileos most famousconclusion about the pendulum has only partial legitimacy, its importanceresides (a) in it being the rst known scientic deduction about thependulum, and (b) in the fact that the insight of approximate isochronismis part of the opus of a very famous seminal character in the history ofphysical science. In these circumstances, the pendulum begins its history asa signicant model in physical science and, as we will see, continues tojustify its importance in science and technology during the succeedingcenturies.

2.2 The simple pendulum

The simple pendulum is an idealization of a real pendulum. It consists of apoint mass, m, attached to an innitely light rigid rod of length l that isitself attached to a frictionless pivot point. See Fig. 2.3. If displaced from itsvertical equilibrium position, this idealized pendulum will oscillate with aconstant amplitude forever. There is no damping of the motion fromfriction at the pivot or from air molecules impinging on the rod. Newtonssecond law, mass times acceleration equals force, provides the equation ofmotion:

mld2

dt2 mg sin , (2:1)

where is the angular displacement of the pendulum from the verticalposition and g is the acceleration due to gravity. Equation (2.1) may besimplied if we assume that amplitude of oscillation is small and thatsin . We use this linearization approximation throughout this chapter.The modied equation of motion is

d2

dt2 g

l 0: (2:2)

The solution to Eq. (2.2) may be written as

0 sin (!t 0), (2:3)where 0 is the angular amplitude of the swing,

! g

l

r(2:4)

is the angular frequency, and 0 is the initial phase angle whose valuedepends on how the pendulum was startedits initial conditions. Theperiod of the motion, in this linearized approximation, is given by

T 2l

g

s, (2:5)

u

m

Fig. 2.3The simple pendulum with a point

mass bob.

The simple pendulum 9

which is a constant for a given pendulum, and therefore lends support toGalileos conclusion of isochronism.The dependence of the period on the geometry of the pendulum and the

strength of gravity has very interesting consequences whichwewill explore.But for the moment we consider further some of the mathematical rela-tionships. Figure 2.4 shows the angular displacement 0 sin (!t 0)and the angular velocity _ 0! cos (!t 0), respectively, as functions oftime. We refer to such graphs as time series. The displacement and velocityare 90 degrees out of phase with each other and therefore when onequantity has a maximum absolute value the other quantity is zero. Forexample, at the bottom of its motion the pendulum has no angular dis-placement yet its velocity is greatest.The relationship between angle and velocity may be represented

graphically with a phase plane diagram. In Fig. 2.5 angle is plotted on thehorizontal axis and angular velocity is plotted on the vertical axis. As timegoes on, a point on the graph travels around the elliptically shaped curve.In effect, the equations for angle and angular velocity are considered tobe parametric equations for which the parameter is proportional to time.Then the orbit of the phase trajectory is the ellipse

2

20

_2

(!0)2 1: (2:6)

Since the motion has no friction nor any forcing, energy is conserved onthis phase trajectory. Therefore the sum of the kinetic and potentialenergies at any time can be shown to be constant as follows. In the line-arized approximation,

E 12ml2 _2 1

2mgl2 (2:7)

and, using Eqs. (2.3) and (2.4), we nd that

E 12mgl20 , (2:8)

which is the energy at maximum displacement.The phase plane is a useful tool for the display of the dynamical prop-

erties of many physical systems. The linearized pendulum is probably oneof the simplest such systems but even here the phase plane graphicis helpful. For example, Eq. (2.6) shows that the axes of the ellipse inFig. 2.5 are determined by the amplitude and therefore the energy ofthe motion. A pendulum of smaller energy than that shown would exhibitan ellipse that sits inside the ellipse of the pendulum of higher energy.See Fig. 2.6. Furthermore the two ellipses would never intersect becausesuch intersection implies that a pendulum can jump from one energy toanother without the agency of additional energy input. This result leadsto a more general conclusion called the no-crossing theorem; namely, thatorbits in phase space never cross. See Fig. 2.7.

Time

du/dt

u

Fig. 2.4Time series for the angular displacement

and the angular velocity, _.

u.

u

Fig. 2.5Phase plane diagram. As time increases

the phase point travels around the

ellipse.

u

u.

Fig. 2.6Phase orbits for pendulums with

different energies, E1 and E2.

Pendulums somewhat simple10

Why should this be so? Every orbit is the result of a deterministicequation of motion. Determinism implies that the orbit is well dened andthat there would be no circumstance in which a well determined particlewould arrive at some sort of ambiguous junction point where its pathwould be in doubt. (Later in the book we will see apparent crossing pointsbut these false crossings are the result of the system arriving at the samephase coordinates at different times.)We introduce one last result about orbits in the phase plane. In Fig. 2.6

there are phase trajectories for two pendulums of different energy. Nowthink of a large collection of pendulums with energies that are between thetwo trajectories such that they have very similar, but not identical, anglesand velocities. This cluster of pendulums is represented by a set of manyphase points such that they appear in the diagram as an approximately solidblock between the original two trajectories. As the group of pendulumsexecutes their individual motions the set of phase points will move betweenthe two ellipses in such a way that the area dened by the boundaries of theset of points is preserved. This preservation of phase area, known asLiouvilles theorem (after Joseph Liouville (18091882)) is a consequenceof the conservation of energy property for each pendulum. In the nextchapter we will demonstrate how such areas decrease when energy is lost inthe pendulums. But for now let us show how phase area conservation istrue for the very simple case when 0 1, 0, and ! 1. In this specialcase, the ellipses becomes circles since the axes are now equal. See Fig. 2.8.A block of points between the circles is bounded by a small polar angleinterval, in the phase space, that is proportional to time. Each point inthis block rotates at the same rate as the motion of its correspondingpendulum progresses. Therefore, after a certain time, all points in theoriginal block have rotated, by the same polar angle, to new positionsagain bounded by the two circles. Clearly, the size of the block has notchanged, as we predicted.The motion of the pendulum is an obvious demonstration of the

alternating transformation of kinetic energy into potential energy andthe reverse. This phenomenon is ubiquitous in physical systems and isknown as resonance. The pendulum resonates between the two states(Miles 1988b). Electrical circuits in televisions and other electronic devicesresonate. The terms resonate and resonance may also refer to a sympathybetween two or more physical systems, but for now we simply think ofresonance as the periodic swapping of energy between two possibleformats.We conclude this section with the introduction of one more mathe-

matical device. Its use for the simple pendulum is hardly necessary but itwillbe increasingly important for other parts of the book. Almost two hundredyears ago, the French mathematician Jean Baptiste Fourier (17681830)showed that periodic motion, whether that of a simple sine wave like ourpendulum, or more complex forms such as the triangular wave thatcharacterizes the horizontal sweep on a television tube, are simple linearsums of sine and cosine waves now known as Fourier Series. That is, let f (t)

?

?

Fig. 2.7If two orbits in phase space intersect,

then it is uncertain which orbit takes

which path from the intersection. This

uncertainty violates the deterministic

basis of classical mechanics.

u

a

u.

Fig. 2.8Preservation of area for conservative

systems. A block of phase points keeps

its same area as time advances.

The simple pendulum 11

be a periodic function such that f (t) f (t (2)=!0), where T (2)=!0is the basic periodicity of the motion. Then Fouriers theorem says that thisfunction can be expanded as

f (t) X1n1

bn cos n!0tX1n1

cn sin n!0t d, (2:9)

where the coefcients bn and cn give the strength of the respective cosineand sine components of the function and d is constant. The coefcientsare determined by integrating f (t) over the fundamental period, T. Theappropriate formulas are

d 1T

Z T=2T=2

f(t) dt, bn 1T

Z T=2T=2

f(t) cos n!0t dt,

cn 1T

Z T=2T=2

f(t) sin n!0t dt:

(2:10)

These Fourier coefcients are sometimes portrayed crudely on stereoequipment as dancing bars in a dynamic bar chart that is meant to portraythe strength of the music in various frequency bands.The use of complex numbers allows Fourier series to be represented

more compactly. Then Eqs. (2.9) and (2.10) become

f(t) Xn1n1

anein!0t, where an !0

2

Z =!0=!0

f(t)ein!0tdt: (2:11)

Example 1 Consider the time series known as the sawtooth, f(t) t whenT2 < t < T2, with the pattern repeated every period, T. Using Eq. (2.11) itcan be shown that

an 0 for n 0,an 1in!0 for n odd integer, and

an 1in!0 for n even integer:Through substitution and appropriate algebraic manipulation we obtain

the nal result:

f(t) 2!0

sin!ot 12sin 2!0t 1

3sin 3!0t

: (2:12)

The original function and the rst three frequency components are shown inFigs. 2.9 and 2.10.

The time variation of the motion of the linearized version of the simplependulum is just that of a single sine or cosine wave and therefore onefrequency, the resonant frequency !0 is present in that motion. Obviously,the machinery of the Fourier series is unnecessary to deduce that result.

0.0 0.5 1.0 1.5 2.0 2.5Time

1

0

1

Am

plitu

de

FirstSecondThirdTotal

Fig. 2.9The rst three Fourier components of

the sawtooth wave. The sum of these

three components gives an

approximation to the sawtooth shape.

1.2

0.8

0.4

0.0

0.4

0.8

1.2a1

a2

a3

a4

a5

a6

Fig. 2.10The amplitudes of several Fourier

components for the sawtoothwaveform.

Pendulums somewhat simple12

However, we now have it available as a tool for more complex periodicphenomena.Fourier, like other contemporary French mathematicians, 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 arrestedalthough soon released. Later when Napoleon went to Egypt, Fourieraccompanied the expedition and coauthored a massive work on everypossible detail of Egyptian life,Description de lEgypt. This is multivolumework included nine volumes of text and twelve volumes of illustrations.During that same campaign, one of Napoleons engineers uncovered theRosetta Stone, so-named for being found near the Rosetta branch of theNile river in 1899. The signicance of this nd was that it led to anunderstanding of ancient Egyptian Hieroglyphics. The stone, was inscri-bed with the same text in three different languages, Greek, demoticEgyptian, andHieroglyphics. OnlyGreek was understood, but the size andthe juxtaposition of the texts allowed for the eventual understanding ofHieroglyphics and the ability to learn much about ancient Egypt. In 1801,the victorious British, realizing the signicance of the Rosetta stone, took itto the British Museum in London where it remains on display and is apopular artifact. Much later, the writings from the Rosetta stone becomethe basis for translating the hieroglyphics on the Rhind Papyrus and theGolenischev Papyrus; these two papyri provide much of our knowledge ofearly Egyptian mathematics. The French Egyptologist Jean Champollion(17901832) who did much of the work in the translation of Hieroglyphicsis 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 tabletswhich he showed to the boy. Fourier explained that no one could readthem. Apparently Champollion replied that he would read them when hewas oldera prediction that he later fullled during his brilliant career ofscholarship (Burton 1999). After his Egyptian adventures, Fourier con-centrated on his mathematical researches. His 1807 paper on the idea thatfunctions could be expanded in trigonometric series was not well receivedby the Academy of Sciences of Paris because his presentation was notconsidered sufciently rigorous and because of some professional jealousyon the part of other Academicians. But eventually Fourier was accepted asa rst rate mathematician and, in later life, acted a friend and mentor to anew generation of mathematicians (Boyer and Merzbach 1991).We have now developed the basic equations for the linearized,

undamped, undriven, very simple harmonic pendulum. There are anamazing number of applications of even this simple model. Let us reviewsome of them.

2.3 Some analogs of the linearized pendulum

2.3.1 The spring

The linearized pendulum belongs to a class of systems known as harmonicoscillators. Probably the most well known realization of a harmonic

Some analogs of the linearized pendulum 13

oscillator is that of a mass suspended from a spring whose restoring forceis proportional to its stretch. That is

Frestoring kx, (2:13)where k is the spring constant and rate at which the springs responseincreases with stretch, x. This force lawwas discovered byRobertHooke in1660. The equation of motion

md2x

dt2 kx 0 (2:14)

is identical in form to that of the linearized pendulum and therefore itssolution has corresponding properties: single frequency periodic motion,resonance, energy conservation and so forth. A schematic drawing of thespring and a graph of its force law are shown in Fig. 2.11.The functional dependence of the spring force (Eq. (2.13)) can be viewed

more generally. Consider any force law that is derived from a smoothpotentialV(x); that is F(x) dV=dx. The potential may be expanded in apower series about some arbitrary position x0which, for simplicity, we willtake as x0 0. Then the series becomes

V(x) V(0) V 0(0)x 12V 00(0)x2 1

6V 000(0)x3 : (2:15)

The rst term on the right side is constant and, as the reference point of apotential, is typically arbitrary and may be set equal to zero. The second,linear, term containsV 0(0) which is the negative of the force at the referencepoint. Since this reference point is, again typically, chosen to be a point ofstable equilibriumwhere the forces are zero, this second term also vanishes.For the spring, this would be the point where the mass attached to thespring hangs when it is not in motion. Thus, the rst nonvanishing term inthe series is the quadratic term 12V

00(0)x2 and comparison of it with thesprings restoring force (Eq. 2.13) leads to the identication

k V 00(0): (2:16)The spring constant is the second derivative of any smooth potential.

Example 2 TheLennardJones potential is often used to describe the electro-static potential energy between two atoms in a molecule or between twomolecules. Its functional form is displayed in Fig. 2.12 and is given by theequation

V(r) ar12

br6, (2:17)

where a and b are constants appropriate to the particular molecule. Thepositive term describes the repulsion of the atoms when they are too closeand the negative term describes the attraction if the atoms stray too far fromeach other. Hence, the two terms balance at a stable equilibrium point asshown in the gure, req (2ab )

1=6. The second derivative of the potential may

m

k

Slope: kx

F

Fig. 2.11A mass hanging from a spring. The

graph shows the dependence of the

extension of the spring on the force

(weight). The linear relationship is

known as Hookes law.

0 3Radius

0.40.30.20.10.00.10.20.30.4

Pote

ntia

l

req

V(r) =r121

r6

1 2 4

1

Fig. 2.12A typical LennardJones potential curve

that can effectively model, for example,

intermolecular interactions. For this

illustration, a b 1.

Pendulums somewhat simple14

be evaluated at req to yield the spring constant of the equivalent harmonicoscillator,

V 00(req) 18b2

a

b

2a

1=3 k: (2:18)

Knowledge of the molecular bond length provides req and observation of thevibrational spectrumof themoleculewill yieldavalue for thespringconstant,k.With just these two pieces of information, the parameters, a and b of theLennardJones potential may be determined.

The linearized pendulum is therefore equivalent to the spring in that theyboth are simple harmonic oscillators each with a single frequency andtherefore a single spectral component. Occasionally we will refer to apendulums equivalent oscillator or equivalent spring, and by this ter-minology we will mean the linearized version of that pendulum.

2.3.2 Resonant electrical circuit

We say that a function f (t) or operator L(x) is linear if

L(x y) L(x) L(y)L(x) L(x): (2:19)

Examples of linear operators include the derivative and the integral. Butfunctions such as sin x or x2 are nonlinear. Because linear models arerelatively simple, physics and engineering often employ linear mathemat-ics, usually with great effectiveness. Passive electrical circuits, consisting ofresistors, capacitors, and inductors are realistically modeled with lineardifferential equations. A circuit with a single inductorL and capacitorC, isshown in Fig. 2.13. The sum of the voltages measured across each elementof a circuit is equal to the voltage provided to a circuit from some externalsource. In this case, the external voltage is zero and therefore the sum of thevoltages across the elements in the circuit is described by the linear dif-ferential equation

Ld2q

dt2 1Cq 0, (2:20)

where q is the electrical charge on the capacitor. The form of Eq. (2.20) isexactly that of the linearized pendulum and therefore a typical solution is

q q0 sin(!t ), (2:21)where the resonant frequency depends on the circuit elements:

! 1

LC

r: (2:22)

The charge q plays a role analogous to the pendulums angular displace-ment and the current i dq=dt in the circuit is analogous to the pendu-lumsangular velocity,d=dt.All the sameconsiderations, about themotionin phase space, resonance, and energy conservation, that previously held

+

+

L

C

q

Fig. 2.13A simple LC (inductor and capacitor)

circuit.

Some analogs of the linearized pendulum 15

for the linearized pendulum, also apply for this simple electrical circuit. In a(q, i) phase plane, the point moves in an elliptical curve around the origin.The charge and current oscillate out of phase with each other. The capa-citor alternately lls with positive and negative charge. The voltage acrossthe inductor is always balanced by the voltage across the capacitor suchthat the total voltage across the circuit always adds to zero as expressed byEq. (2.20). As with the spring, we will return to this electrical analog withadditional complexity. For now, we turn to some applications and com-plexities of the linearized pendulum.

2.3.3 The pendulum and the earth

From ancient times thinkers have speculated about, theorized upon,calculated, and measured the physical properties of the earth (Bullen1975). About 900bc, the Greek poet Homer suggested that the earthwas a convex dish surrounded by the Oceanus stream. The notion thatthe earth was spherical seems to have made its rst appearance in Greeceat the time of Anaximander (610547bc). Aristotle, the universalist thin-ker, quoted contemporary mathematicians in suggesting that thecircumference of the earth was about 400,000 stadiaone stadium beingabout 600 Greek feet. Mensuration was not a precise science at the timeand the unit of the stadium has been variously estimated as 178.6 meters(olympic stadium), 198.4 m (BabylonianPersian), 186 m (Italian) or212.6 m (PhoenicianEgyptian). Using any of these conversion factorsgives an estimate that is about twice the present measurement of theearths circumference, 4:0086 104 km. Later Greek thinkers somewhatrened the earlier values. Eratosthenes (276194bc), Hipparchus(190125bc), Posidonius (13551bc), and Claudius Ptolemy (ad100161)all worked on the problem. However the Ptolemaic result was too low. Itis rumored that a low estimate of the distance to India, based on thePtolemys result, gave undue encouragement to Christopher Columbus1500 years later.

In China the astronomer monk Yi-Hsing (ad683727) had a largegroup of assistants measure the lengths of shadows cast by the sun andthe altitudes of the pole star on the solstice and equinox days at thirteendifferent locations in China. He then calculated the length L of a degreeof meridian arc (earths circumference/360) as 351.27 li (a unit of the TangDynasty) which, with present day conversion, is about 132 km, an estimatethat is almost 20% too high.The pendulum clock, invented by the Dutch physicist and astronomer

Christiaan Huygens (16291695) and presented on Christmas day, 1657,provided a powerful tool formeasurement of the earths gravitational eld,shape, and density. The daily rotation of the earth was, by then, anaccepted fact and Huygens, in 1673, provided a theory of centrifugalmotion that required the effective gravitational eld at the equator to beless than that at the poles. Furthermore, the centrifugal effect shouldalso have the effect of fattening the earth at the equator, thereby further

Pendulums somewhat simple16

weakening gravity at the surface near the equator. In 1687 Newtonpublished his universal law of gravity in the Principia. It is the existenceof the relationship between gravity and the length of the pendulum(Eq. (2.5)), established through the work of Galileo and Huygens, thatmakes the pendulum a useful tool for the measurement of the gravitationaleld and therefore a tool to infer the earths shape and density. The rstrecorded use of the pendulum in this context is usually attributed to themeasurements of Jean Richer, the French astronomer, made in 1672 .Richer (16301696) found that a pendulum clock beating out seconds inParis at latitude 49 North lost about 2(1/2) minutes per day near theequator in Cayenne at 5 North and concluded that Cayenne was furtherfrom the center of the earth than was Paris. Newton, on hearing of thisresult ten years later by accident at a meeting of the Royal Society, used itto rene his theory of the earths oblateness (Bullen 1975). However,Richers result also helped lead to the eventual demise of the idea of using apendulum clock as a reliable timing standard for the measurement oflongitude (Matthews 2000).A clever bit of theory by Pierre Bougeur (16981758), a French pro-

fessor of hydrography and mathematics, allowed the pendulum to be aninstrument for estimating the earths density, (Bouguer 1749). In 1735Bouguer was sent, by the French Academy of Sciences, to Peru, tomeasurethe length of a meridian arc, L near the equator. (A variety of such mea-surements at different latitudes would help to determine the earths oblate-ness.) But while in Peru he made measurements of the oscillations of apendulum, which in Paris beat out seconds whereas in Quito, (latitude0.25 South) the period was different. His original memoir is a little con-fusing as towhether hemaintained a constant length pendulumorwhether,as his data suggests, he modied the length of the pendulum to keep timewith his pendulum clock that he adjusted daily. At any rate, he used thependulum to measure the gravitational eld. But more than this he mademeasurements of the gravitational eld close to sea level and then on top ofthe Cordilleras mountain range. In this way Bouguer was able to estimatethe relative size of the mean density of the earth.In order to appreciate the cleverness of Bouguers method, we derive his

result. Consider the schematic diagram of the earth with the heightincrement (themountain range) shown in Fig. 2.14. The acceleration due togravity at the surface of the earth is readily shown to be

g0 GMEa2

43Ga, (2:23)

where a is the earths radius, is the mean density, and G is the universalgravitational constant. Now consider the acceleration due to gravity on themountain range. There are two effects. First, the gravitational eld isreduced by the fact that the eld point is further from the center of theearth, and second, the eld is enhanced by the gravitational pull of themountain range. The rst effect is found through a simple ratio usingNewtons law of gravity, but the second effect is a little more involved and

a

As9

s

h

Fig. 2.14The little bump on the earths surface

represents a whole mountain range.

Some analogs of the linearized pendulum 17

requires the use of a fundamental relation in the theory of gravitationalelds; Gauss law. It is expressed mathematically asZ

g n dA 4GZ dV, (2:24)

where g is the acceleration due to gravity (expressed as a vector) and n is theoutward unit normal vector. The integral on the left side is calculated overa closed surface and the integral on the right side is a volume integralthroughout the inside of the closed surface boundary. Essentially the uxor amount of gravitational eld coming out of the surface is proportionalto the mass contained inside the surface. In the diagram, the mountainrange is approximated by a pill box with height h and top and bottomareas A. We suppose that h is much less than any lateral dimension andtherefore assume that the gravitational eld is directed only out of the topand the bottom of the pill box. Then Eq. (2.24) becomes

2gA 4G 0Ah (2:25)or

g 2G 0h, (2:26)where 0 is the density of the mountain range as determined by samplingthe local soil materials. With these equations we can now write an expres-sion for the acceleration due to gravity as measured on top of themountainrange:

g 43G

a3

(a h)2 2G0h: (2:27)

Since h a, the rst term on the right can be approximated using thebinomial expansion and then the ratio of the two measurements of thegravitational eld is found to be

g

g0 1 2h

a 3h2a

0

: (2:28)

The two corrections terms on the right side of the equation are the rst ofseveral corrections that were eventually incorporated into experiments ofthis or similar types. The rst term 2h/a is the so-called free air term and theother term is referred to as the Bouguer term. The point of Eq. (2.28) isthat, with data on the relative accelerations due to gravity, it shouldbe possible to calculate the ratio of the density of themountainousmaterialto that of the rest of the earth. Bouguers pendulum measurements con-vinced him that the earths mean density was about four times that ofthe mountains, a ratio not too different from a modern value of 4.7. InBouguers own words

Thus it is necessary to admit that the earth is much more compact below thanabove, and in the interior than at the surface . . .Those physicists who imagined agreat void in the middle of the earth, and who would have us walk on a kind of verythin crust, can think so no longer. We can make nearly the same objections to

Pendulums somewhat simple18

Woodwards theory of great masses of water in the interior. (page 33 of (Bouguer

1749)).

Bouguers experiment was the rst of many of this type. A commonvariant on the mountain range experiment was to measure the difference ingravitational eld at the top and the bottom of a mine shaft. In this case,the extra structure was not just a mountain range but a spherical shellabove the radius of the earth to the bottom of the shaft. See Fig. 2.15. Anequation similar to Eq. (2.28) holds although the Bouguer term must bemodied as

gtopgbottom

1 2 ha 3 h

a

0

(2:29)

because of the shape of the spherical shell of density 0, and the radius a ismeasured from the center of the earth to bottom of the mine shaft. Coalmines were widely available in England and the seventh astronomer royaland Lucasian professor of mathematics at Cambridge, George Airy (18011892) was one ofmany to attempt this type of experiment. His early effortsin 1826 and 1828 in Cornwall were frustrated by oods and re. But muchlater in 1854 he successfully applied his techniques at a Harton coal-pit inSunderland and obtained a value for the earths density of 6:6 gm=cm3(Bullen 1975, p. 16).Pendulum experiments continued to be improved. Von Sterneck

explored gravitational elds at various depths inside silver mines inBohemia and in 1887 invented a four pendulum device. Two pairs of 1/2second pendulums were placed at right angles. Each pendulum in a givenpair oscillated out of phase with its partner, thereby reducing exure in thesupport structure that ordinarily contributed a surprising amount of errorto measurements. The two mutually perpendicular pairs provided a checkon each other. Von Sternecks values for the mean density of the earthranged from 5.0 to 6.3 gm/cm3. The swing of the pendulums in a pair iscompared with a calibrated 1/2 second pendulum clock by means of anarrangement of lights and mirrors as observed through a telescope.Because they are slightly out of phase, the gravity pendulum and the clockpendulum eventually get out of phase by a whole period. The number ofcounts between such coincidences is observed and used in calculating theprecision of the gravity pendulum period. Accuracies as high as 2 107were claimed for the apparatus.Other types of pendulums have also been used in geological exploration,

but they are based upon pendulums that are more involved than the simplependulum that is the fundamental ingredient of the experiments andequipment described above.

2.3.4 The military pendulum

Since the mid-twentieth century physics has had a strong relationship withthe engineering of military hardware. Yet there are precursors to thismodern connection.BenjaminRobin (17071751), aBritishmathematician

h

a

s9

s

Fig. 2.15A schematic of the earth with a

mine shaft of depth, h.

Some analogs of the linearized pendulum 19

and military engineer gave a giant boost to the modern science of artil-lery with the 1742 publication of his book, New Principles of Gunnery.One of his contributions was a method for determining the muzzle velocityof a projectile; the apparatus is illustrated in Fig. 2.16. (Even today, under-graduate physics majors do an experiment with a version of this methodusing an apparatus known as the Blackwood ballistic pendulumBlackwood was a professor of physics in the early twentieth century atthe University of Pittsburgh.)With a relatively modern apparatus a bullet is red into a pendulum

consisting of a large wooden bob suspended by several ropes. The pro-jectile is trapped in the bob, causing the bob to pull laterally againstthe ropes and therefore rise to some measurable height. See Fig. 2.17.Application of the elementary laws of conservation of energy andmomentum produce the required value of projectile muzzle velocity.Here is the simple analysis. Prior to the moment of collision between the

projectile of massm and the pendulum bob of massM, the projectile has avelocity v. After the collision, the projectile quickly embeds in the bob andimparts a velocityV to the bob.Momentum before and after the collision ispreserved so that

mv (Mm)V: (2:30)After the particle is embedded in the bob, the kinetic energy of the com-bination of projectile and bob thrusts the pendulum outward and upwardto a height h. All the kinetic energy is transformed to potential energy andtherefore

1

2(Mm)V2 (Mm)gh: (2:31)

Mutual solution of Eqs. (2.30) and (2.31) yields the muzzle velocity of theprojectile,

v Mmm

2gh:

p(2:32)

The beauty of this result is that it bypasses the need to have any sort ofmeasure of the energy lost as the projectile is trapped by the pendulum bob.That lost kinetic energy simply produces heat in the pendulum.One wonders if the many students who perform this laboratory

experiment each year are aware that they are replicating early militaryresearch.

2.3.5 Compound pendulum

Themodel of a simple pendulum requires that all mass be concentrated at asingle point. Yet a real pendulum will have some extended mass distribu-tion as indicated in Fig. 2.18. Such a pendulum is called a compoundpendulum. If Ip is the moment of inertia about the pivot point, l is thedistance from the pivot to the center of mass, and m is the mass of the

Fig. 2.16Robins 1742 ballistic pendulum. (From

Taylor (1941) with permission from

Dover).

h

m

Fig. 2.17Schematic diagram of the Blackwood

ballistic pendulum used in

undergraduate laboratories.

Pendulums somewhat simple20

pendulum, then Newtons second law prescribes the following equationof motion:

Ipd2

dt2mgl sin 0, (2:33)

and for small angular displacements we again substitute for sin Thelinearized equation of motion is

Ipd2

dt2mgl 0 (2:34)

with period equal to

T 12

Ip

mgl

s: (2:35)

This expression reverts to that for the simple pendulumwhen all themass isconcentrated at the lowest point.

2.3.6 Katers pendulum

The formulas for the period of the simple pendulum and the compoundpendulum both contain a term for g, the acceleration due to gravity, andtherefore one should be able to time the oscillations of the small amplitudependulum and arrive at an estimate of the local gravitational eld. Yetwithout special effort the results obtained tend to be inaccurate. Forexample, it is often difcult to determine the appropriate length of thependulum as there is ambiguity in the measurement at the pivot or at thebob. At the suggestion of the German astronomer F. W. Bessel (17841847), Captain Henry Kater (17771835) of the British Army invented areversible pendulum in 1817 that signicantly increased the accuracy of themeasurement of g Katers pendulum, shown schematically in Fig. 2.19,consists of a rod with two pivot points whose positions along the rod areadjustable. In principle, the determination of g is made by adjusting thepivot points until the periods of small oscillation about both positionsare equal. In practice, it is difcult to adjust the pivot pointsusually knifeedgesand instead counterweights are attached to the rod and are easilypositioned along the rod until the periods are equal. In this way, the pivotpositions are dened by xed knife edges that provide the possibility ofaccurate measurement. Once the periods are found to be equal andmeasured, the acceleration due to gravity is calculated from the formula

T 2h1 h2

g

s, (2:36)

where h1 and h2 are the respective distances from the pivots to the center ofmass of the pendulum. But more importantly their sum (h1 h2) is easilymeasurable as the distance between the two knife edge pivot points.Equation (2.36) is not obvious and its derivation is of some interest.

Referring to Fig. 2.19, the pendulum, of mass m, may be suspended about

mg

p

u

Fig. 2.18A compound pendulum with an

arbitrary distribution of mass.

mg

h1

h2

p1

p2

u

Fig. 2.19The Kater reversing pendulum.

Some analogs of the linearized pendulum 21

either point P1 or point P2. The distances of these suspension points fromthe center of mass are h1 and h2, respectively. The moments of inertia of thependulum about each of the pivots are denoted as I1 and I2. Therefore thelinearized equations of motion corresponding to the two pivot points are

I1d2

dt2mgh1 0 I2 d

2

dt2mgh2 0: (2:37)

The moments of inertia may be expanded using the parallel axis theoremsuch that

I1 mk2 mh21 I2 mk2 mh22, (2:38)where k is the radius of gyration, an effective radius of the system about thecenter of mass such that mk2 is equal to the moment of inertia about thecenter of mass. Solution of Eq. (2.37) leads to periods of

T1 2k2 h21gh1

sT2 2

k2 h22gh2

s: (2:39)

With a little algebra we see that the periods are equal if

h1h2(h1 h2) k2(h1 h2): (2:40)While it would seem easiest to set h1 equal to h2, realization of this con-dition is difcult to achieve with good accuracy in a physical conguration.Instead, the counterweights are used to establish the other algebraiccondition;

k2 h1h2 (2:41)and therefore

T1 T2 2h1 h2

g

s: (2:42)

Example 3 Consider a Kater pendulum in the form of a rod of length L andmass m. Suppose, not very realistically, that we can arbitrarily position thepivot points rather than achieve equality of swing periods with counterweights. Now let us suppose that the pendulum is swung about a pivot locatedat one end of the rod.We ask ourselves where the other pivot (on the other halfof the rod) could be located that would give an equal swing period. (Thisexample is due to Peters (1999).) The moment of inertia of the rod is(1=12) mL2 about its center. By the parallel axis theorem the moment ofinertia about one end is (1=12) mL2 m(L=2)2 (1=3) mL2. Referringto Eq. (2.35), the period of an oscillation for the pivot located at oneend becomes TA (1=2)

(I=mgL=2)

p (1=2) (2L=3g)p . Let x be thedistance from the center along the other half of the rod where the otherpivot point is located. By the parallel axis theorem, the moment of inertiaabout this point is (1=12) mL2 mx2 so that the period is TB (1=2)((1=12)mL2 mx2)=mgx

p. Setting TATB leads to a quadratic expres-

sion for x with the two roots L/2 and L/6. The root at L/2 is obvious anduninteresting and therefore we choose xL/6. Substitution of this root

Pendulums somewhat simple22

into the equation leads to TB (1=2)((1=12)mL2 mx2)=mgx

p

(1=2)(2L=3g)

p TA as expected. As noted previously, the pivot points cannot be set exactly and some adjustments are required, using small counterweights, in order to obtain equality of periods.

In practice, the lengths in Eq. (2.41) are difcult to predict accuratelyand the experimenter uses a convergence process to arrive at equality ofperiods. The counterweights are moved systematically until equality isachieved.With this type of pendulum theNational Bureau of Standards, in1936, determined the acceleration due to gravity at Washington, DC asg 980:080 0:003 cm=s2 (Daedalon 2000).After its invention, many of the pendulum gravity experiments were

done with theKater reversing pendulum. One of the original pendulums,number 10, constructed by a certain Thomas Jones, rests in the ImperialScience Museum in London. The display card reads as follows.

This pendulumwas taken together withNo. 11, whichwas identical, . . . on a voyagelasting from 18281831. During this time Captain Henry Foster swung it at twelvelocations on the coasts and islands of the South Atlantic. Subsequently it was used

in the Euphrates Expedition, of 18356, then taken to Antarctic by James Rossin 1840.

2.4 Some connections

One of the fascinating aspects of the history of the pendulum is theremarkable number of famous and not-so-famous physical scientists thathave some connection to the pendulum. This phenomenon will come intosharper relief as our story unfolds. We have mentioned a few of thesepeople; here are some others. Marin Mersenne (15881648) , a friar of theorder of Minims in Paris, proposed the use of the pendulum as a timingdevice to Christiaan Huygens thereby inspiring the creation of Huygenspendulum clock. Mersenne is perhaps better known as the inventor ofMersenne numbers. These numbers are generated by the formula

2p 1, (2:43)where p is prime. Most, but not all, of the numbers generated by this for-mula are also prime. Jean Picard (16201682), a professor of astronomy atthe College de France in Paris, introduced the use of pendulum clocks intoobservational astronomy and thereby enhanced the precision of astro-nomical data. Picard is perhaps better known for being the rst to accur-ately measure the meridian distance L and his observations, like Richersobservations were used byNewton in calculating the earths shape. RobertHooke (16351703) well known for the linear law of elasticity, Eq. (2.13),for his invention of the microscope, a host of other inventions, and hiscontroversies with Newton, was one of the rst to suggest, in 1666, thatthe pendulum could be used to measure the acceleration due to gravity.EdmondHalley(16561742) ,astronomerroyal,ofHalleyscometfame,wasanother user of the pendulum. In 1676 Halley sailed to St. Helenas island,the southernmost British possession, located in the south Atlantic, in order

Some connections 23

to make a star catalog for the southern hemisphere. As a friend of Hooke,he was aware ofHookes suggested use of the pendulum tomeasure gravityand did make such measurements while on St. Helena. (While Halley isfamous for having his name applied to the comet, he probably rendered asignicantly more important service to mankind by pressing for and n-ancially supporting the publication of Newtons Principia.) In the nextcentury, Sir Edward Sabine (17881883) , an astronomer with Sir WilliamParry in the search for the northwest passage (through the Arctic oceanacross the north of Canada) spent the years from 1821 to 1825 determiningmeasurements of the gravitational eld along the coasts of North Americaand Africa, and, of course, in the Arctic, with the pendulum.The American philosopher Charles Saunders Peirce (18391914) makes

a surprising appearance in this context. Known for his contributions tologic and philosophy, Peirce rarely held academic position in these bran-ches of learning, but made his living with the US Coast and GeodeticSurvey. Between 1873 and 1886, Pierce conducted pendulum experimentsat a score of stations in Europe andNorth America in order to improve thedetermination the earths ellipticity. However, his relationship with theSurvey administration was fractious, and he resigned in 1891. And nally,in the twentieth century, we note the work of Felix Andries VeningMeinesz (18871966), a Dutch geophysicist who, as part of his Ph.D.(1915) dissertation, devised a pendulum apparatus which, somewhat likeVon Sternecks device, used the concept of pairs of perpendicularlyoriented pendulums swinging out of phase with each other. (See Fig. 2.20).In this way Vening Meinesz eliminated a horizontal acceleration term

due to the vibration of peaty subsoil that seemed to occur in many placeswhere gravity was measured. Vening Meinesz apparatus was also espe-cially tted for measurements on or under water and contained machinerythat compensated for the motion of the sea. Aside from the interruptioncaused by the Second World War, some version of this device was used onsubmarines from 1923 until the late 1950s (Vening 1929).In the next chapter we add some complexity to the pendulum. We

include friction and then compensate for the energy loss with an externalsource of energy. Eventually, we also relax the condition of small ampli-tude motion and therefore the equations of motion become nonlinear,a signicant complication in our discussion. However the small amplitudemotion of the linearized pendulum will predominate in three of thechapters; those on the Foucault pendulum, the torsion pendulum (which iswell modeled as linear), and the pendulum clock. Obviously, the linearizedpendulum is the basis of important applications.

2.5 Exercises1. In a later chapter we discuss the Foucault pendulum that was the rst explicit

demonstration of the rotation of the earth. The original Foucault pendulumwas67 meters in length. Calculate the frequency and period of its motion. The planeof oscillation of the pendulum rotated through a full 360 degrees in 31.88 hours.How many oscillations does the pendulum make in that time?

Fig. 2.20Vening Meinesz pendulum. Four

pendulums arranged in mutually

perpendicular pairs are visible.

(Courtesy of the Society of Exploration

Geophysicists Geoscience center. Photo

#2004 by Bill Underwood.)

Pendulums somewhat simple24

2. In the early days of gravity measurement by pendulum oscillation, a secondspendulum had a length of about 1m. This connection between the meter and thesecond was thought to have some special signicance. What was the actualperiod of the seconds pendulum? From your result how do you think theperiod of the pendulum was initially dened?

3. A particle undergoing uniform acceleration from a standing start at the positionx 0 has the following parametric equations (or time series) for position andvelocity:

v atx 1

2at2:

Determine the equation for its orbit in an x, v phase space and sketch the orbit.4. Consider the phase orbit given by Eq. (2.6). Form the phase space diagram such

that the x-axis is and the y-axis is _/! Then the phase orbit becomes a circle of

radius 0. Note also that 0 cos!t. Therefore the phase point traces out acircular orbit with a polar angle !t. We are now ready to easily prove thatareas in phase space are preserved in time. Proceed as follows. Consider twoboundary orbits in phase space dened by two pendulums of different ampli-tudes (energies), 0(1) and 0(2). These orbits are two concentric circles. Nowimagine a region between these two orbits bounded on the other sides by angles1!t1 and 2!t2 Using polar coordinates calculate the area of this regionand show that for some later times t1t and t2t, the area still only dependsupon the difference, t2 t1. That is, the area is preserved in time, and the systemis conservative. See Fig. 2.21.

5. Find the Fourier series for the periodic function,

f (t) 1 :0 < t < T=2f (t) 1 :T=2 < t < T:

6. The complete restoring force of the pendulum is F mg sin . Variousapproximations may be obtained using a Taylor series expansion in which theexpansion variable is the length along the arc of the pendulums swing, s l.That is

F(s) F(s0) F 0(s0)(s s0) F 00(s0)(s s0)2=2! F 000(s0)(s s0)3=3! where F 0 dF/ds. Express F in terms of s. Let s0 0 and show that the rstnonvanishing term in the expansion is the usual small angle linear approxima-tion, Fmg. Now let s0 l/4, and show that the linear approximation, inthe region of /4, is

F mg2

p 4 1

2p

:

7. Determine equations for the constants a and b in the LennardJones potential,in terms of given values of the molecular spring constant, k and the equilibriumbond length, req Note that the force is zero at r req.

8. Derive Eq. (2.29) for the ratio of densities 0/ where 0 is the density nearthe surface of the earth (above themine shaft), and is the average density of theearth. For this derivation try the following sequence of calculations. First cal-culate g at the bottom of themine shaft usingGauss law, and remember that theearth at a radius above that of the bottom of the shaft contributes nothing tothe gravitational eld. Then useGauss law to calculate the gravitational eld ontop of the earth by dividing the earth into two parts: one at a depth below theshaft with density , and the shell above the bottom of the shaft with density 0.Finally, examine the ratio of gtop/gbottom and use the binomial expansion interms of h/a where needed. Neglect any terms that are more than rst degree inthe ratio h/a.

a=v (t2+ t)

a=v (t1+ t)

a=v (t1)

a=v (t2)

u0(1) u0(2)u

u

Fig. 2.21Figure for problem 4.

Exercises 25

9. Figure 2.22 shows aKater pendulumwith two attachedmasses,M and 2MThepivot points are just inside the ends of the bar (mass m) at a distance fromthe ends. The smaller mass is xed at a distance of from the right pivot point.The larger mass is located a variable distance x from the left point. The point ofthis exercise is to nd the location of the mass 2M such that the pendulum willoscillate with equal period from either pivot point.

(a) Find the center of mass xx of the system in terms of the quantities shownin Fig. 2.22.

(b) Find h1 and h2.(c) Check that h1 h2L 2.(d) Use the condition that h1 h2 to nd the appropriate value of x.

10. For the example in the text, h1L/2 and h2L/6. Using Eq. (2.42) show thatthese values lead to the correct result for th