calle c.i. superstrings and other things

Multiple copying is permitted inaccordance with the terms of licences issued bythe Copyright LicensingAgencyunder the terms of its agreement with the Committee of Vice-Chancellors and Pr

S U P E R S T R I N G S

A G U I D E T O P H Y S I C S

About the Author

Carlos I Calle received his PhD in theoretical nuclearphysics from Ohio University He is a senior researchscientist at NASA KennedySpace Center where heleads the electromagnetic physics research group DrCalle is currentlyworking on the problem of electro-static phenomena on planetarysurfaces, particularly

on Mars and the Moon, developing instrumentationfor future planetaryexploration missions He is theprincipal investigator for the electrostatic studies ofMartian soil and dust and for the electrometer calibra-tion project for the Mars Surveyor mission He is alsoproject manager for the studyof the electrostaticproperties of lunar soil and dust

His earlier research work involved the development

of a theoretical model for a microscopic treatment ofparticle scattering He also introduced one-particleexcitation operators in a separable particle-holeHamiltonian for the calculation of particle excitations

As a professor of physics for many years, he taught thewhole range of college physics courses He has pub-lished over eightyscienti®c papers and been invited

to participate in international scienti®c conferences

He has been the recipient of ten research grants fromNSF, from NASA, and from private foundations

NASA Kennedy Space Center

Institute of Physics Publishing

Bristol and Philadelphia

All rights reserved No part of this publication maybe reproduced,stored in a retrieval system or transmitted in anyform or byanymeans,electronic, mechanical, photocopying, recording or otherwise, withoutthe prior permission of the publisher Multiple copying is permitted inaccordance with the terms of licences issued bythe Copyright LicensingAgencyunder the terms of its agreement with the Committee of Vice-Chancellors and Principals.

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A catalogue record for this book is available from the British Library.ISBN 0 7503 0707 2

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Published byInstitute of Physics Publishing, whollyowned byTheInstitute of Physics, London

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To Dr Luz Marina Calle, Fellow NASA Scientist and Wife

and to our son Daniel

The scienti®c method: learning from our mistakes 7

Frontiers of physics: Very small numbers 19Pioneers of physics: Measuring the circumference of the Earth 20

PART 2: THE LAWS OF

Pioneers of physics: Galileo's method 39

3 T H E L A W S O F M E C H A N I C S :

Pioneers of physics: Galileo's dialog with Aristotle 47Galileo formulates the Law of Inertia 48Physics in our world: The Leaning Tower of Pisa 50Newton's First Law: Law of inertia 52Physics in our world: Car seat belt 54Newton's Second Law: Law of force 56Newton's Third Law: Law of action and reaction 59

Units of work and energy66The concept of energy66Pioneers of Physics: James Prescott Joule (1818±1889) 67

Ef®ciency82Pioneers of physics: The physicists' letters 83

Physics in our world: Automobile ef®ciency 86

Physics in our world: Air bags 90

Elastic and inelastic collisions 94

6 R O T A T I O N A N D T H E U N I V E R S A L L A W

The frontiers of physics: CD-ROM drives 101

Physics in our world: Twisting cats 106

Origins of our view of the universe 110Kepler's laws of planetarymotion 114Newton's law of universal gravitation 118The frontiers of physics: Measuring the distance to the Moon 124Spacecraft and orbital motion 125The frontiers of physics: The Global Positioning Satellite System 128

PART 3: THE STRUCTURE OF

MATTER

7 A T O M S : B U I L D I N G B L O C K S O F T H E

The underlying structure of matter 133

Physics in our world: Thermography 204Heat capacity205Heat of fusion and heat of vaporization 207

Physics in our world: Instant ice cream 213Humidity213

The unusual expansion of water 218

11 T H E L A W S O F T H E R M O D Y N A M I C S 221

The four laws of thermodynamics 221

Physics in our world: Automobile engines 224The zeroth law of thermodynamics 225The ®rst law of thermodynamics 226The second law of thermodynamics 229The third law of thermodynamics 233The frontiers of physics: Entropy that organizes? 234Entropyand the origin of the Universe 235Entropyand the arrow of time 239

PART 5: ELECTRICITY AND

The frontiers of physics: Electrostatics on Mars 258

Storing electrical energy262The frontiers of physics: Storing single electrons 264Physics in our world: Inkjet printers 265

13 A P P L I E D E L E C T R I C I T Y 267

Electric current and batteries 268

Physics in our world: Magneto-optical drives 295Electric currents and magnetism 295

A moving charge in a magnetic ®eld 298

Physics in our world: Avian magnetic navigation 307

Physics in our world: Microwave ovens 317

PART 6: WAVES

The principle of superposition 325

The frontiers of physics: Chaos in the brain 336

Physics in our world: Telephone tones 344

The ear 347The frontiers of physics: Electronic ear implants 352

The frontiers of physics: Gradient-index lenses 389

The frontiers of physics: Arti®cial vision 403

The electromagnetic spectrum 411

Addition of velocities 464

20 T H E G E N E R A L T H E O R Y O F R E L A T I V I T Y 469

The principle of equivalence 469

The perihelion of Mercury483The gravitational time dilation 485The frontiers of physics: Orbiting clocks 488

The Bohr model of the atom revisited 503Physics in our world: Using photons to detect tumors 504

The frontiers of physics: Knowledge and certainty 522Physics in our world: Electron microscopes 523The frontiers of physics: Quantum teleportation 530

Pioneers of physics: Gell-Mann's quark 568

25 S U P E R F O R C E : E I N S T E I N ' S D R E A M 571

Supersymmetry and superstrings 585The creation of the universe 588The ®rst moments of the universe 591The frontiers of physics: The cosmic background explorer 594

P R E F A C E

As a research scientist at NASA KennedySpace Center working

on planetaryexploration, I am veryfortunate to be able to ence ®rst hand the excitement of discovery As a physicist, it is notsurprising that I ®nd science in general and physics in particularcaptivating I have written this book to tryto conveymyexcite-ment and fascination with physics to those who are curiousabout nature and who would like to get a feeling for the thrillsthat scientists experience at the moment of discovery

experi-The advances in physics that have taken place during thetwentieth centuryhave been astounding One hundred yearsago, Max Planck and Albert Einstein introduced the concept ofthe quantum of energythat made possible the development ofquantum mechanics This revolutionarytheoryopened thedoors for the breathtaking pace of innovation and discoverythat we have witnessed during the last ®ftyyears

At the beginning of the new century, physics continues itsinexorable pace toward new discoveries An exciting newtheorymight give us the ``theoryof everything,'' the uni®cation

of all the forces of nature into one single force which wouldreveal to us how the universe began and perhaps how it will end.Although these exciting new theories are highlymathemati-cal, their conceptual foundations are not dif®cult to understand

As a college professor for manyyears, I had the occasion toteach physics to nonscience students and to give public lectures

on physics topics In those lectures and presentations, I kept themathematics to a minimum and concentrated on the concepts.The idea for this book grew out of those experiences

This book is intended for the informed reader who is ested in learning about physics It is also useful to scientists inother disciplines and to professionals in non-scienti®c ®elds.The book takes the reader from the basic introductoryconcepts

inter-to discussions about the current theories about the structure ofmatter, the nature of time, and the beginning of the universe.Since the book is conceptual, I have kept simple mathematical for-mulas to a minimum I have used short, simple algebraic deriva-tions in places where theywould serve to illustrate the discoveryprocess (for example, in describing Newton's incredible beautifuldiscoveryof the universal law of gravitation) These short foraysinto elementaryalgebra can be skipped without loss of continu-ity The reader who completes the book will be rewarded with

a basic understanding of the fundamental concepts of physicsand will have a verygood idea of where the frontiers of physicslie at the present time

I have divided the book into seven parts Part 1 starts withsome introductoryconcepts and sets the stage for our studyofphysics Part 2 presents the science of mechanics and the study

of energy Part 3 follows with an introduction to the structure

of matter, where we learn the storyof the atom and its nucleus.The book continues with thermodynamics in Part 4, the concep-tual development of electricityand magnetism in Part 5, wavesand light (Part 6), and ®nally, in Part 7, with the rest of thestoryof modern physics, from the development of quantumtheoryand relativityto the present theories of the structure ofmatter

Acknowledgments

I wish to thank ®rst mywife, Dr Luz Marina Calle, a fellow NASAresearch scientist and myinvaluable support throughout themanyyears that writing this book took She witnessed all theups and downs, the dif®culties, setbacks, and the slow progress

in the long project She read the entire manuscript and offeredmanysuggestions for clari®cation, especiallyin the chapterswhere, as a physical chemist, she is an expert

I wish to thank Professor Karen Parshall, of the UniversityofVirginia, who verycarefullyand thoroughlyread the ®rst draft ofthe ®rst four chapters and made manysuggestions I also thankProfessors George H Lenz, Scott D Hyman, Joseph Giammarco,and Robert L Chase in the physics department at Sweet BriarCollege, who read all or part of the manuscript and offered

manycomments I am grateful to Karla Faulconer for manyof theillustrations that appear in the book For their help with differentaspects of the preparation of the manuscript, I am indebted toGwen Hudson, Rebecca Harvey, and Rachelle Raphael I wouldespeciallylike to acknowledge the invaluable help of mysonDaniel, now a software engineer at Digital Paper, who read theentire manuscript, made manyimportant suggestions and wasmyearlytest for the readabilityof manydif®cult sections.

No book can be written without the important peer reviewprocess The criticisms, corrections and, sometimes, praisemade the completion of this book possible Over a dozenuniversityphysics professors reviewed this book during thedifferent stages of its development I wish to thank them fortheir invaluable advice The work of two reviewers was particu-larlyimportant in the development of the book I appreciate thecomprehensive reviews of Professor Michael J Hones at VillanovaUniversity, who reviewed the manuscript four times, offeringcriticism and advice everytime Professor KirbyW Kemper atFlorida State University, reviewed the book several times andsuggested changes, corrections, and better ways to describe orexplain a concept The book is better because of them

Finally, I wish to thank Nicki Dennis, Simon Laurenson andVictoria Le Billon at IOP Publishing, and Graham Saxby, for theirunderstanding and for their ef®ciencyin converting mymanu-script into this book

Carlos I Calle

KennedySpace Center, Florida

funda-It is not dif®cult to imagine that, some thirty thousand yearsago, during a cold, dark spring night, a young child, moved per-haps by the pristine beauty of the starry sky, looked at his motherand, in a language incomprehensible to any of us today, askedher: ``Mother, who made the world?''

To wonder how things come about is, of course, a universalhuman quality As nearas we can tell, human beings have beenpreoccupied with the origin and nature of the world for as long

as we have been human Each of us echoes the words of thegreat Austrian physicist Erwin SchroÈdinger, ``I know not whence

I came norwhitherI go norwho I am,'' and seeks the answers.Here lies the excitement that this quest for answers brings toourminds Today, scientists have been able to pierce a few of theveils that cloud the fundamental questions that whisperin ourminds with a new and wonderful way of thinking which is

®rmly anchored in the works of Galileo, Newton, Einstein,Bohr, SchroÈdinger, Heisenberg, Dirac and many others whom

we shall meet in our incursion into the world of physics

Physics, then, attempts to describe the way the universeworks at the most basic level Although it deals with a greatvariety of phenomena of nature, physics strives for explanationswith as few laws as possible Let us, through a few examples,taste some of the ¯avorof physics

We all know that if we drop a sugar cube in water, the sugardissolves in the waterand as a result the waterbecomes thicker,denser; that is, more viscous We, however, are not likely to pay agreat deal of attention to this well-known phenomenon Oneinquisitive mind did.

One year after graduating from college, the young AlbertEinstein considered the same phenomenon and did, indeed,pay attention to it Owing to his rebellious character, Einsteinhad been unable to ®nd a university position as he had wantedand was supporting himself with temporary jobs as tutor or as

a substitute teacher While substituting for a mathematics teacher

in the Technical School in Winterthur, near Zurich, from May toJuly 1901, Einstein started thinking about the sweetened water

Figure 1.1 The laws of physics apply to a falling snow¯ake (courtesy

W P Wirgin), the explosion of a star or the eruption of a volcano (courtesyNASA)

problem ``The idea may well have come to Einstein as he washaving tea,'' writes a former collaborator of Einstein.

Einstein simpli®ed the problem by considering the sugarmolecules to be small hard bodies swimming in a structureless

¯uid This simpli®cation allowed him to perform calculationsthat had been impossible until then and that explained how thesugarmolecules would diffuse in the water, making the liquidmore viscous

This was not suf®cient forthe twenty-two-year-old scientist

He looked up actual values of viscosities of different solutions ofsugar in water, put these numbers into his theory and obtainedfrom his equations the size of sugar molecules! He also found

a value forthe numberof molecules in a certain mass of anysubstance (Avogadro's number) With this number, he couldcalculate the mass of any atom Einstein wrote a scienti®c paperwith his theory entitled ``A New Determination of the Sizes ofMolecules.''

Figure 1.2 Albert Einstein

On the heels of this paper, Einstein submitted for publicationanother important paper on molecular motion, where heexplained the erratic, zigzag motion of individual particles ofsmoke Again, always seeking the fundamental, Einstein wasable to show that this chaotic motion gives direct evidence ofthe existence of molecules and atoms ``My main aim,'' hewrote later, ``was to ®nd facts that would guarantee as far aspossible the existence of atoms of de®nite ®nite size.''

Almost a century earlier, Joseph von FraunhoÈfer, an ous German physicist, discovered that the apparent continuity ofthe sun's spectrum is actually an illusion This seemingly unre-lated discovery was actually the beginning of the long and tortu-ous road toward the understanding of the atom The eleventh andyoungest child of a glazier, FraunhoÈferbecame apprenticed to aglass maker at the age of twelve Three years later, a freak acci-dent turned the young lad's life around; the rickety boardinghouse he was living in collapsed and he was the only survivor.Maximilian I, the elector of Bavaria, rushed to the scene andtook pity of the poorboy He gave the young man eighteenducats With this small capital, FraunhoÈferwas able to buybooks on optics and a few machines with which he started hisown glass-working shop While testing high-quality prismsFraunhoÈfer found that the spectrum formed by sunlight after itpassed through one of his prisms was missing some colors; itwas crossed by numerous minuscule black lines, as in ®gure 1.3(colorplate) FraunhoÈfer, intrigued, continued studying thephenomenon, measuring the position of several hundred lines

illustri-He placed a prism behind the eyepiece of a telescope and ered that the dark lines in the spectrum formed by the light fromthe stars did not have quite the same pattern as that of sunlight

discov-He later discovered that looking at the light from a hot gasthrough a prism produced a set of bright lines similar to thepattern of dark lines in the solar spectrum

Today we know that the gaps in the spectrum that hoÈfer discovered are a manifestation of the interaction betweenlight and matter The missing colors in the spectrum are deter-mined by the atoms that make up the body emitting the light

Fraun-In the spring of 1925 a twenty-four-year old physicist namedWerner Heisenberg, suffering from severe hay fever, decided totake a two week vacation on a small island in the North Sea,

away from the ¯owers and the pollen During the previous year,Heisenberg had been trying to understand this interactionbetween light and matter, looking for a mathematical expressionfor the lines in the spectrum He had decided that the problem ofthe relationship between these lines and the atoms could be ana-lyzed in a simple manner by considering the atom as if it were anoscillating pendulum In the peace and tranquility of the island,Heisenberg was able to work out his solution, inventing themechanics of the atom Heisenberg's new theory turned out to

be extremely powerful, reaching beyond the original purpose ofobtaining a mathematical expression for the spectral lines

In 1984, this idea of thinking about the atom as oscillationstook a new turn John Schwarz of the California Institute ofTechnology and Michael B Green of the University of Londonproposed that the fundamental particles that make up the atomare actually oscillating strings The different particles that scien-tists detect are actually different types or modes of oscillation ofthese strings, much like the different ways in which a guitarstring vibrates This clever idea, which was incredibly dif®cult

to implement, produced a theory of enormous beauty andpowerwhich explains and solves many of the dif®culties thatprevious theories had encountered The current version of thetheory, called superstring theory ± which we will study in moredetail in chapter25 ± promises to unify all of physics and help

us understand the ®rst moments in the life of the universe Stillfar from complete, superstring theory is one of the most activeareas of research in physics at the present time

In all these cases, the scientists considered a phenomenon ofnature, simpli®ed its description, constructed a theory of its beha-viorbased on the knowledge acquired by otherscientists in thepast, and used the new theory not only to explain the phenom-enon, but also to predict new phenomena This is the way physics

is done This book shows how we can also do physics, and share

in its excitement

The scienti®c method: learning from our mistakes

In contrast to that of many other professionals, the work of ascientist is not to produce a ®nished product No scienti®c

theory will ever be a correct, ®nished result ``There could be nofairer destiny for any theory,'' wrote Albert Einstein, ``thanthat it should point the way to a more comprehensive theory inwhich it lives on, as a limiting case.''

Science is distinguished from other human endeavor by itsempirical method, which proceeds from observation or experiment.The distinguished philosopherof science Karl R Poppersaid thatthe real basis of science is the possibility of empirical disproof

A scienti®c theory cannot be proved correct It can, however, bedisproved

According to the scienti®c method, a scientist formulates atheory inspired by the existing knowledge The scientist usesthis new theory to make predictions of the results of futureexperiments If when these experiments are carried out the pre-dictions disagree with the results of the experiments the theory

is disproved; we know it is incorrect If, however, the resultsagree with the forecasts of the theory, it is the task of the scientists

to draw additional predictions from the theory, which can betested by future experiments No test can prove a theory, butany single test can disprove it

In the 1950s, a great variety of unpredicted subatomic ticles discovered in laboratories around the world left physicistsbewildered The picture that scientists had of the structure ofmatter up to the 1940s ± as we will learn in more detail in chapters

par-7 and 8 ± was relatively simple and fairly easy to understand:matterwas made of atoms, which were composed of a tinynucleus surrounded by a cloud of electrons The nucleus, inturn, was made up of two kinds of particles, protons and neu-trons The new particles being discovered did not ®t this simplescheme Two theories were formulated to explain their existence.The ®rst one proposed a ``particle democracy,'' in which no par-ticle was any more fundamental than any other This theory was

so well received by the scienti®c community in the United Satesthat one of the proponents of the second theory, Murray Gell-Mann of the California Institute of Technology decided to publishhis paper in a European journal where he felt the opposition to hisnew ideas would not be so great Gell-Mann and independentlyGeorge Zweig, also of Caltech, proposed that many of the grow-ing number of particles and in particular the proton and theneutron were actually made up of smaller, indivisible particles

which Gell-Mann called quarks Different combinations of quarks,

in groups of two or three, were responsible for many of these ticles According to their theory, the growing number of new par-ticles being discovered was not a problem anymore Whatmattered was that the objects of which these particles weremade of were simple and small in number

par-Which theory was correct? In 1959 Stanford University built

a large particle accelerator which, among other things, coulddetermine whether or not quarks existed Seven years later,experiments carried out at the Stanford Linear AcceleratorLaboratory, SLAC, allowed physicists to determine the presence

of the quarks inside protons and neutrons Since then, manyexperiments have corroborated the Stanford results; the quark

is accepted today as one of the fundamental constituents ofmatter and the ``particle democracy'' theory is no longer viable

We shall see in the ®nal chapters of this book that these newtheories of matter are far from complete Nevertheless, the knowl-edge obtained from these theories has given us not only a betterunderstanding of the universe we live in but has also producedthe modern technological world based largely on the computerchip

We can summarize the scienti®c method by saying that wecan learn from our mistakes Scienti®c knowledge progresses byguesses, by conjectures which are controlled by criticism, bycritical tests These conjectures or guesses may survive the tests;but they can never be established as true ``The very refutation

of a theory,'' writes Popper, ``is always a step forward thattakes us nearer to the truth And this is how we learn from ourmistakes.''

Physics and other sciences

Physicists often become interested in phenomena normallystudied by scientists in otherscienti®c disciplines, and applytheir knowledge of physics to these problems with great success.The recent formulation of the impact theory of mass extinctions is

a good illustration of physicists becoming involved in other ti®c ®elds and of the way working scientists apply the scienti®cmethod to theirwork

scien-In 1980, the Nobel prize winning physicist Luis Alvarez andhis son Walter, a professor of geology at the University of Califor-nia at Berkeley, reported in a paper published in the journalScience that some 65 million years ago a giant meteorite crashedinto the earth and caused the extinction of most species Thedinosaurs were the most famous casualties Alvarez and hiscollaborators based their theory on their study of the geologicalrecord Walter Alvarez had told his father that the 1-cm-thickclay layerthat separates the Italian limestone deposits of theCretaceous period ± the last period of age of reptiles ± from those

of the Tertiary period ± the ®rst period of the age of mammals,

Figure 1.4 An unorthodox theory of the extinction of the dinosaurs.(Cartoon by Sydney Harris.)

was laid down during precisely the time when the great majority

of the small swimming animals in the marine waters of that regionhad disappeared What made it even more exciting was the factthat this time also coincided with the disappearance of the dino-saurs

The layer of clay ± observed worldwide and known as theK-T boundary layer ± contains an unusually high concentration

of the element iridium This element is present in very smallamounts in the earth's crust but is much more abundant inmeteorites The father and son team thought that by measuringthe amount of iridium present in the clay they could determinehow long the layer had taken to form They assumed that iridiumcould have rained down on the earth from meteoritic dust at afairly steady rate during the thousand years that it took toform If that were the case, they could measure the amount ofiridium in the clay and in the rocks above the clay (formedlater) and below (formed earlier) and determine the time it hadtaken forthe iridium to accumulate To that effect, they enlistedthe help of Frank Asaro and Helen Michel, nuclear chemists atthe Lawrence Berkeley Laboratory Asaro and Michel showedthat the clay layer contains three hundred times as much iridium

as the layers above and below

The source of this unusual amount of iridium had to beextraterrestrial, Luis Alvarez reasoned Meteorites, which are

Figure 1.5 K-T boundary layer with a high concentration of iridium.(Courtesy Alessandro Montanari.)

extraterrestrial, have fallen on the earth since its formation If theiridium came from the meteorites, why this sudden increase inthe meteorite rate during this particular time and why did itdecrease again to normal levels? What was so special about thisparticular time in the history of the earth? More importantly,why did it coincide with the extinction of about 50 percent ofthe species in existence then?

The Alvarez team ®rst proposed that the iridium could havecome from the explosion of a supernova near the solar system.Astrophysicists had proposed that the mass extinctions couldhave been caused by such an explosion Since these tremendousexplosions produce heavy elements Luis Alvarez proposed ana-lyzing the samples taken from the clay for their presence.Detailed measurements revealed no heavy elements, however,and the supernova idea had to be abandoned

While Walter Alvarez returned to Italy to collect more claysamples, his father worked on theory, inventing ``a new schemeevery week for six weeks and [shooting] them down one byone,'' as he wrote later Luis Alvarez then considered the possi-bility of an asteroid or a comet passing through the atmosphere,breaking up into dust which would eventually fall to the ground,like the comet that broke up over Tunguska, Siberia, in 1908.Calculations showed him that a larger asteroid, of 10 kilometers

in diameter, for example, would not break up into pieces TheTunguska comet was smaller

Alvarez then concluded that some 65 million years ago, a kilometer comet or asteroid struck the earth, disintegrated, andthrew dust into the atmosphere The dust remained in the atmos-phere for several years, blocking sunlight, turning day into night,and preventing photosynthesis, the process by which, in thepresence of light, plants convert water, carbon dioxide, and miner-als into oxygen and othercompounds Without plants to eat,animals starved to death We see the remnants of dust today asthe global K-T boundary layer between the Cretaceous andTerciary layers Alvarez calculated the diameter of the objectfrom the known concentration of iridium in meteorites and hisgroup's data on the iridium content of the Italian clay samples.Otherscientists proposed the idea that intense volcanic erup-tions could account forthe mass extinctions These scientistsfound high levels of iridium in tiny airborne particles released

10-by the Kilauea volcano in Hawaii and concluded that iridiumfrom the inner earth can reach the surface For a few years afterthey were proposed, both ideas could be used to explain theK-T extinctions However, different predictions could be drawnfrom the two competing ideas and scientists scurried to ®ndnew evidence in support of the different predictions Recent ®nd-ings, however, appear to con®rm the predictions of the impacttheory.

According to the scienti®c method, however, no theory canever be proved correct One of the theories will eventually beshown to be incorrect, leaving the remaining theories stronger,but not proven ``You will never convince some [scientists] that

an impact killed the dinosaurs unless you ®nd a dinosaur ton with a crushed skull and a ring of iridium around the hole,''joked a scientist at a conference on the subject

skele-Sizes of things: measurement

Most work in physics depends upon observation and surement To describe the phenomena encountered in natureand to be able to make observations, physicists must agree on aconsistent set of units

mea-Throughout history, several different systems of units weredeveloped It began with the Babylonians and the Egyptians,thousands of years ago The earliest recorded unit of mea-surement, the cubit, based on the length of the arm, appeared inEgyptian papyrus texts According to Genesis, Noah's Ark was

300 cubits long (about 150 m) Because the length of the armvaries from person to person, so did the cubits used among vari-ous civilizations The Egyptians used a short cubit of 0.45 m and aroyal cubit of 0.524 m The ancient Romans used the mille passus,

1000 double steps by a Roman legionary, which was equal to

5000 Roman feet In the 15th century, Queen Bess of Englandadded 280 feet to the mile to make it eight ``furrow-longs'' orfurlongs

In 1790, Thomas Jefferson proposed a system based on units

of 10 where 10 feet would be a decad, 10 decads a road, 10 roads afurlong, and 10 furlongs a mile Congress did not approveJefferson's system

In France, however, the French Revolution brought an est in science and anotherbase 10 system, the metric system, wasborn This system, based on the meter, from the Greek metron,meaning ``measure'', was more scienti®c Instead of usinghuman anatomy, the meter, as approved by the French NationalConvention in 1795, was de®ned as 1/10 000 000 of the length ofEarth's meridian between the Equator and the North Pole (®gure1.6).

inter-Once the meterwas de®ned, a unit of volume, the liter, could

be de®ned by cubing a tenth of a meter From the liter, thekilogram as a unit of mass was derived Multiples of 10 providedlarger units indicated by Greek pre®xes, and for smaller units,Latin pre®xes were used

Due to the consistency and uniformity of the system andthe easiness of de®ning new units merely by adding a Greekora Latin pre®x, the metric system was adopted in Europe inthe 19th century Today, an expanded version of the system,

SI units, for Le SysteÁme International d'UniteÂs, is used by 95percent of the world's population and is the of®cial system inscience In Table 1.1 we list the standard pre®xes used in the SIsystem

Notice in Table 1.1 that for large and small numbers, it iseasierto use scienti®c notation In the scienti®c notation, numbersare written as a number between 1 and 10 multiplied by a power

of 10 The radius of the earth, for example, which is 6380 km, can

be written in scienti®c notation as 6:380  103km To see why,

Figure 1.6 The meterwas originally de®ned as the 1/10 000 000 of thelength of the Earth's meridian from the North Pole to the Equator

note that we can write the number 1000 as follows:

1000 ˆ 10  10  10 ˆ 103:The radius of the earth is, then,

6380 km ˆ 6:38  1000 km ˆ 6:38  103km:

Fundamental units

All mechanical properties can be expressed in terms of threefundamental physical quantities: length, mass, and time The SIfundamental units are:

The General Conference on Weights and Measures held in Paris in

1983 de®ned the meteras the distance traveled by light throughspace in 1/299 792 458 seconds Notice that the unit of length isde®ned with such high precision in terms of the unit of time This

is possible because the second is known to betterthan 1 part in 10trillion The 1967 General Conference on Weights and Measures

Table 1.1 Powers of ten pre®xes

de®ned the second as the duration of 9 192 631 770 periods tions of one oscillation) of a particular radiation emitted by thecesium atom The device that permits this measurement is thecesium clock, an instrument of such high precision that it wouldlose orgain only 3 seconds in one million years (®gure 1.8).The last fundamental mechanical quantity is mass Mass is ameasure of the resistance that an object offers to a change in itscondition of motion For an object at rest with respect to us,mass is a measure of the amount of matter present in the object.The standard unit of mass is the standard kilogram, a solid plati-num-iridium cylinder carefully preserved at the Bureau ofWeights and Measures in SeÁvres, near Paris The kilogram isnow derived from the meter, which is derived from the second.

(dura-A copy of the standard kilogram, the Prototype Kilogram No

20, is kept at the National Bureau of Standards in Washington,

DC A high precision balance, especially designed for theNational Bureau of Standards, allows the comparison of themasses of otherbodies within a few parts in a billion

Figure 1.8 A cesium atomic clock at the National Institute of Standardsand Technology in Washington, DC (Courtesy National Institute ofStandards and Technology.)

The mass of an atom cannot be measured by comparisonwith the standard kilogram with such a high degree of precision.The masses of atoms, however, can be compared with each otherwith high accuracy For this reason, the masses of atoms are given

in atomic mass units (amu) The mass of carbon in these units hasbeen assigned a value of 12 atomic mass units In kilograms, anatomic mass unit is

1 amu ˆ 1:660 540 2  10ÿ27kg:

Physics and mathematics

Physics and mathematics are closely intertwined Mathematics is

an invention of the human mind inspired by our capacity to deal

(Cartoon by Sydney Harris.)

with abstract ideas; physics deals with the real material world.Yet, mathematical concepts invented by mathematicians whodid not foresee their applications outside the abstract world ofmathematics have been applied by physicists to describe natural

Frontiers of physics: Very small numbers

What does a mass like 1:660 540 2  10ÿ27kg mean? Supposethat you start with one grain of salt, which has a mass ofabout one ten-thousandth of a gram and with a very precisecutting instrument you divide it into ten equal parts, takeeach one of the tenths, divide them into ten new equalparts, and so on You will not arrive at single electrons thisway because, as we shall see in chapters 7 and 8, the electron

is one of the several constituents of atoms Although atomscan be split, you cannot do it with a cutting instrument.Suppose, however, that we divide the grain of salt intothe smallest amounts of salt possible, single molecules ofsalt One single molecule of table salt has a mass of about

9  10ÿ23g Let's round this number up to 10ÿ22g If yourinstrument takes one second, say, to take each piece of saltand divide it into ten equal parts, how long would it take

to end up with individual molecules of salt? The answeris

3  1010years

Astrophysicists estimate that the age of the universe is ofthe order of 1010years It would take our hypothetical instru-ment roughly the age of the universe to arrive at a singlemolecule of salt!

Grains of salt, magni®ed 100 times (Courtesy V Cummings, NASA.)

Pioneers of physics: Measuring the

circumference of the Earth

The meter, as we saw, was de®ned in 1795 as 1/10 000 000 ofthe length of the earth's meridian from the Equator to theNorth Pole For that de®nition to make sense, an accurateknowledge of the Earth's dimensions was needed That is,the actual length of the meridian from the Equator to theNorth Pole had to be known with good precision How did

we come to know the Earth's dimensions before the advent

of twentieth century technology?

The dimensions of the Earth have been known since thetime of the ancient Greeks The Greek astronomer Eratos-thenes, who lived in the third century BC in Alexandria(Egypt), came up with a very clever method for obtainingthe circumference of the Earth Eratosthenes had heard that

in the city of Syene, an ancient city on the Nile, neartoday's Aswan, on the ®rst day of summer, the sun shone

on the bottom of a vertical well at noon However, in hisnative Alexandria, the sun's rays did not fall verticallydown but at an angle of 78 to the vertical This angle of 78was about one-®ftieth of a circle and that meant that the

phenomena ``It is a mystery to me,'' wrote the Nobel Prizewinning physicist Sheldon Glashow, ``that the concepts of mathe-matics (things like the real and complex numbers, the calculusand group theory), which are purely inventions of the humanimagination, turn out to be essential for the description of thereal world.''

Physicists, on the otherhand, have invented powerful ematical techniques in their search to understand the physicalworld Newton developed the calculus to solve the problem ofthe attraction that the earth exerts on all objects on its surface.Mathematicians latercontinued the development of calculusinto what it is today

math-Mathematics is then the instrument of physics; the only guage in which the nature of the world can be understood Nonethe less, in this book we are interested in the concepts of physics.These concepts can usually be described with words and exam-ples In some instances, however, there is no substitute for theelegance and conciseness of a simple formula In these cases,

lan-we shall considersuch a formula to see how it explains new cepts and how they can be linked to otherconcepts alreadylearned The reader should always keep in mind that our purpose

con-is to understand the physical phenomenon, not the mathematicsthat describes it

distance between Syene and Alexandria must be one-®ftieth

of the earth's circumference

During Eratosthene's time, the distance between thesetwo cities was estimated to be 5000 stadia So the circumfer-ence of the earth was 50 times this distance or 250 000stadia Although the exact length of that Greek unit is notknown, we do know that the length of a Greek stadiumvaried between 154 and 215 meters If we use an averagevalue of 185 m, the result is only about 15% larger thanmodern measurements, a remarkable achievement

comprehen-we might be able to discover where it has been and predict where

it will be some time in the future, provided that the presentconditions are maintained In physics, we are interested in thedescription of the motion of the different bodies that we observe,such as automobiles, airplanes, basketballs, sound waves, elec-trons, planets, and stars

To study how objects move, we need to begin by studyinghow a simple object moves An object without moving parts,such as a ball or a block, is simpler than one with separateparts because we do not need to worry about the motions ofthe parts, and we can concentrate on how the object moves as awhole A ball can roll and a block can slide on a surface Whichone is simpler? It would be easier for us if we did not have todecide beforehand either the shape of the object or its internalstructure Physicists simplify the problem by considering themotion of a point, an ideal object with no size, and therefore nointernal structure and no shape

We will consider ®rst the motion of a point However, inour illustrations and examples we might refer to the motion

of real objects, like cars, baseballs, rockets or people If we donot consider the internal structure of the object, and do notallow it to rotate, this object behaves like a point for ourpurposes

Uniform motion

``My purpose is to set forth a very new science dealing with a veryancient subject,'' wrote Galileo in his Two New Sciences He con-tinued: ``There is, in nature, perhaps nothing older than motion,concerning which the books written by philosophers are neitherfew nor small; nevertheless I have discovered by experimentsome properties of it which are worth knowing and which havenot hitherto been either observed or demonstrated.''

Galileo, one of the ®rst modern scientists and the ®rst one tounderstand the nature of motion, was born in Pisa the same yearthat Shakespeare was born in England and three days beforeMichelangelo died The year was 1564 His full name was GalileoGalilei, following a Tuscan custom of using a variation of thefamily name as the ®rst name of the eldest son

His father, a renowned musician, wanted his son to be aphysician, a far more lucrative profession in those days Thus,

he entered the University of Pisa to study medicine Upon ing a lecture on geometry which encouraged him to study thework of Archimedes, the young medical student decided thatscience and mathematics seemed far more interesting thanmedicine Galileo talked to his father about letting him switch.Fortunately for the world his father consented

hear-Galileo became well known throughout Italy for his scienti®cability and at the age of 26 was appointed Professor of Mathe-matics at the University of Pisa There he dug deeply into funda-mental science He also made some enemies, especially amongthe older and more respected professors, who did not like theiropinions and views challenged by the young and tactless Galileo.Partly because of this, and partly because the Republic of Venicewas, in 1600, the hub of the Mediterranean, which in turn was thecenter of the world, Galileo accepted a position as Professor ofMathematics at Padua, where he began the work in astronomythat was to bring him immortal fame

Galileo's work on mechanics was published as Discourses andMathematical Demonstrations Concerning Two New Sciences Pertain-ing to Mechanics and Local Motion, which appeared in 1638 Inthe chapter ``De Motu Locali'' or ``Change of Position'', he writes:

The discussion is divided into three parts; the ®rst part deals withmotion which is steady or uniform; the second treats of motion as

we ®nd it accelerated in nature; the third deals with the so-calledviolent motions and with projectiles.

Galileo then goes on to explain what ``motion which is steady oruniform'' is:

By steady or uniform motion, I mean one in which the distancestraversed by the moving particle during any equal interval of time,are themselves equal

Figure 2.1 is an example of uniform motion; it shows severalpositions of an athlete running along a straight 100-m track at asteady pace The marks alongside the track show that therunner moves equal distances of 10 meters in equal intervals of

6 seconds

Average speed

The runner in ®gure 2.1 travels 10 meters in six seconds or 100meters in 60 seconds (1 minute) We can say that the runnertravels at 100 meters per minute Average speed is de®ned asthe total distance traveled divided by the time taken to travelthis distance If we use the letter d to indicate distance, and theletter t to indicate time, we can write the average speed, v, as

v ˆdistance traveledtime taken ˆdt

Figure 2.1 Several positions of a runner running along a straight track

where the bar above the letter v is used to indicate that this is theaverage value The runner of our example travels a distance of

100 meters in one minute The average speed of the runner,then, is

v ˆdt ˆ100 m1 min ˆ 100 m=min:

Figure 2.2 shows a multiple-exposure photograph of a disk

of solid carbon dioxide (``dry ice'') in uniform motion The disk,resting on the polished surface of a table, is given a gentlepush With the room darkened, the shutter of a camera set on atripod is kept open while at equal intervals of time a strobe is

®red Since the only source of light comes from the strobe ±which for this experiment was ®red at 0.10 second intervals ±the ®lm records the position of the disk as it slides on the table.The meter rule shows that the disk moves 13 cm between ¯ashes.The disk, then, traverses equal distances of 13 cm in equal inter-vals of 0.10 s or 130 centimeters in 1.0 second We can say thatthe disk travels at an average speed v ˆ 130 centimeters persecond

The units of speed are units of distance divided by units oftime Speed can thus be given in miles per hour, kilometers perhour, meters per second, feet per minute, etc The SI unit ofspeed is the meter per second (m/s)

In both of those cases, the speed did not change The runnerand the disk were moving at a uniform or constant speed, at leastfor the intervals that were considered However, few motions areuniform The most common situation is that of variable speed Ifyou drive from your dorm to the movies, you start from rest,speed up to 30 miles per hour and probably drive at that speed

Figure 2.2 Multiple-exposure photograph of a disk of ``dry ice'' moving

on a smooth surface (Illustration from PROJECT PHYSICS, copyright #

1981 by Holt, Rinehart and Winston, Inc Reprinted by permission of thepublisher.)