GRAVITYGEORGEGAMOWUniversityof ColoradoIllustrations bythe Authort c. 1.......----------Published by AnchorBooksDoubleday & Company, Inc.GardenCity, NewYork1962TYPOGRAPHYBYSUSANSIENLibrary of Congress Catalog Card Number62-8840Copyright 1962 by Educational Services IncorporatedAll Rights ReservedPrinted intheUnited States of AmericaFirst EditionToQUIGGNEWTONwhoreads all my booksTHE SCIENCE STUDY SERIESTheScienceStudySeriesofferstostudentsandtothegeneral publicthe writingof distinguishedauthors onthe most stirring and fundamental topics of science, fromthe smallest known particles to the whole universe. Someofthebookstell oftheroleof science inthe worldofman, his technologyandcivilization. Others are bio-graphicalin nature, telling the fascinatingstoriesof thegreatdiscoverersandtheirdiscoveries. Alltheauthorshave been selected both for expertness in the fieldsthey discuss and for ability to communicate their specialknowledgeandtheirownviews inaninterestingway.The primarypurpose of these books is to provide asurvey within the grasp of the young student or the lay-man. Manyof the books, it is hoped, will encouragethe reader tomake his owninvestigations of naturalphenomena.The Series, which now offers topics in all the sciencesand theirapplications, hadits beginning in aproject torevise the secondary schools' physics curriculum. At theMassachusetts Instituteof Technologyduring 1956agroup of physicists, highschool teachers, journalists,apparatus designers, film producers, and other specialistsorganizedthePhysical ScienceStudyCommittee, nowoperatingas a part of Educational Services Incorpo-rated, Watertown, Massachusetts. They pooled theirknowledgeandexperiencetowardthedesignand crea-12THE SCIENCE STUDYSERIEStion of aids tothelearning of physics. Initially their ef-fort was supported by the National Science Foundation,which has continued to aid the program. The FordFoundation, theFund fortheAdvancement ofEduca-tion, and the Alfred P. SloanFoundation have alsogivensupport. TheCommitteehascreatedatextbook,anextensivefilmseries, alaboratoryguide, especiallydesignedapparatus, andateacher's sourcebook.TheSeriesisguided byaBoardofEditors, consist-ing ofBruceF. Kingsbury, ManagingEditor; JohnH.Durston, General Editor; Paul F. Brandwein, theCon-servation Foundationand Harcourt, Brace &WorId,Inc.; Francis L. Friedman, Massachusetts InstituteofTechnology; Samuel A. Goudsmit, Brookhaven Na-tional Laboratory; Philippe LeCorbeiller, Harvard Uni-versity; GerardPiel, ScientificAmerican;andHerberts. Zim, Simon and Schuster, Inc.PREFACEGravityrules the Universe. It holds together the onehundredbillionstars of ourMilkyWay; it makes theEarth revolve around the Sun and the Moon around theEarth; it makes ripenedapples anddisabledairplanesfall totheground. Therearethreegreat namesinthehistory of man's understanding of gravity: GalileoGalilei, whowasthe first tostudy indetail the processof freeandrestricted fall; Isaac Newton, whofirst hadthe idea of gravity as a universal force; and AlbertEinstein, who said that gravity is nothing but the curva-tureof thefour-dimensional space-time continuum.Inthisbookweshallgothroughall threestages ofthe development, devoting one chapter to Galileo'spioneeringwork, six chapters toNewton's ideas andtheirsubsequent development, onechapter toEinstein,andonechapter topost-Einsteinianspeculations con-cerningtherelationbetween gravityandother physicalphenomena.The emphasis on the "classics" in thisout-linegrows fromthe fact that the theoryof universalgravityis aclassical theory. It is veryprobable thatthere is ahiddenrelationbetweengravityontheonehand and the electromagneticfield andmaterial par-ticles on the other, but nobody is prepared today tosaywhat kind of relation it is. And there is noway of fore-telling how soonany furtherimportant progresswill bemadeinthisdirection.14PREFACEConsidering the "classical" part of the theory ofgravitation, theauthorhad tomakeanimportant deci-sionabouttheuseofmathematics. WhenNewtonfirstconceivedthe ideaof Universal Gravity, mathematicswas not yet developedto a degree that couldpermithimtofollowall theastronomical consequencesof hisideas. Thus Newton had to develop hisown mathemati-calsystem, nowknownas thedifferential andintegralcalculus, largely in order toanswerthe problems raisedby his theory of universal gravitation. Therefore itseems reasonable, and not only from the historical pointof view, to include in this book a discussion of theelementary principles of calculus, a decision whichaccounts for a rather large number of mathematicalformulas inthethirdchapter. Thereader whohasthegrit toconcentrateonthat chapterwill certainlyprofitby it asa basis forhisfurther studyof physics. On theother hand, thosewhoarefrightenedbymathematicalformulas canskipthat chapter without muchdamageto ageneral understandingof thesubject. But if youwant to learn physics, please do try to understandChapter 3!George GamowUniversityof ColoradoJanuary13, 1961CONTENTSTHESCIENCESTUDYSERIES 11PREFACE 13CHAPTER 1 HowThingsFall 19CHAPTER2 The Apple and the Moon 35CHAPTER 3 Calculus 47CHAPTER4 PlanetaryOrbits 61CHAPTER5 TheEarthasaSpinningTop 71CHAPTER 6 TheTides 79CHAPTER7 Triumphs of Celestial Mechanics 93CHAPTER8 EscapingGravity 103CHAPTER9 Einstein's Theoryof Gravity 115CHAPTER 10Unsolved Problems of Gravity 133GravityandQuantumTheory 141Antigravity 143INDEX 147GRAVITYChapter 1HOWTHINGS FALLThe notionof "up" and "down" dates backto timeimmemorial, andthe statement that "everythingthatgoesupmust comedown"couldhavebeencoinedbyaNeanderthal man. Inoldentimes, whenit was be-lievedthat theworldwas fiat, "up"was thedirectiontoHeaven, theabode of the gods, while"down"wasthe directiontotheUnderworld.Everythingwhichwasnot divinehadanaturaltendencytofall down, andafallenangel from Heavenabove would inevitably finishinHell below. And, although great astronomers ofancient Greece, like Eratosthenes and Aristarchus,presented the most persuasive arguments that the Earthwasround, thenotionof absoluteup-and-downdirec-tionsinspacepersistedthroughtheMiddleAges andwas usedtoridiculetheideathat theEarthcouldbespherical. Indeed, itwasarguedthat if the Earthwereround, then the antipodes, the people living on theopposite side of theglobe, would fall off the Earth intoempty spacebelow, and, far worse, all ocean waterwould pour off theEarthinthesame direction.WhenthesphericityoftheEarthwasfinallyestab-lished in the eyes of everyone by Magellan's round-the-worldtrip, thenotionof up-and-downas anabsolutedirectioninspacehadtobemodified. The terrestrialglobewasconsidered toberestingat thecenterofthe22 ljKAYITYUniversewhile all thecelestial bodies, beingattachedto crystal spheres, circled around it. This concept oftheUniverse, or cosmology, stemmedfromtheGreekastronomer Ptolemy and thephilosopher Aristotle.Thenatural motionof all material objects was towardthecenter of the Earth,and only Fire, which had somethingdivine in it, defied the rule, shooting upward from burn-ing logs. For centuries Aristotelian philosophy andscholasticism dominated human thought. Scientific ques-tions were answered by dialectic arguments (i.e., byjust talking), andnoattempt was madeto check, bydirect experiment, the correctness of the statementsmade. For example, it was believedthat heavybodiesfall fasterthanlight ones, butwehavenorecordfromthosedaysofanattempt tostudythemotionof fallingbodies. Thephilosophers' excusewasthatfreefall wastoo fastto be followedby the human eye.Thefirst trulyscientificapproachtothequestionofhow things fall was made by the famousItalian scientistGalileoGalilei (1564-1642) at thetimewhenscienceand art began to stir fromtheir dark sleep of the MiddleAges. According to the story, which is colorful butprobablynot true, it all startedone daywhen youngGalileowasattending aMassintheCathedral of Pisa,andabsent-mindedlywatchedacandelabrumswingingtoandfroafter anattendant hadpulledit tothesideto light the candles (Fig. 1). Galileo noticed thatalthough the successive swings became smaller andsmaller as the candelabrumcametorest, the time ofeach swing (oscillation period) remained the same.Returninghome, he decidedtocheckthis casual ob-servationbyusingastonesuspendedonastringandmeasuring theswing period bycounting hispulse. Yes,he was right; the period remained almost the same whilethe swings became shorter and shorter. Being of aninquisitive turn of mind, Galileo started a series ofHOWTHINGS FALL 23Fig. 1. Acandelabrum(a) anda stoneona rope (b)swingwiththesameperiodifthesuspensionsareequallylong.experimellts, using stones of different weights andstringsof different lengths. These studiesledhim toanastonishingdiscovery. Althoughtheswingperiod de-pendedonthestring'slength(beinglongerfor longer24GRAVITYstrings), itwas quiteindependent oftheweight ofthesuspendedstone. This observationwas definitelycon-tradictorytotheaccepteddogmathat heavy bodiesfallfaster than light ones. Indeed, the motion of a pendulumisnothing butthefreefall of a weightdeflected fromavertical directionbya restrictionimposedbyastring,whichmakes theweight movealonganarcofacirclewiththecenter inthesuspensionpoint (Fig. 1).If light and heavy objects suspendedon strings ofequal length and deflected by the sameangletake equaltime tocomedown, thentheyshouldalsotakeequaltimetocomedown if dropped simultaneously fromthesameheight. Toprovethisfact totheadherentsof theAristotelian school, Galileo climbed theLeaning Towerof Pisa or someother tower (or perhaps deputizedapupiltodoit) anddroppedtwoweights, alight andaheavyone, whichhit thegroundat the sametime, tothegreatastonishmentof his opponents (Fig. 2).Thereseemstobenoofficial recordconcerningthisdemonstration, butthefactisthat Galileowasthemanwhodiscoveredthat the velocityof free fall does notdependon themassof thefallingbody. Thisstatementwas later proved by numerous, much more exact experi-ments, and, 272years after Galileo's death, was usedby Albert Einstein as the foundation of his relativ-istic theory of gravity, to be discussed later in thisbook.It is easy to repeat Galileo's experiment withoutvisiting Pisa. Just take a coin and a small piece of paperanddropthemsimultaneously fromthesameheight tothefloor. Thecoinwill godownfast, whilethe pieceofpaper will lingerin theair for amuchlongerperiodof time. But if you crumple the piece of paper and roll itintoalittleball, it will fall almost as fast as thecoin.If youhadalongglasscylinder evacuatedof air, youwouldseethat acoin, anuncrumpledpiece of paper,HOWTHINGS FALLFig. 2. Galileo'sexperimeot inPisa.25andafeather wouldfall insidethecylinder at exactlythe samespeed.Thenext step taken by Galileo in the study of fallingbodies was tofindamathematical relationbetweenthetimetakenbythefall andthedistancecovered. Sincethefree fall is indeedtoofast tobe observedindetail26 GRAVITYbythe humaneye, andsince Galileo didnot possesssuch modern devicesasfast moviecameras, hedecidedto"dilute"the forceof gravity by lettingballs madeofdifferent materialsroll downaninclinedplaneinsteadof falling straight down. Heargued correctly that, sincetheinclinedplaneprovides apartial support toheavyobjects placed on it, theensuing motion should be simi-lar to free fall except that the time scale would belengthened by a factor depending on the slope. Tomeasuretimehe usedawater clock, adevicewithaspigot that couldbeturned onandoff. Hecould meas-ureintervalsof time byweighingtheamounts of waterthat poured out the spigot indifferent intervals.Galileomarked the successive position of the objects rollingdown an inclined planeatequal intervalsof time.You will not find it difficult to repeat Galileo'sexperiment and check on the results he obtained. *Take a smooth board 6 feet long and lift one endofit2inchesfromthefloor, placingunder it acoupleof books (Fig. 3a). The slope of the boardwill be_2_ =~ , andthiswill alsobethefactor bywhich6X 12 36thegravityforce actingontheobject will bereduced.Nowtakeametal cylinder (whichislesslikelytorolloff the board than a ball) and let it go, without pushing,fromthe top end of the board. Listen to a ticking clockor a metronome(such as music students use) and markthe position of the rolling cylinder at the end of the first,second, third, and fourth seconds. (The experimentshouldberepeatedseveral times togetthesepositionsexact.) Under these conditions, consecutive distancesfrom the topend will be 0.53, 2.14,4.82,8.5,and13.0* Not beingan experimentalist. the author is not able tosay, onthebasis ofhis ownexperience, howeasyit is todoGalileo'sexperiment. Hehasheardfromvarioussources, how-ever, that this is, infact, not so easy andwould recommendthat thereaders ofthis booktrytheirskill at it.HOWTHINGS FALL 27inches. Wenotice, as Galileodid, thatdistancesattheend of the second, third, and fourth seconds arerespec-tively4, 9, 16, and 25timesthedistanceat theendofthe firstsecond. Thisexperiment provesthat theveloc-6''(j 'D------------r:/lFig. 3. (a) Arollingcylinderonaninclinedplane; (b)Galileo'smethodofintegration.ityof free fall increases insucha waythat thedis-tances covered by a moving object increase as thesquaresof the time of travel. (4= 22; 9= 32; 16=42; 2S= 52) Repeat the experiment with awoodencylinder, and a still lighter cylinder made of balsa wood,andyouwill findthat thespeedoftravel andthedis-28 GRAVITYtancescoveredat theendofconsecutivetimeintervalsremain the same.Theproblemthat thenfacedGalileo wastofindthelawofchangeofvelocitywithtime, whichwouldleadto the distance-time dependence stated above. In hisbookDialogueConcerningTwoNewSciences Galileowrotethatthedistancescoveredwouldincreaseas thesquares of timeif thevelocityof motionwas propor-tional tothefirst power oftime. InFig. 3bwegiveasomewhat modernized form of Galileo's argument. Con-sider a diagraminwhichthe velocity of motion v isplottedagainst time t. If ~ I is directlyproportional totwe will obtaina straight line runningfrom (0;0) to(t;v). Letus nowdivide thetimeinterval from0 totintoalarge number of veryshort time intervals anddraw vertical linesasshownin thefigure, thusforminga large number of thin tall rectangles. We nowcanreplacethesmooth slopecorrespondingtothecontinu-ousmotionof theobject witha kindof staircaserepre-sentingajerkymotioninwhichthe velocityabruptlychanges by small increments and remains constant forashort timeuntil thenext jerktakes place. If wemakethetime intervalsshorterandshorterand their numberlarger andlarger, the difference between the smoothslope and thestaircase will become lessand less notice-ableandwill disappear whenthe number of divisionsbecomes infinitelylarge.During each short time interval the motion isassumed to proceed with a constant velocity corre-sponding to that time, and thedistance coveredisequalto this velocity multiplied by the time interval. But sincethe velocityisequalto theheightof thethinrectangle,andthetimeinterval toitsbase, this product is equaltotheareaoftherectangle.Repeating the same argument for each thin rectangle,we come to the conclusion that the total distanceHOWTHINGS FALL 29coveredduringthe time interval (o,t) is equal to thearea ofthestaircaseor, in thelimit, totheareaofthetriangle ABC. Butthisareaisone-half oftherectangleABeDwhich, initsturn, isequaltotheproduct of itsbaset byitsheight u. Thus, wecanwritefor thedis-tance covered during the timet:1s = - vt2wherevisthevelocityat thetimet. But, accordingtoourassumption, uisproportionaltot sothat:v =atwhere a isa constant knownasaccelerationortherateof changeof velocity. Combiningthetwoformulas, weobtain:1 ')s= - at-2which proves that distance covered increases as thesquareoftime.Themethod of dividing a givengeometrical figureinto a large number of small parts and considering whathappens when the number of these parts becomesinfinitelylargeandtheir sizeinfinitelysmall, was usedinthethirdcenturyB.C. bytheGreekmathematicianArchimedesinhis derivationofthevolumeof aconeand other geometricalbodies. ButGalileowasthefirstto apply themethodto mechanical phenomena, thuslayingthefoundationfor thedisciplinewhichlater, inthehands ofNewton, grewintooneof themost im-portant branches of themathematical sciences.Another important contribution of Galileo to theyoungscienceof mechanics was the discoveryof theprinciple of superposition of motion. If we should throwastoneina horizontal direction, andif therewerenogravity, thestonewouldmovealongastraight lineas30GRAVITYa ball does on a billiard table. If, on the other hand,wejust droppedthestone, itwouldfall vertically withtheincreasing velocity we have described. Actually, wehave the superposition of two motions: the stone moveshorizontally with constant velocity and at the sametime fallsinanaccelerated way. Thesituation is repre-o39Fig. 4. Acombination of horizontal motion with con-stantvelocity, and vertical, uniformlyacceleratedmotion.sentedgraphically in Fig. 4wherethenumberedhori-zontal andvertical arrows represent distancescoveredinthetwokindsofmotion. Theresultingpositionsofthe stone can also be given by the single (white-headed)arrows, which become longer and longer and tumaround the point of origin.HOWTHINGS FALL 31Arrows like these that show the consecutive positionsof a moving object inrespect tothe pointof originarecalleddisplacement vectors andare characterized bytheirlengthandtheir directioninspace.Iftheobjectundergoesseveralsuccessivedisplacements, eachbeingdescribed by the corresponding displacement vector,thefinalposition can be describedby a single displace-ment vector called thesum of the original displacementvectors. You just draw each succeeding arrow beginningfrom the end of the previous one(Fig. 4), and connectthe end of the last arrow with the beginning of the firstby a straight line. In plain words foratrivial example,aplanewhichflewfromNewYorkCitytoChicago,fromChicagotoDenver, andfromDenver toDallas,could have gone from New York City to Dallas by flyinga straightcourse betweenthetwocities. Analtemativeway of adding twovectorsis todraw both arrows fromthesamepoint, completetheparallelogramanddrawitsdiagonalasshown in Fig. Saandb. Comparingthetwodrawings, one is easilypersuadedthat theybothlead to the same result.The notion of displacement vectors and theiradditionscanbeextendedtoothermechanical quanti-ties which have a certain direction in space. Imagine anaircraft carrier making so many knotson a north-north-west course, andasailorrunningacrossitsdeckfromstarboard to port at the speed of so manyfeet perminute. Both motions canbe represented by arrowspointing inthedirectionofmotionandhavinglengthsproportional to the corresponding velocities (whichmust of coursebeexpressed inthesameunits). Whatisthe velocityofthesailor in respect towater?All wehave todoistoadd the two velocity vectorsaccordingto the rules, i.e., by constructingthe diagonal of aparallelogram defined by the twooriginal vectors.Forces, too, canberepresentedbyvectors showing32Q.F2GRAVITY\.\."B'-- c__- - > - - _ ~ AC.Fig. 5. (a)and(b)Two ways to add vectors; (c) Forcesacting on a cylinder placed on an inclined plane.the direction of the actingforce and the amount ofeffort applied,andcan be addedaccordingto thesamerule. Let us consider, for example, the vector of thegravity forceactingon anobjectplacedon aninclinedplane(Fig. 5c). This vector is directed vertically down,ofcourse, but reversingthemethodofaddingvectors,we can represent it as the sumof two (or more)vectors pointing in given directions. Inour example wewant onecomponent topoint inthe directionof theHOWTHINGS FALL33inclined plane andanother perpendicular to it asshowninthefigure. We noticethat therectangular trianglesABC (geometry of the inclined plane), and abc(formedby vectors F, Fpand Ft ) are similar, having equalangles at Aand a respectively. It follows from Euclidiangeometry thatF2 BC-=-F ACandthisequationjustifiesthestatement wemadecon-cerning Galileo'sexperiment withthe inclined plane.Using the data obtainedby experiments with inclinedplanes, onecanfindthattheaccelerationof freefall isinches ( bbl f'li . h h386.2 --,,- you pro a yare amI ar WIt t es e c ~equivalent expression "32.2 feet per second per sec-ond") or inthemetricsystem, 981 cm. This valuesec2variesslightlywiththelatitudeontheEarth'ssurface,and thealtitudeabove sea level.Chapter 2THE APPLE ANDTHE MOONThe story that Isaac Newton discoveredthe LawofUniversal Gravity by watching anapple fall from a tree(Fig. 6)mayor may not beas legendaryasthestoriesabout Galileo's watching the candelabrum in theCathedral of Pisa or dropping weights from the LeaningTower, but it enhances the role of applesin legendandhistory. Newton's applerightfullyhasaplacewiththeapple of Eve, which resulted in the expulsion fromParadise, theappleof Paris, whichstartedtheTrojanWar, andthe appleof WilliamTell, whichfiguredintheformationof one of the world's most stable andpeace-loving countries. There is no doubt that when thetwenty-three-year-old Isaac was contemplating thenatureof gravity, he hadampleopportunity toobservefallingapples; atthetimehewasstaying on afarminLincolnshire to avoid the Great Plague, which de-scendedonLondonin1665andledtothetemporaryclosing of Cambridge University. In his writings Newtonremarks: "During thisyearIbegantothinkofgravityextendingtotheorbof theMoon, andcomparedtheforce requisite tokeeptheMooninher orbwiththeforcesof gravity at the surface of the Earth." Hisargu-ments concerningthissubject, givenlaterinhis bookMathematical Principles of Natural Philosophy, runroughlyasfollows: If, standingonthetopof a moun-38GRAVITYFig. 6. Isaac Newton on theLincolnshire farm.THE APPLE AXD THE MOON39tain we shoot a bullet in a horizontal direction, itsmotionwill consist of twocomponents: a) horizontalmotion with theoriginal muzzle velocity; b) anacceler-atedfree fall under the actionof gravityforce. As aresultof superpositionofthesetwomotions, the bulletwill describeaparabolictrajectoryandhit thegroundsomedistance away. IftheEarthwerefiat, thebulletwould always hit the Earth even though the impactmight be veryfar away fromthe gun. But since theEarthis round, its surface continuouslycurves underthe bullet'spath, and, at acertain limitingvelocity, thebullet's curvingtrajectorywill followthecurvatureofthe globe. Thus, if there were no air resistance, thebullet wouldnever fall to the groundbut wouldcon-tinuecircling the Earthat a constantaltitude. Thiswasthe first theory of an artificial satellite, andNewtonillustratedit with a drawingverysimilar tothoseweseetodayinpopular articles onrockets andsatellites.Ofcourse, thesatellites arenot shot fromthetopsofmountains, but arefirst liftedalmostverticallybeyondthelimit of theterrestrial atmosphere, andthengiventhe necessaryhorizontal velocityfor circular motion.Consideringthe motionof theMoonas acontinuousfall whichall thetimemissestheEarth, Newton couldcalculate the force of gravity acting on the Moon'smaterial. This calculation, in somewhat modernizedform, runsasfollows:Consider the Moon moving along a circular orbitaround the Earth (Fig. 7). Its position at a certainmoment is M, and its velocity perpendicular to theradiusof theorbit is lJ. If the Moonwere not attractedbytheEarth, it wouldmovealongastraight line, and,afterashorttimeinterval, 11t, would bein position M'with MM' = vtlt. But there is another component ofthe Moon's motion; namely, the free fall toward theEarth. Thus, itstrajectorycurvesand, instead of arriv-40 GRAVITYFig. 7. Calculation of theacceleration of theMoon.ing at M', it arrives at the point M" on its circular orbit,and the stretch M'M" is the distance it has fallen towardthe Earth during the time interval M. Now, consider theright triangleEMM' andapplytoit the Pythagoreantheorem, whichsaysthat inaright trianglethesquareon theside oppositethe right angleisequal tothesumof thesquaresontheothertwosides. Weobtain(EM" + M"M')2= EM2 + MM'2or, openingtheparenthesis:EM"2 + 2EM" . M"M' + M"M'2= EM2 +MM'2THE APPLE AND THE MOON 41Since EA1" = EM,we canceltheseterms on bothsidesof the equation, and, dividingby 2EM weobtain:__ Ivf"M'2 MM'2MilM' +2EM = 2EMNowcomes an important argument. If we considershorter and shorter time intervals, M"M' becomescorrespondingly smaller, and both terms on the leftside come closer and closer to zero. But, since thesecond term contains the square of M"M' it goes to zerofaster than the first; in fact, if M"M' takes the values:I10;itssquarebecomes:I_.100 'I1000; etc.I_.100 'I I10,000 ; I 000 000; etc., ,Thus, for sufficiently small time intervals we mayneglect thesecondtermontheleft as comparedwiththefirst, andwrite:__ MM'2M"M'=---=2EMwhich will be exactly correct, of course, only whenM"M' is infinitesimallysmall.SinceMM'= vAt andEM= R, we canrewritetheabove asM"M' = ~ ( ~ ) At2In discussing Galileo's studies of the law of fall, we haveseen that the distance traveled duringthe time inter-val At is ~ a A t 2 , where a is the acceleration, so that,242GRAVITYu2comparing the two expressions, we conclude that -Rrepresents the acceleration a with which the Mooncontinuouslyfalls towardtheEarth, missingit all thetime.Thus we can write for that acceleration:a=~ = (iYR=w2Rwherevw="Ris theangular velocity of the Moon in its orbit. Angularvelocityw(theGreekletter omega) of anyrotationalmotion is very simply connected with the rotationperiodT. Infact, wecanrewritetheformula as:211"v vW=-- = 211"-211"R swhere s= 271R is the total length of the orbit. Ap-sparently the rotation period T is equal to - so thatvtheformulabecomes:211"w=-TThe Moon takes 27.3days, or2.35.106seconds, for acompleterevolutionaroundtheEarth. Substituting thisvalue forT in theexpression, weget:1W= 2.67 . 10-6-secUsingthisvalueforwandtakingR= 384,400km=3.844.1010cm, Newton obtainedfor the accelerationof thefallingMoonthevalue0.27cm/sec2whichis3640timessmallerthantheacceleration981cm/sec2onthesurfaceof the Earth. Thus, it became clearthatTHE APPLE AND THE MOON 43the force of gravitydecreases withthe distance fromthe Earth, but whatisthelaw governingthisdecrease?Thefallingappleisat thedistance6371 Ianfromthecenter of the Earth andthe Moonis at thedistance384,400 Ian, i.e., 60.1 times farther. Comparing thetwo ratios 3640and 60.1, Newton noticed that thefirstis almost exactly equal to the square of the second. Thismeant that the law of gravity is very simple: the force ofattractiondecreases as the inverse squareof the dis-tance.But, iftheEarthattracts the apple andthe Moon,whynot assume that theSun attracts the Earth andother planets, keepingthemintheir orbits?Andthenthere shouldalsobe anattractionbetweenindividualplanets disturbing, in turn, their motions around thecentral body of the system. And, if so, two applesshould also attract each other, though the force betweenthemmaybe tooweakto be noticedbyour senses.Clearlythis force of universal gravitational attractionmust dependonthemasses of theinteracting bodies.According to one of the basic lawsof mechanicsformu-lated by Newton, a given force acting on a certainmaterial bodycommunicatestothisbodyanaccelera-tionwhich is proportional to the force andinverselyproportional tothemassofthebody;indeed, it takestwiceasmucheffort tobringuptothesamespeedabodywithdouble mass. Thus, fromGalileo's findingthat all bodies, independent of their weight, fall withthesameaccelerationinthefieldofgravity, onemustconclude that the forces pullingthemdownare pro-portional to their mass; Le., to the resistance to accelera-tion. And, if so, gravitational force might also beexpected to be proportional to the mass of anotherbody. Gravitational attractionbetweenthe Earth andtheMoonis verylargebecausebothbodies areverymassive. The attraction between the Earth and an apple44GRAVITYismuchweakerbecausetheappleissosmall, andtheattractionbetweentwoapplesmustbequitenegligible.Byusingargumentsof that kind, Newtoncametotheformulation of the Law oj Universal Gravity, accordingtowhicheverytwomaterialobjectsattract eachotherwith a jorce proportional to the product oj their masses,and inverselyproportional tothesquare oj the distancebetween them. If we write M1and M2 for the masses oftwointeracting bodies, and Rforthedistancebetweenthem, theforceofgravitational interactionwill beex-pressedbyasimpleformula:F=GMIM2R2whereG(forgravity) isauniversalconstant.Newton did not livetowitnessa direct experimentalproof of hislawof attractionbetween twobodieseachnot muchlarger thananapple, but three-quarters ofa century after his death another talented Britisher,Henry Cavendish, demonstrated the proof beyond argu-ment. Inorder toprovetheexistenceof gravitationalattraction between everyday-sized bodies, Cavendishused very delicate equipment that in his day representedthe height of experimental skill but which can be foundtoday in most physics lecture rooms to impressNewton's law of gravity on the minds of freshmen. Theprincipleof the Cavendishbalanceisshown inFig. 8.A light bar with two small spheres attached at each endis suspended on a long thread as thin asa cobweb, andplaced inside aglass box to keep air currents fromdisturbingit. Outsidetheglassboxaresuspendedtwoverymassivesphereswhichcanbe rotatedaroundthecentral axis. After the systemcomes to the state ofequilibrium, the position of the large sphere is changed,andit isobservedthat thebarwiththesmall spheresturns through a certain angle asa result of gravitationalTHE APPLE AND THE MOON45~ .Fig. 8. TheprincipleoftheCavendishbalance(a), and(b) Boys' modification.attractiontothelargesphere. Measuringthedeflectionangle andknowing the resistance of the thread to atwist, Cavendishcould estimate theforce withwhich46GRAVITYmassive spheres actedonthe little ones. Fromtheseexperiments hefoundthat thenumerical valueofthecoefficient G in Newton's formula is6.66X10-8 if thelengths, masses, andtimearemeasuredincentimeters,grams,and seconds. Using this value,onecancalculatethatthegravityforcebetweentwoapplesplacedcloseto each other is equivalent to the weight of one-billionthofan ounce!Amodifiedformofthe Cavendishexperiment wasperformed later bythe British physicist C. V. Boys(1855-1944).* After having balanced two equalweights on the scales (Fig. 8) he placed a massivesphereunder oneof theplates, andobservedaslightdeflection; theattraction of theterrestrial globeonthatweightwasaugmented bytheattraction of themassivesphere. The observed deflection permitted Boys tocalculate the ratio of the mass of the sphere to themassof theEarth; theEarth, hefound, weighs 6.1024kilograms (kg).* Author of Soap Bubbles and the Forces Which MouldThem, ScienceStudySeries.Chapter 3CALCULUSIt mayseemhardtounderstandthat Newton, havingobtained the basic ideas of Universal Gravityin thevery beginning of his scientific career, should havewithheld publication for about twenty years until hewas abletopresent acompletemathematicalformula-tionoftheTheoryofUniversal Gravityinhisfamousbook, Philosophiae Naturalis Principia Mathematica,publishedin1687.Thereason forsucha longdelaywasthat, althoughNewtonhadclear ideas concerningthephysical lawsof gravity, helackedthemathematical methodsneces-saryfor the development of all the consequences ofhis fundamental law of interaction between the materialbodies. The mathematical knowledge of his time wasinadequate forthe solution of the problems which arosein connection with gravitational interaction betweenmaterial bodies. For example, inthetreatment of theEarth-Moon problem described in the previous chapter,Newton had to assume that the force of gravity isinversely proportional to the square of the distancebetweenthecentersofthesetwobodies. But whenanapple is attracted by the terrestrial globe, the forcepullingit downis composedof aninfinite number ofdifferent forces causedby the attraction of rocks at50GRAVITYvarious depths under the roots of the apple tree, bytherocksoftheHimalayasandtheRockyMountains,bythewatersofthePacificOcean, andbythemoltencentral ironcore of the Earth. Inorder tomakethepreviouslygivenderivationoftheratioofforces withwhich the Earth acts onthe apple andontheMoonmathematicallyimmaculate, Newtonhadtoprovethatall these forcesadd up toa single force whichwould bepresent if all themass oftheEarthwereconcentratedinits center.Thisproblem,similar tobut much more complicatedthan Galileo's problemconcerning the motion of aparticle withconstantly increasing velocity, wasbeyondthemathematical resources of Newton's time, andhehad to develop his own mathematics. In doing so he laidthe foundation for what is now known asthe calculus ofinfinitesimals, or, simply, Calculus. This branch ofmathematics, whichistoday anabsolute"must"in thestudy of all physical sciences and is becoming more andmore important in biology and other fields, differs fromclassicalmathematicaldisciplinesby usingamethod inwhichthe lines, the surfaces,and the volumesof classi-cal geometryaredividedintoaverylargenumber ofvery small parts, and one considers the interrelationsinthe limiting case when the size of each subdivision goesto zero. We alreadyhave encounteredsuch kinds ofargumentationinNewton's derivationoftheaccelera-tion of the Moon(p.39)where the second termon theleftsideoftheequationcan be neglectedascomparedwith the first termif we consider the change of theMoon's position during a vanishingly short time interval.Letus considerageneralkindofmotion in whichthecoordinatexofamovingobjectisgivenasafunctionof time, t. Ineverydaylanguagethis means that thevalue ofxchanges insomeregular way as thevalue oft changes. Inthesimplest casexmaybeproportionaltot, andwewrite:CALCULUSx = At51in which the Ais a constant that makes the twosides oftheequationequal.Thiscaseistrivial. Wetaketwomomentsof timetand t +f:..twhere at isa small increment which is laterto be made equal to zero. The distance traveledduringthis timeinterval is apparently:A(t +at)- At= Aatand, dividing it by at wegetexactly A.Inthiscasewedonot even needtomake at infinitesimallysmall sinceit cancels out of the equation. Thus, we get for thetimerateofchangeofx, or the"fluxionofx," as Newtoncalledit:x=Awhereadot placedabovethevariabledenotes its rateof change.Letusnowtakeasomewhat morecomplicatedcasegivenby:x= At2Taking again the values of x for t and t +at, weobtain:A(t +at)2- At2and, openingthe parenthesis gives:At2+ 2Atat + at2- At2= 2Atat + at2Dividing thisby at, we get a two-term expression:2At +atWhen at becomes infinitesimally small the last term52 GRAVITYdisappearsandwehavefor thefluxionof x = At2 :1: = 2AtTurningtothecaseofx = AtSwehave tocalculatetheexpression:A(t +A.t)s - AtSMultiplying(t+At) byitself three timesand subtract-ing AtS, we get:A(tS +3t2A.t +3tA.t2+A.tS) - Ata=3At2At +3AtA.t2+A ~ t SanddividingbyAt:3At2+3Atllt +AA.t2When A.t becomes infinitesimally small, the last twotermsvanishand weobtain forthe fluxionof x = AtS:oX =3At2We can go on with x = At4, x = At5, etc.,obtainingthefluxions 4AtS, 5At4, etc. It is easyto notice thegeneral rule: the fluxion of x = Atllwhere n is anintegernumberisequal tonAtll-l.Intheforegoingexampleswecalculatethefluxionsof the quantities which are changing in direct pro-portion to time, tothe square of time, thecube of time,etc. But what aboutquantitieswhichchangeininverseproportiontovarious powers oftime?Weknowfromalgebrathat:1t-1 = -'t 'It-S= t3; etc.Using these negative exponents and proceeding asCALCULUSbefore, wefindthat thefluxionsof53x=At-I; x=At-2; x=At-3; etc.are:X = -At-2; x = -2Ar3; X =-3At-4; etc.The minus sign here stands because, in the case ofinverse proportionality, variable quantities decreasewithtime, andtherateofchange is negative. But thegeneral rule for calculating the fluxions remains thesameas inthe caseofdirect proportionality: toget theexpression of the fluxionwe multiply the original powerjunctionbyitsexponent, andreduce the valueof theexponent byoneunit. Theresults oftheforegoingdis-cussionaresummarizedinatablebelow:Ix =~ =At-3; Ar-2; At-I; At;At2; At3; At\ etc.-3At-4; -2At-3; -At-2; A; 2At; 3At2; 4At3; etc.While xin Newton's notations represents the rate ofchangeof x, x representstherate of change of thisrateof change. Thus, for example, if x = At3,x = 3At2and x =13At21 = 3A 2t= 6AtSimilarly X, which is the rateof changeof therateofchange of the rate of change, will be, inthe same case:x=16Atl=6AWe can nowtry these simple rules on Galileo'sformula for the free fall of material bodies. In Chapter 1wefound that the distancescoveredat the time t isgivenby1s= - at22Since thevelocity v is the rate of changeof position, we54have:GRAVITY1v =S= - a . 2t =at2whichsays that the velocityis simply proportional totime.For theaccelerationa, which isthe rateof changeofvelocity(or therateofchangeoftherateofchangeof position), we have:a=s=v=awhichis, ofcourse, atrivial result.Before we leave this subject, we must notice thatNewton'sfluxionnotationsareveryseldomusedinthebooks of today. At the same time that Newton wasdevelopinghismethod of fluxionsnow knownasdiffer-ential calculus, aGermanmathematician, GottfriedW.Leibniz, wasworkingalongthesamelines using, how-ever, asomewhat different terminology andsystemofnotations. What Newton called first, second, etc.,fluxions, Leibniz calledfirst, second, etc., derivatives,and, insteadofwritingx; x;x; etc., he wrote:dx d2x d3xdt ; dt2; dt3; etc.But the mathematical content of the two systems is,of course, thesame.While differential calculus considers the relationbetween the parts of geometrical figures when theseparts become infinitelysmall, integral calculus has anexactlyoppositetask: theintegrationof infinitelysmallparts into geometrical figures of final size. We en-counteredthismethod inChapter 1whenwe describedGalileo'smethodof addingupaverylargenumber ofverythinrectangles, theareaofwhichrepresentedthemotionof a particleduringa veryshort timeinterval.Similar methods were used before Galileo by Greekmathematiciansforfindingvolumesof cones andotherI'DCALCULUS'U=fCt )').755c.,Fig. 9. Integrationof (a) arbitraryfunction; (b) quad-ratic function; (c) cubic function.simplegeometrical figures, but thegeneral methodforsolvingsuchkinds of problemswas not known.Tounderstandtherelationbetweendifferential andintegralcalculus,let usconsider themotionofapoint56GRAVITYwhose velocity is given by the functionl)(t) as shown inFig. 9. Using thesameargumentsasinthesimple caserepresentedinFig. 3, we concludethat thedistancestraveled during the timet is given by thearea under thevelocity curve. The rate of change of s at any particularmoment is given by the velocity of motion at thatmoment, sothat we canwrite:dss= v or - =vdtin Newton'sandLeibniz' notations, respectively. Thus,if tl isgivenasthefunctionoftime, smust besuchafunctionof timethat itsfluxion(orderivative) isequaltov. Inthecase ofuniformlyacceleratedmotion:v= atso that we have tofinda function of time the fluxion ofwhich isequaltoat. Consultingthetableonp. 53, wefind that the fluxion of At2is 2At,so that the derivativeo f ~ At2is equal to At. Thus, writinga insteadof A, we2 1findthat s= -a(!'. This is, of course, the same result2that Galileo hadobtained frompurely geometricalcon-siderations.But letusconsidertwomorecomplicated cases, oneinwhichthevelocityincreases as the square of timeandanother inwhichit increasesasthecubeoftime.For thesetwocaseswemust write:v= bt2and v =et3Thesetwocases are representedbygraphs inFig. 9,and, just as in the previous simple example, thedistances traveledare representedbythe areas underthe curves. But, since we have here curving and notstraight lines, thereisnosimplegeometrical ruleshow-ing how to find these areas. Using Newton's method, welookagaininto the table onp. 53 andfind that theCALCULUS 57derivatives of At3andA t ~ are3At'2and 4At3, differingfromthe given expressions of the velocities only bythe numerical coefficients. Thus, putting 3A= band4A=c, we find for the areas under the twocurves:1So = - bt33and1s = - ct4c 4Themethodis quitegeneral, andcanbeusedfor anypower of t, as well asfor morecomplicated expressionssuchas:v = at +bt'2 + et3for whichweget:1 1 1s= -at2+ - bt3+ - ct4234Fromthediscussionweseethatintegral calculusisareverseof differential calculus: the problemhereistofindthe unknownfunctionwhosederivativeis equalto a givenfunction. Thus, we can nowrewrite thetableon p. 53, changing the order of the two lines, andchangingthenumerical coefficients, intheform:x=At-4; Al3; At-2; A- At; At2; At3; etc. ,A A A A Ax=- - t-3 - - t-2 -At-I. At - t2- t3- t4 etc3 ' 2' "2'3'4"We say thatxis an integral of x. In Newton's notationsone writes:x=(x)'wheretheprimeaccent placedoutsidetheparenthesiscounteractsthedot abovethex. InLeibniz' notations,wewrite:X=f xdtwhere the symbol in the front on the right side is nothingbut an elongated S standing forthewordsum.58GRAVITYLet us applythisnewtabletothesameoldexampleof a uniformly accelerated motion. Since accelerationisconstant, wewrite:x=aoro=afromwhichit follows that:x=Jadt = atIntegrating a second timeand consulting our new table,weobtain:x=J at . dt =~ t2Le., thesameresult as obtainedbefore. Ifaccelerationisnot constantbut, let ussay, proportional totime, wehave:x = bt. J Ix = btdt = 2.bt2x =J! bt2dt =~ J t 2 d t =~ t32 2 6Thus, in this case the distance covered by a movingobjectwouldincreaseasthecubeof time.The elementary formulation of differential andintegral calculus can beextended into three dimensions,whenall threecoordinates x, y, and z arepresent, butthis we leave to the reader who found the previousdiscussiontooeasy.Having developed the basic principles of calculus,Newton applied them to the solution of problems whichstoodintheway of hisTheoryof Universal Gravity-inthefirst place, tothe problemof thegravityforceCALCULUS 59exerted by the body of the Earthonany small materialobjectatany distancefromitscenter. For thispurposehe divided the Earthinto thin concentric shells, andA~ .Fig. 10. (a) Gravitational force exerted by sphericalshell on the outside point; (b) same thing, onlyinsideinstead of outside.considered their gravitational action separately (Fig.10). Inorder touse integral calculus we must dividethesurfacesoftheshellsintoalargenumber ofsmall60GRAVITYregions of equal areas, and then calculate the gravi-tational force exerted by each region on the object 0 inaccordancewiththeinversesquarelaw. Thisanalysisleads to a very large numberof force vectorsapplied tothe point 0,and these vectorsshould be integratedac-cording to the rules of vector addition. The actualsolution of that problemis beyondthe elementary prin-ciples we have discussed, but Newton managed tosolveit. The result was that when the point 0wasoutside thespherical shell, all the vectorsadded up to forma singlevectorequal tothegravity force whichwouldexist ifthe entire mass ofthespherical shell were concentratedinits center. In thecasewherepoint0 wasinsidetheshell, the sum of all vectors wasexactly zero, so that nogravity force was acting on the object. This result solvedNewton's trouble concerning the attractive force exertedby the Earth on an apple, and justifiedthe LawofUniversalGravityhehadstatedwhenhewas a youngman contemplating the riddlesof nature intheorchardonthe Lincolnshirefarm.Chapter4PLANETARYORBITSNowthat wehave learnedalittle bit of calculus, wecan try toapply it to the motion of natural and aliificialcelestial bodiesunder theforceof gravity. Let us firstcalculate howfast a rocket shouldbe shot fromthesurfaceof the Earthinorder to escape the bondofterrestrial gravity. Consider furnituremoverswho haveto move a grand piano fromthe street level toa certainfloor in a high apartment building. Everybody will agree(andespeciallythefurnituremovers) that tobring agrand piano up three floorstakes three times more workthan to bringit up one floor. The work of carryingheavypieces of furniture is alsoproportional totheirweight,and in carrying up six chairs onedoessix timesmoreworkthanincarryingupjustonechair.Thisisall veryinconsequential, ofcourse, butwhatabout theworknecessarytoraisearocket sufficientlyhightoput itintoaprescribedorbit, or theworkofbringingit still higher sothat it woulddropdownontheMoon?Insolvingproblemsof thiskind, wemustrememberthat theforce ofgravitydecreaseswiththedistance from the center of the Earth; the higherwe liftthe object the easierit will beto lift it still higher.Fig. 11showsthechangeofgravitationalforcewiththedistancefromthecenter oftheEarth. Inordertocalculate the total work needed to bring an object64GRAVITYfromthe surface of the Earth (distance Rofromthecenter) tothedistance R, takingintoaccount thecon-tinuous decrease of gravitational force, we divide the>-I->