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Newtonian Physics

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Newtonian Physics
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Science, Physics, Modern Physics, Newton, Scientific Method, Metric System, Metric Unit of Force, Scientific Notation, Motion in One Dimension, Velocity and Relative Motion, Types of Motion, Distance and Time, Graphs of Motion, Velocity, Inertia, Scaling of Area and Volume, Galileo, Scaling Applied to Biology, Addition of Velocities, Kepler’s Laws, Newton’s …
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This is an introductory physics textbook designed for use in a typical one year survey course. Contents: 1) Scaling and Order-of-Magnitude Estimates. 2) Velocity and Relative Motion. 3) Acceleration and Free Fall. 4) Force and Motion. 5) Analysis of Forces. 6) Newton’s Laws in Three Dimensions. 7) Vectors. 8) Vectors and Motion. 9) Circular Motion. 10) Gravity. This is book 1 in the Light and Matter series of free introductory physics textbooks.
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Copyright 1998-2003 Benjamin Crowell. This book is licensed under the Creative Commons Attribution-ShareAlike license, version 1.0, http://creativecommons.org/licenses/by-sa/1.0/, except for those photographs and drawings of which I am not the author, as listed in the photo credits. If you agree to the license, it grants you certain privileges that you …
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Book1intheLightandMatterseriesoffreeintroductoryphysicstextbooks www.lightandmatter.com

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The LightandMatter seriesof introductoryphysicstextbooks: 1NewtonianPhysics 2ConservationLaws 3VibrationsandWaves 4ElectricityandMagnetism 5Optics 6TheModernRevolutioninPhysics

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BenjaminCrowell www.lightandmatter.com

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Fullerton,California www.lightandmatter.com copyright1998-2008BenjaminCrowell rev.October12,2008 ThisbookislicensedundertheCreativeCommonsAttribution-ShareAlikelicense,version1.0, http://creativecommons.org/licenses/by-sa/1.0/,except forthosephotographsanddrawingsofwhichIamnot theauthor,aslistedinthephotocredits.Ifyouagree tothelicense,itgrantsyoucertainprivilegesthatyou wouldnototherwisehave,suchastherighttocopythe book,ordownloadthedigitalversionfreeofchargefrom www.lightandmatter.com.Atyouroption,youmayalso copythisbookundertheGNUFreeDocumentation Licenseversion1.2,http://www.gnu.org/licenses/fdl.txt, withnoinvariantsections,nofront-covertexts,andno back-covertexts. ISBN0-9704670-1-X

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ToPaulHerrschaftandRichMuller.

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BriefContents 0IntroductionandReview19 1ScalingandOrder-of-MagnitudeEstimates43 MotioninOneDimension 2VelocityandRelativeMotion69 3AccelerationandFreeFall91 4ForceandMotion123 5AnalysisofForces145 MotioninThreeDimensions 6Newton'sLawsinThreeDimensions175 7Vectors187 8VectorsandMotion199 9CircularMotion215 10Gravity229

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Contents Preface..............15 0IntroductionandReview 0.1TheScienticMethod......19 0.2WhatIsPhysics?........22 Isolatedsystemsandreductionism,24. 0.3HowtoLearnPhysics......25 0.4Self-Evaluation.........27 0.5BasicsoftheMetricSystem....27 Themetricsystem,27.|Thesecond,28.| Themeter,29.|Thekilogram,30.| Combinationsofmetricunits,30. 0.6TheNewton,theMetricUnitofForce31 0.7LessCommonMetricPrexes...31 0.8ScienticNotation........32 0.9Conversions..........33 Shouldthatexponentbepositive,or negative?,34. 0.10SignicantFigures.......35 Summary.............38 Problems.............40 1ScalingandOrder-ofMagnitudeEstimates 1.1Introduction..........43 Areaandvolume,43. 1.2ScalingofAreaandVolume....45 Galileoonthebehaviorofnatureonlarge andsmallscales,46.|Scalingofareaand volumeforirregularlyshapedobjects,49. 1.3 ? ScalingAppliedtoBiology....53 Organismsofdierentsizeswiththesame shape,53.|Changesinshapetoaccommodatechangesinsize,55. 1.4Order-of-MagnitudeEstimates...57 Summary.............60 Problems.............61 IMotioninOneDimension 2VelocityandRelativeMotion 2.1TypesofMotion.........69 Rigid-bodymotiondistinguishedfrommotionthatchangesanobject'sshape, 69.|Center-of-massmotionasopposedto rotation,69.|Center-of-massmotionin onedimension,73. 2.2DescribingDistanceandTime...73 Apointintimeasopposedtoduration, 74.|Positionasopposedtochangein position,75.|Framesofreference,76. 2.3GraphsofMotion;Velocity....76 Motionwithconstantvelocity,76.| Motionwithchangingvelocity,77.| Conventionsaboutgraphing,78. 2.4ThePrincipleofInertia......80 Physicaleectsrelateonlytoachangein 10

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velocity,80.|Motionisrelative,81. 2.5AdditionofVelocities.......83 Additionofvelocitiestodescriberelative motion,83.|Negativevelocitiesinrelative motion,83. 2.6GraphsofVelocityVersusTime..85 2.7 R ApplicationsofCalculus....86 Summary.............87 Problems.............89 3AccelerationandFreeFall 3.1TheMotionofFallingObjects...91 Howthespeedofafallingobjectincreases withtime,93.|Acontradictioninaristotle'sreasoning,94.|Whatisgravity?,94. 3.2Acceleration..........95 Denitionofaccelerationforlinear v )]TJ/F20 9.9626 Tf 10.819 0 Td [(t graphs,95.|Theaccelerationofgravityis dierentindierentlocations.,96. 3.3PositiveandNegativeAcceleration.98 3.4VaryingAcceleration.......102 3.5TheAreaUndertheVelocity-Time Graph...............105 3.6AlgebraicResultsforConstant Acceleration............107 3.7 ? BiologicalEffectsofWeightlessness110 Spacesickness,110.|Eectsoflongspace missions,111.|Reproductioninspace, 112.|Simulatedgravity,112. 3.8 R ApplicationsofCalculus....112 Summary.............114 Problems.............115 4ForceandMotion 4.1Force.............124 Weneedonlyexplainchangesinmotion, notmotionitself.,124.|Motionchanges duetoaninteractionbetweentwoobjects., 125.|Forcescanallbemeasuredonthe samenumericalscale.,125.|Morethan oneforceonanobject,126.|Objectscan exertforcesoneachotheratadistance., 126.|Weight,126.|Positiveandnegative signsofforce,127. 4.2Newton'sFirstLaw.......127 Moregeneralcombinationsofforces,129. 4.3Newton'sSecondLaw......131 Ageneralization,132.|Therelationship betweenmassandweight,132. 4.4WhatForceIsNot........135 Forceisnotapropertyofoneobject., 135.|Forceisnotameasureofanobject's motion.,135.|Forceisnotenergy.,135.| Forceisnotstoredorusedup.,136.| Forcesneednotbeexertedbylivingthings ormachines.,136.|Aforceisthedirect causeofachangeinmotion.,136. 4.5InertialandNoninertialFramesof Reference.............137 Summary.............140 Problems.............141 5AnalysisofForces 5.1Newton'sThirdLaw.......145 Amnemonicforusingnewton'sthirdlaw correctly,147. 11

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5.2ClassicationandBehaviorofForces150 Normalforces,153.|Gravitationalforces, 153.|Staticandkineticfriction,153.| Fluidfriction,157. 5.3AnalysisofForces........158 5.4TransmissionofForcesbyLow-Mass Objects..............161 5.5ObjectsUnderStrain......163 5.6SimpleMachines:ThePulley...164 Summary.............166 Problems.............168 IIMotioninThreeDimensions 6Newton'sLawsinThree Dimensions 6.1ForcesHaveNoPerpendicular Effects..............175 Relationshiptorelativemotion,177. 6.2CoordinatesandComponents...179 Projectilesmovealongparabolas.,181. 6.3Newton'sLawsinThreeDimensions181 Summary.............183 Problems.............184 7Vectors 7.1VectorNotation.........187 Drawingvectorsasarrows,189. 7.2CalculationswithMagnitudeand Direction.............190 7.3TechniquesforAddingVectors..192 Additionofvectorsgiventheir components,192.|Additionofvectors giventheirmagnitudesanddirections, 192.|Graphicaladditionofvectors,192. 7.4 ? UnitVectorNotation......194 7.5 ? RotationalInvariance......194 Summary.............196 Problems.............197 8VectorsandMotion 8.1TheVelocityVector.......200 8.2TheAccelerationVector.....202 8.3TheForceVectorandSimple Machines.............205 8.4 R CalculusWithVectors.....206 Summary.............210 Problems.............211 12

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9CircularMotion 9.1ConceptualFrameworkforCircular Motion..............215 Circularmotiondoesnotproduceanoutwardforce,215.|Circularmotiondoesnot persistwithoutaforce,216.|Uniformand nonuniformcircularmotion,217.|Onlyan inwardforceisrequiredforuniformcircularmotion.,218.|Inuniformcircularmotion,theaccelerationvectorisinward,219. 9.2UniformCircularMotion.....221 9.3NonuniformCircularMotion....224 Summary.............225 Problems.............226 10Gravity 10.1Kepler'sLaws.........230 10.2Newton'sLawofGravity.....232 Thesun'sforceontheplanetsobeysan inversesquarelaw.,232.|Theforcesbetweenheavenlybodiesarethesametypeof forceasterrestrialgravity.,233.|Newton's lawofgravity,234. 10.3ApparentWeightlessness....237 10.4VectorAdditionofGravitational Forces..............238 10.5WeighingtheEarth.......241 10.6 ? EvidenceforRepulsiveGravity.243 Summary.............245 Problems.............247 Appendix1:Exercises 252 Appendix2:PhotoCredits 265 Appendix3:HintsandSolutions 266 13

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Preface WhyaNewPhysicsTextbook? WeAmericansassumethatoureconomicsystemwillalwaysscampertoprovideuswiththeproductswewant.Specialordersdon't upsetus!IwantmyMTV!Thetruthismorecomplicated,especiallyinoureducationsystem,whichispaidforbythestudents butcontrolledbytheprofessoriate.Witnesstheperversesuccess ofthebloatedsciencetextbook.ThenewspaperscontinuetocompareoursystemunfavorablytoJapaneseandEuropeaneducation, wheredepthisemphasizedoverbreadth,butwecan'tseemtocreateaphysicstextbookthatcoversamanageablenumberoftopics foraone-yearcourseandgiveshonestexplanationsofeverythingit toucheson. Thepublisherstrytopleaseeverybodybyincludingeveryimaginabletopicinthebook,butenduppleasingnobody.Thereiswide agreementamongphysicsteachersthatthetraditionalone-yearintroductorytextbookscannotinfactbetaughtinoneyear.One cannotsurgicallyremoveenoughmaterialandstillgracefullynavigatetherestofoneofthesekitchen-sinktextbooks.Whatisfar worseisthatthebooksaresocrammedwithtopicsthatnearlyall theexplanationiscutoutinordertokeepthepagecountbelow 1100.Vitalconceptslikeenergyareintroducedabruptlywithan equation,likearst-datekissthatcomesbeforehello." Themovementtoreformphysicstextsissteamingahead,but despiteexcellentbookssuchasHewitt's ConceptualPhysics fornonsciencemajorsandKnight's Physics:AContemporaryPerspective forstudentswhoknowcalculus,therehasbeenagapinphysics booksforlife-sciencemajorswhohaven'tlearnedcalculusorare learningitconcurrentlywithphysics.Thisbookismeanttoll thatgap. LearningtoHatePhysics? Whenyoureadamysterynovel,youknowinadvancewhatstructure toexpect:acrime,somedetectivework,andnallytheunmasking oftheevildoer.LikewisewhenCharlieParkerplaysablues,yourear expectstohearcertainlandmarksoftheformregardlessofhowwild someofhisnotesare.Surveysofphysicsstudentsusuallyshowthat theyhave worse attitudesaboutthesubjectafterinstructionthan before,andtheircommentsoftenboildowntoacomplaintthatthe personwhostrungthetopicstogetherhadnotlearnedwhatAgatha ChristieandCharlieParkerknewintuitivelyaboutformandstructure:studentsbecomeboredanddemoralizedbecausethemarch throughthetopics"lacksacoherentstoryline.Youarereadingthe rstvolumeoftheLightandMatterseriesofintroductoryphysics textbooks,andasimpliedbyitstitle,thestorylineoftheseries isbuiltaroundlightandmatter:howtheybehave,howtheyare Preface 15

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dierentfromeachother,and,attheendofthestory,howthey turnouttobesimilarinsomeverybizarreways.Hereisaguideto thestructureoftheone-yearcoursepresentedinthisseries: 1NewtonianPhysics Matter movesatconstantspeedina straightlineunlessaforceactsonit.Thisseemsintuitivelywrong onlybecausewetendtoforgettheroleoffrictionforces.Material objectscanexertforcesoneachother,eachchangingtheother's motion.Amoremassiveobjectchangesitsmotionmoreslowlyin responsetoagivenforce. 2ConservationLaws Newton'smatter-and-forcespictureof theuniverseisneasfarasitgoes,butitdoesn'tapplyto light whichisaformofpureenergywithoutmass.Amorepowerful world-view,applyingequallywelltobothlightandmatter,isprovidedbytheconservationlaws,forinstancethelawofconservation ofenergy,whichstatesthatenergycanneverbedestroyedorcreated butonlychangedfromoneformintoanother. 3VibrationsandWaves Light isawave.Welearnhowwaves travelthroughspace,passthrougheachother,speedup,slowdown, andarereected. 4ElectricityandMagnetism Matter ismadeoutofparticles suchaselectronsandprotons,whichareheldtogetherbyelectrical forces. Light isawavethatismadeoutofpatternsofelectricand magneticforce. 5Optics Devicessuchaseyeglassesandsearchlightsuse matter lensesandmirrorstomanipulate light 6TheModernRevolutioninPhysics Untilthetwentieth century,physiciststhoughtthat matter wasmadeoutofparticles and light waspurelyawavephenomenon.Wenowknowthatboth lightandmatteraremadeofbuildingblockswithacombinationof particleandwaveproperties.Intheprocessofunderstandingthis apparentcontradiction,wendthattheuniverseisamuchstranger placethanNewtonhadeverimagined,andalsolearnthebasisfor suchdevicesaslasersandcomputerchips. ANotetotheStudentTakingCalculusConcurrently Learningcalculusandphysicsconcurrentlyisanexcellentidea| it'snotacoincidencethattheinventorofcalculus,IsaacNewton, alsodiscoveredthelawsofmotion!Ifyouareworriedabouttaking thesetwodemandingcoursesatthesametime,letmereassureyou. Ithinkyouwillndthatphysicshelpsyouwithcalculuswhilecalculusdeepensandenhancesyourexperienceofphysics.Thisbook isdesignedtobeusedineitheranalgebra-basedphysicscourseor acalculus-basedphysicscoursethathascalculusasacorequisite. Thisnoteisaddressedtostudentsinthelattertypeofcourse. Artcriticsdiscusspaintingswitheachother,butwhenpainters 16

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gettogether,theytalkaboutbrushes.Artneedsbothawhy" andahow,"conceptsaswellastechnique.Justasitiseasierto enjoyanoilpaintingthantoproduceone,itiseasiertounderstand theconceptsofcalculusthantolearnthetechniquesofcalculus. Thisbookwillgenerallyteachyouthe concepts ofcalculusafew weeksbeforeyoulearntheminyourmathclass,butitdoesnot discussthe techniques ofcalculusatall.Therewillthusbeadelay ofafewweeksbetweenthetimewhenacalculusapplicationisrst pointedoutinthisbookandtherstoccurrenceofahomework problemthatrequirestherelevanttechnique.Thefollowingoutline showsatypicalrst-semestercalculuscurriculumside-by-sidewith thelistoftopicscoveredinthisbook,togiveyouaroughideaof whatcalculusyourphysicsinstructormightexpectyoutoknowat agivenpointinthesemester.Thesequenceofthecalculustopics istheonefollowedby CalculusofaSingleVariable ,2nded.,by Swokowski,Olinick,andPence. NewtonianPhysics 0-1introduction review 2-3velocityandacceleration limits 4-5Newton'slaws thederivativeconcept 6-8motionin3dimensions techniquesforndingderivatives;derivativesoftrigonometricfunctions 9circularmotion thechainrule 10gravity localmaximaandminima ConservationLaws 1-3energy concavityandthesecond derivative 4momentum 5angularmomentum theindeniteintegral VibrationsandWaves 1-2vibrations thedeniteintegral 3-4waves thefundamentaltheoremof calculus Preface 17

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TheMarsClimateOrbiterispreparedforitsmission.Thelaws ofphysicsarethesameeverywhere,evenonMars,sothe probecouldbedesignedbased onthelawsofphysicsasdiscoveredonearth.Thereisunfortunatelyanotherreasonwhythis spacecraftisrelevanttothetopicsofthischapter:itwasdestroyedattemptingtoenterMars' atmospherebecauseengineers atLockheedMartinforgottoconvertdataonenginethrustsfrom poundsintothemetricunitof forcenewtonsbeforegivingthe informationtoNASA.Conversionsareimportant! Chapter0 IntroductionandReview Ifyoudropyourshoeandacoinsidebyside,theyhitthegroundat thesametime.Whydoesn'ttheshoegetthererst,sincegravityis pullingharderonit?Howdoesthelensofyoureyework,andwhy doyoureye'smusclesneedtosquashitslensintodierentshapesin ordertofocusonobjectsnearbyorfaraway?Thesearethekinds ofquestionsthatphysicstriestoansweraboutthebehavioroflight andmatter,thetwothingsthattheuniverseismadeof. 0.1TheScienticMethod Untilveryrecentlyinhistory,noprogresswasmadeinanswering questionslikethese.Worsethanthat,the wrong answerswritten bythinkersliketheancientGreekphysicistAristotlewereaccepted withoutquestionforthousandsofyears.Whyisitthatscientic knowledgehasprogressedmoresincetheRenaissancethanithad inalltheprecedingmillenniasincethebeginningofrecordedhistory?Undoubtedlytheindustrialrevolutionispartoftheanswer. Buildingitscenterpiece,thesteamengine,requiredimprovedtech19

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b / Asatiricaldrawingofan alchemist'slaboratory.H.Cock, afteradrawingbyPeterBrueghel theElderthcentury. a / Scienceisacycleoftheoryandexperiment. niquesforpreciseconstructionandmeasurement.Earlyon,itwas consideredamajoradvancewhenEnglishmachineshopslearnedto buildpistonsandcylindersthatttogetherwithagapnarrower thanthethicknessofapenny.Butevenbeforetheindustrialrevolution,thepaceofdiscoveryhadpickedup,mainlybecauseofthe introductionofthemodernscienticmethod.Althoughitevolved overtime,mostscientiststodaywouldagreeonsomethinglikethe followinglistofthebasicprinciplesofthescienticmethod: Scienceisacycleoftheoryandexperiment. Scientictheoriesarecreatedtoexplaintheresultsofexperimentsthatwere createdundercertainconditions.Asuccessfultheorywillalsomake newpredictionsaboutnewexperimentsundernewconditions.Eventually,though,italwaysseemstohappenthatanewexperiment comesalong,showingthatundercertainconditionsthetheoryis notagoodapproximationorisnotvalidatall.Theballisthen backinthetheorists'court.Ifanexperimentdisagreeswiththe currenttheory,thetheoryhastobechanged,nottheexperiment. Theoriesshouldbothpredictandexplain. Therequirementof predictivepowermeansthatatheoryisonlymeaningfulifitpredicts somethingthatcanbecheckedagainstexperimentalmeasurements thatthetheoristdidnotalreadyhaveathand.Thatis,atheory shouldbetestable.Explanatoryvaluemeansthatmanyphenomena shouldbeaccountedforwithfewbasicprinciples.Ifyouanswer everywhy"questionwithbecausethat'sthewayitis,"thenyour theoryhasnoexplanatoryvalue.Collectinglotsofdatawithout beingabletondanybasicunderlyingprinciplesisnotscience. Experimentsshouldbereproducible. Anexperimentshould betreatedwithsuspicionifitonlyworksforoneperson,oronly inonepartoftheworld.Anyonewiththenecessaryskillsand equipmentshouldbeabletogetthesameresultsfromthesame experiment.Thisimpliesthatsciencetranscendsnationalandethnicboundaries;youcanbesurethatnobodyisdoingactualscience whoclaimsthattheirworkisAryan,notJewish,"Marxist,not bourgeois,"orChristian,notatheistic."Anexperimentcannotbe reproducedifitissecret,soscienceisnecessarilyapublicenterprise. Asanexampleofthecycleoftheoryandexperiment,avitalstep towardmodernchemistrywastheexperimentalobservationthatthe chemicalelementscouldnotbetransformedintoeachother,e.g., leadcouldnotbeturnedintogold.Thisledtothetheorythat chemicalreactionsconsistedofrearrangementsoftheelementsin dierentcombinations,withoutanychangeintheidentitiesofthe elementsthemselves.Thetheoryworkedforhundredsofyears,and wasconrmedexperimentallyoverawiderangeofpressuresand temperaturesandwithmanycombinationsofelements.Onlyin thetwentiethcenturydidwelearnthatoneelementcouldbetransformedintooneanotherundertheconditionsofextremelyhigh pressureandtemperatureexistinginanuclearbomborinsideastar. 20 Chapter0IntroductionandReview

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Thatobservationdidn'tcompletelyinvalidatetheoriginaltheoryof theimmutabilityoftheelements,butitshowedthatitwasonlyan approximation,validatordinarytemperaturesandpressures. self-checkA Apsychicconductsseancesinwhichthespiritsofthedeadspeakto theparticipants.Hesayshehasspecialpsychicpowersnotpossessed byotherpeople,whichallowhimtochannelthecommunicationswith thespirits.Whatpartofthescienticmethodisbeingviolatedhere? Answer,p.266 Thescienticmethodasdescribedhereisanidealization,and shouldnotbeunderstoodasasetprocedurefordoingscience.Scientistshaveasmanyweaknessesandcharacterawsasanyother group,anditisverycommonforscientiststotrytodiscreditother people'sexperimentswhentheresultsruncontrarytotheirownfavoredpointofview.Successfulsciencealsohasmoretodowith luck,intuition,andcreativitythanmostpeoplerealize,andthe restrictionsofthescienticmethoddonotstieindividualityand self-expressionanymorethanthefugueandsonataformsstied BachandHaydn.Thereisarecenttendencyamongsocialscientiststogoevenfurtherandtodenythatthescienticmethodeven exists,claimingthatscienceisnomorethananarbitrarysocialsystemthatdetermineswhatideastoacceptbasedonanin-group's criteria.Ithinkthat'sgoingtoofar.Ifscienceisanarbitrarysocial ritual,itwouldseemdiculttoexplainitseectivenessinbuilding suchusefulitemsasairplanes,CDplayers,andsewers.Ifalchemy andastrologywerenolessscienticintheirmethodsthanchemistryandastronomy,whatwasitthatkeptthemfromproducing anythinguseful? DiscussionQuestions Considerwhetherornotthescienticmethodisbeingappliedinthefollowingexamples.Ifthescienticmethodisnotbeingapplied,arethe peoplewhoseactionsarebeingdescribedperformingausefulhuman activity,albeitanunscienticone? A AcupunctureisatraditionalmedicaltechniqueofAsianoriginin whichsmallneedlesareinsertedinthepatient'sbodytorelievepain. Manydoctorstrainedinthewestconsideracupunctureunworthyofexperimentalstudybecauseifithadtherapeuticeffects,sucheffectscould notbeexplainedbytheirtheoriesofthenervoussystem.Whoisbeing morescientic,thewesternoreasternpractitioners? B Goethe,aGermanpoet,islesswellknownforhistheoryofcolor. Hepublishedabookonthesubject,inwhichhearguedthatscientic apparatusformeasuringandquantifyingcolor,suchasprisms,lenses andcoloredlters,couldnotgiveusfullinsightintotheultimatemeaning ofcolor,forinstancethecoldfeelingevokedbyblueandgreenorthe heroicsentimentsinspiredbyred.Washisworkscientic? C Achildaskswhythingsfalldown,andanadultanswersbecauseof gravity.TheancientGreekphilosopherAristotleexplainedthatrocksfell Section0.1TheScienticMethod 21

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becauseitwastheirnaturetoseekouttheirnaturalplace,incontactwith theearth.Aretheseexplanationsscientic? D Buddhismispartlyapsychologicalexplanationofhumansuffering, andpsychologyisofcourseascience.TheBuddhacouldbesaidto haveengagedinacycleoftheoryandexperiment,sinceheworkedby trialanderror,andevenlateinhislifeheaskedhisfollowerstochallenge hisideas.Buddhismcouldalsobeconsideredreproducible,sincethe Buddhatoldhisfollowerstheycouldndenlightenmentforthemselves iftheyfollowedacertaincourseofstudyanddiscipline.IsBuddhisma scienticpursuit? 0.2WhatIsPhysics? Givenforoneinstantanintelligencewhichcouldcomprehend alltheforcesbywhichnatureisanimatedandtherespective positionsofthethingswhichcomposeit...nothingwouldbe uncertain,andthefutureasthepastwouldbelaidoutbefore itseyes. PierreSimondeLaplace Physicsistheuseofthescienticmethodtondoutthebasic principlesgoverninglightandmatter,andtodiscovertheimplicationsofthoselaws.Partofwhatdistinguishesthemodernoutlook fromtheancientmind-setistheassumptionthattherearerulesby whichtheuniversefunctions,andthatthoselawscanbeatleastpartiallyunderstoodbyhumans.FromtheAgeofReasonthroughthe nineteenthcentury,manyscientistsbegantobeconvincedthatthe lawsofnaturenotonlycouldbeknownbut,asclaimedbyLaplace, thoselawscouldinprinciplebeusedtopredicteverythingabout theuniverse'sfutureifcompleteinformationwasavailableabout thepresentstateofalllightandmatter.Insubsequentsections, I'lldescribetwogeneraltypesoflimitationsonpredictionusingthe lawsofphysics,whichwereonlyrecognizedinthetwentiethcentury. Mattercanbedenedasanythingthatisaectedbygravity, i.e.,thathasweightorwouldhaveweightifitwasneartheEarth oranotherstarorplanetmassiveenoughtoproducemeasurable gravity.Lightcanbedenedasanythingthatcantravelfromone placetoanotherthroughemptyspaceandcaninuencematter,but hasnoweight.Forexample,sunlightcaninuenceyourbodyby heatingitorbydamagingyourDNAandgivingyouskincancer. Thephysicist'sdenitionoflightincludesavarietyofphenomena thatarenotvisibletotheeye,includingradiowaves,microwaves, x-rays,andgammarays.Thesearethecolors"oflightthatdonot happentofallwithinthenarrowviolet-to-redrangeoftherainbow thatwecansee. self-checkB Attheturnofthe20thcentury,astrangenewphenomenonwasdiscoveredinvacuumtubes:mysteriousraysofunknownoriginandnature. 22 Chapter0IntroductionandReview

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c / Thistelescopepictureshows twoimagesofthesamedistant object,anexotic,veryluminous objectcalledaquasar.Thisis interpretedasevidencethata massive,darkobject,possibly ablackhole,happenstobe betweenusandit.Lightraysthat wouldotherwisehavemissedthe earthoneithersidehavebeen bentbythedarkobject'sgravity sothattheyreachus.Theactual directiontothequasarispresumablyinthecenteroftheimage, butthelightalongthatcentralline doesn'tgettousbecauseitis absorbedbythedarkobject.The quasarisknownbyitscatalog number,MG1131+0456,ormore informallyasEinstein'sRing. Theseraysarethesameastheonesthatshootfromthebackofyour TV'spicturetubeandhitthefronttomakethepicture.Physicistsin 1895didn'thavethefaintestideawhattherayswere,sotheysimply namedthemcathoderays,afterthenamefortheelectricalcontact fromwhichtheysprang.Aercedebateraged,completewithnationalisticovertones,overwhethertherayswereaformoflightorofmatter. Whatwouldtheyhavehadtodoinordertosettletheissue? Answer,p.266 Manyphysicalphenomenaarenotthemselveslightormatter, butarepropertiesoflightormatterorinteractionsbetweenlight andmatter.Forinstance,motionisapropertyofalllightandsome matter,butitisnotitselflightormatter.Thepressurethatkeeps abicycletireblownupisaninteractionbetweentheairandthe tire.Pressureisnotaformofmatterinandofitself.Itisas muchapropertyofthetireasoftheair.Analogously,sisterhood andemploymentarerelationshipsamongpeoplebutarenotpeople themselves. Somethingsthatappearweightlessactuallydohaveweight,and soqualifyasmatter.Airhasweight,andisthusaformofmatter eventhoughacubicinchofairweighslessthanagrainofsand.A heliumballoonhasweight,butiskeptfromfallingbytheforceofthe surroundingmoredenseair,whichpushesuponit.Astronautsin orbitaroundtheEarthhaveweight,andarefallingalongacurved arc,buttheyaremovingsofastthatthecurvedarcoftheirfall isbroadenoughtocarrythemallthewayaroundtheEarthina circle.Theyperceivethemselvesasbeingweightlessbecausetheir spacecapsuleisfallingalongwiththem,andtheoorthereforedoes notpushupontheirfeet. OptionalTopic:ModernChangesintheDenitionofLightand Matter Einsteinpredictedasaconsequenceofhistheoryofrelativitythatlight wouldafterallbeaffectedbygravity,althoughtheeffectwouldbeextremelyweakundernormalconditions.Hispredictionwasborneout byobservationsofthebendingoflightraysfromstarsastheypassed closetothesunontheirwaytotheEarth.Einstein'stheoryalsoimplied theexistenceofblackholes,starssomassiveandcompactthattheir intensegravitywouldnotevenallowlighttoescape.Thesedaysthere isstrongevidencethatblackholesexist. Einstein'sinterpretationwasthatlightdoesn'treallyhavemass,but thatenergyisaffectedbygravityjustlikemassis.Theenergyinalight beamisequivalenttoacertainamountofmass,givenbythefamous equation E = mc 2 ,where c isthespeedoflight.Becausethespeed oflightissuchabignumber,alargeamountofenergyisequivalentto onlyaverysmallamountofmass,sothegravitationalforceonalight raycanbeignoredformostpracticalpurposes. Thereishoweveramoresatisfactoryandfundamentaldistinction betweenlightandmatter,whichshouldbeunderstandabletoyouifyou havehadachemistrycourse.Inchemistry,onelearnsthatelectrons Section0.2WhatIsPhysics? 23

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d / Reductionism. obeythePauliexclusionprinciple,whichforbidsmorethanoneelectron fromoccupyingthesameorbitaliftheyhavethesamespin.ThePauli exclusionprincipleisobeyedbythesubatomicparticlesofwhichmatter iscomposed,butdisobeyedbytheparticles,calledphotons,ofwhicha beamoflightismade. Einstein'stheoryofrelativityisdiscussedmorefullyinbook6ofthis series. Theboundarybetweenphysicsandtheothersciencesisnot alwaysclear.Forinstance,chemistsstudyatomsandmolecules, whicharewhatmatterisbuiltfrom,andtherearesomescientists whowouldbeequallywillingtocallthemselvesphysicalchemists orchemicalphysicists.Itmightseemthatthedistinctionbetween physicsandbiologywouldbeclearer,sincephysicsseemstodeal withinanimateobjects.Infact,almostallphysicistswouldagree thatthebasiclawsofphysicsthatapplytomoleculesinatesttube workequallywellforthecombinationofmoleculesthatconstitutes abacterium.Somemightbelievethatsomethingmorehappensin themindsofhumans,oreventhoseofcatsanddogs.Whatdierentiatesphysicsfrombiologyisthatmanyofthescientictheories thatdescribelivingthings,whileultimatelyresultingfromthefundamentallawsofphysics,cannotberigorouslyderivedfromphysical principles. Isolatedsystemsandreductionism Toavoidhavingtostudyeverythingatonce,scientistsisolatethe thingstheyaretryingtostudy.Forinstance,aphysicistwhowants tostudythemotionofarotatinggyroscopewouldprobablyprefer thatitbeisolatedfromvibrationsandaircurrents.Eveninbiology, whereeldworkisindispensableforunderstandinghowlivingthings relatetotheirentireenvironment,itisinterestingtonotethevital historicalroleplayedbyDarwin'sstudyoftheGalapagosIslands, whichwereconvenientlyisolatedfromtherestoftheworld.Any partoftheuniversethatisconsideredapartfromtherestcanbe calledasystem." Physicshashadsomeofitsgreatestsuccessesbycarryingthis processofisolationtoextremes,subdividingtheuniverseintosmaller andsmallerparts.Mattercanbedividedintoatoms,andthebehaviorofindividualatomscanbestudied.Atomscanbesplitapart intotheirconstituentneutrons,protonsandelectrons.Protonsand neutronsappeartobemadeoutofevensmallerparticlescalled quarks,andtherehaveevenbeensomeclaimsofexperimentalevidencethatquarkshavesmallerpartsinsidethem.Thismethod ofsplittingthingsintosmallerandsmallerpartsandstudyinghow thosepartsinuenceeachotheriscalledreductionism.Thehopeis thattheseeminglycomplexrulesgoverningthelargerunitscanbe 24 Chapter0IntroductionandReview

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betterunderstoodintermsofsimplerrulesgoverningthesmaller units.Toappreciatewhatreductionismhasdoneforscience,itis onlynecessarytoexaminea19th-centurychemistrytextbook.At thattime,theexistenceofatomswasstilldoubtedbysome,electronswerenotevensuspectedtoexist,andalmostnothingwas understoodofwhatbasicrulesgovernedthewayatomsinteracted witheachotherinchemicalreactions.Studentshadtomemorize longlistsofchemicalsandtheirreactions,andtherewasnowayto understandanyofitsystematically.Today,thestudentonlyneeds torememberasmallsetofrulesabouthowatomsinteract,forinstancethatatomsofoneelementcannotbeconvertedintoanother viachemicalreactions,orthatatomsfromtherightsideoftheperiodictabletendtoformstrongbondswithatomsfromtheleft side. DiscussionQuestions A I'vesuggestedreplacingtheordinarydictionarydenitionoflight withamoretechnical,morepreciseonethatinvolvesweightlessness.It's stillpossible,though,thatthestuffalightbulbmakes,ordinarilycalled light,doeshavesomesmallamountofweight.Suggestanexperiment toattempttomeasurewhetheritdoes. B Heatisweightlessi.e.,anobjectbecomesnoheavierwhenheated, andcantravelacrossanemptyroomfromthereplacetoyourskin, whereitinuencesyoubyheatingyou.Shouldheatthereforebeconsideredaformoflightbyourdenition?Whyorwhynot? C Similarly,shouldsoundbeconsideredaformoflight? 0.3HowtoLearnPhysics Forasknowledgesarenowdelivered,thereisakindofcontractoferrorbetweenthedelivererandthereceiver;forhe thatdeliverethknowledgedesirethtodeliveritinsuchaform asmaybebestbelieved,andnotasmaybebestexamined; andhethatreceivethknowledgedesirethratherpresentsatisfactionthanexpectantinquiry. FrancisBacon Manystudentsapproachasciencecoursewiththeideathatthey cansucceedbymemorizingtheformulas,sothatwhenaproblem isassignedonthehomeworkoranexam,theywillbeabletoplug numbersintotheformulaandgetanumericalresultontheircalculator.Wrong!That'snotwhatlearningscienceisabout!There isabigdierencebetweenmemorizingformulasandunderstanding concepts.Tostartwith,dierentformulasmayapplyindierent situations.Oneequationmightrepresentadenition,whichisalwaystrue.Anothermightbeaveryspecicequationforthespeed Section0.3HowtoLearnPhysics 25

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ofanobjectslidingdownaninclinedplane,whichwouldnotbetrue iftheobjectwasarockdriftingdowntothebottomoftheocean. Ifyoudon'tworktounderstandphysicsonaconceptuallevel,you won'tknowwhichformulascanbeusedwhen. Moststudentstakingcollegesciencecoursesforthersttime alsohaveverylittleexperiencewithinterpretingthemeaningofan equation.Considertheequation w = A=h relatingthewidthofa rectangletoitsheightandarea.Astudentwhohasnotdeveloped skillatinterpretationmightviewthisasyetanotherequationto memorizeandplugintowhenneeded.Aslightlymoresavvystudentmightrealizethatitissimplythefamiliarformula A = wh inadierentform.Whenaskedwhetherarectanglewouldhave agreaterorsmallerwidththananotherwiththesameareabut asmallerheight,theunsophisticatedstudentmightbeataloss, nothavinganynumberstopluginonacalculator.Themoreexperiencedstudentwouldknowhowtoreasonaboutanequation involvingdivision|if h issmaller,and A staysthesame,then w mustbebigger.Often,studentsfailtorecognizeasequenceofequationsasaderivationleadingtoanalresult,sotheythinkallthe intermediatestepsareequallyimportantformulasthattheyshould memorize. Whenlearninganysubjectatall,itisimportanttobecomeas activelyinvolvedaspossible,ratherthantryingtoreadthrough alltheinformationquicklywithoutthinkingaboutit.Itisagood ideatoreadandthinkaboutthequestionsposedattheendofeach sectionofthesenotesasyouencounterthem,sothatyouknowyou haveunderstoodwhatyouwerereading. Manystudents'dicultiesinphysicsboildownmainlytodicultieswithmath.Supposeyoufeelcondentthatyouhaveenough mathematicalpreparationtosucceedinthiscourse,butyouare havingtroublewithafewspecicthings.Insomeareas,thebrief reviewgiveninthischaptermaybesucient,butinotherareas itprobablywillnot.Onceyouidentifytheareasofmathinwhich youarehavingproblems,gethelpinthoseareas.Don'tlimpalong throughthewholecoursewithavaguefeelingofdreadaboutsomethinglikescienticnotation.Theproblemwillnotgoawayifyou ignoreit.Thesameappliestoessentialmathematicalskillsthatyou arelearninginthiscourseforthersttime,suchasvectoraddition. Sometimesstudentstellmetheykeeptryingtounderstanda certaintopicinthebook,anditjustdoesn'tmakesense.Theworst thingyoucanpossiblydointhatsituationistokeeponstaring atthesamepage.Everytextbookexplainscertainthingsbadly| evenmine!|sothebestthingtodointhissituationistolook atadierentbook.Insteadofcollegetextbooksaimedatthesame mathematicallevelasthecourseyou'retaking,youmayinsome casesndthathighschoolbooksorbooksatalowermathlevel 26 Chapter0IntroductionandReview

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giveclearerexplanations. Finally,whenreviewingforanexam,don'tsimplyreadback overthetextandyourlecturenotes.Instead,trytouseanactive methodofreviewing,forinstancebydiscussingsomeofthediscussionquestionswithanotherstudent,ordoinghomeworkproblems youhadn'tdonethersttime. 0.4Self-Evaluation Theintroductorypartofabooklikethisishardtowrite,because everystudentarrivesatthisstartingpointwithadierentpreparation.OnestudentmayhavegrownupoutsidetheU.S.andsomay becompletelycomfortablewiththemetricsystem,butmayhave hadanalgebracourseinwhichtheinstructorpassedtooquickly overscienticnotation.Anotherstudentmayhavealreadytaken calculus,butmayhaveneverlearnedthemetricsystem.Thefollowingself-evaluationisachecklisttohelpyougureoutwhatyou needtostudytobepreparedfortherestofthecourse. Ifyoudisagreewiththisstatement... youshouldstudythissection: Iamfamiliarwiththebasicmetric unitsofmeters,kilograms,andseconds,andthemostcommonmetric prexes:milli-m,kilo-k,and centi-c. section0.5BasicoftheMetricSystem Iknowaboutthenewton,aunitof force section0.6Thenewton,theMetric UnitofForce Iamfamiliarwiththeselesscommonmetricprexes:mega-M, micro,andnano-n. section0.7LessCommonMetric Prexes Iamcomfortablewithscienticnotation. section0.8ScienticNotation Icancondentlydometricconversions. section0.9Conversions Iunderstandthepurposeanduseof signicantgures. section0.10SignicantFigures Itwouldn'thurtyoutoskimthesectionsyouthinkyoualready knowabout,andtodotheself-checksinthosesections. 0.5BasicsoftheMetricSystem Themetricsystem Unitswerenotstandardizeduntilfairlyrecentlyinhistory,so whenthephysicistIsaacNewtongavetheresultofanexperiment withapendulum,hehadtospecifynotjustthatthestringwas37 7 = 8 incheslongbutthatitwas 7 = 8 Londonincheslong."The Section0.4Self-Evaluation 27

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inchasdenedinYorkshirewouldhavebeendierent.Evenafter theBritishEmpirestandardizeditsunits,itwasstillveryinconvenienttodocalculationsinvolvingmoney,volume,distance,time,or weight,becauseofalltheoddconversionfactors,like16ouncesin apound,and5280feetinamile.Throughthenineteenthcentury, schoolchildrensquanderedmostoftheirmathematicaleducationin preparingtodocalculationssuchasmakingchangewhenacustomer inashopoeredaone-crownnoteforabookcostingtwopounds, thirteenshillingsandtuppence.Thedollarhasalwaysbeendecimal, andBritishmoneywentdecimaldecadesago,buttheUnitedStates isstillsaddledwiththeantiquatedsystemoffeet,inches,pounds, ouncesandsoon. EverycountryintheworldbesidestheU.S.hasadoptedasystemofunitsknowninEnglishasthemetricsystem."Thissystem isentirelydecimal,thankstothesameeminentlylogicalpeoplewho broughtabouttheFrenchRevolution.IndeferencetoFrance,the system'socialnameistheSystemeInternational,orSI,meaning InternationalSystem.ThephraseSIsystem"isthereforeredundant. ThewonderfulthingabouttheSIisthatpeoplewholivein countriesmoremodernthanoursdonotneedtomemorizehow manyouncesthereareinapound,howmanycupsinapint,how manyfeetinamile,etc.Thewholesystemworkswithasingle, consistentsetofprexesderivedfromGreekthatmodifythebasic units.Eachprexstandsforapoweroften,andhasanabbreviation thatcanbecombinedwiththesymbolfortheunit.Forinstance, themeterisaunitofdistance.Theprexkilo-standsfor10 3 ,soa kilometer,1km,isathousandmeters. Thebasicunitsofthemetricsystemarethemeterfordistance, thesecondfortime,andthegramformass. Thefollowingarethemostcommonmetricprexes.Youshould memorizethem. prexmeaningexample kilo-k10 3 60kg=aperson'smass centi-c10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 28cm=heightofapieceofpaper milli-m10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 1ms=timeforonevibrationofaguitar stringplayingthenoteD Theprexcenti-,meaning10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 ,isonlyusedinthecentimeter; ahundredthofagramwouldnotbewrittenas1cgbutas10mg. Thecenti-prexcanbeeasilyrememberedbecauseacentis10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 dollars.TheocialSIabbreviationforsecondsiss"notsec" andgramsareg"notgm". Thesecond Thesunstoodstillandthemoonhalteduntilthenationhad takenvengeanceonitsenemies... 28 Chapter0IntroductionandReview

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e / PopeGregorycreatedour modernGregoriancalendar,with itssystemofleapyears,tomake thelengthofthecalendaryear matchthelengthofthecycle ofseasons.Notuntil1752did ProtestantEnglandswitchedto thenewcalendar.Someless educatedcitizensbelievedthat theshorteningofthemonthby elevendayswouldshortentheir livesbythesameinterval.Inthis illustrationbyWilliamHogarth, theleaetlyingontheground reads,Giveusourelevendays. Joshua10:12-14 Absolute,true,andmathematicaltime,ofitself,andfromits ownnature,owsequablywithoutrelationtoanythingexternal... IsaacNewton WhenIstatedbrieyabovethatthesecondwasaunitoftime, itmaynothaveoccurredtoyouthatthiswasnotreallymuchof adenition.Thetwoquotesabovearemeanttodemonstratehow muchroomforconfusionexistsamongpeoplewhoseemtomeanthe samethingbyawordsuchastime."Therstquotehasbeeninterpretedbysomebiblicalscholarsasindicatinganancientbeliefthat themotionofthesunacrosstheskywasnotjustsomethingthat occurredwiththepassageoftimebutthatthesunactuallycaused timetopassbyitsmotion,sothatfreezingitintheskywouldhave somekindofasupernaturaldeceleratingeectoneveryoneexcept theHebrewsoldiers.Manyancientculturesalsoconceivedoftime ascyclical,ratherthanproceedingalongastraightlineasin1998, 1999,2000,2001,...Thesecondquote,fromarelativelymodern physicist,maysoundalotmorescientic,butmostphysiciststodaywouldconsiderituselessasadenitionoftime.Today,the physicalsciencesarebasedonoperationaldenitions,whichmeans denitionsthatspellouttheactualstepsoperationsrequiredto measuresomethingnumerically. Nowinanerawhenourtoasters,pens,andcoeepotstellusthe time,itisfarfromobvioustomostpeoplewhatisthefundamental operationaldenitionoftime.Untilrecently,thehour,minute,and secondweredenedoperationallyintermsofthetimerequiredfor theearthtorotateaboutitsaxis.Unfortunately,theEarth'srotationisslowingdownslightly,andby1967thiswasbecomingan issueinscienticexperimentsrequiringprecisetimemeasurements. Thesecondwasthereforeredenedasthetimerequiredforacertainnumberofvibrationsofthelightwavesemittedbyacesium atomsinalampconstructedlikeafamiliarneonsignbutwiththe neonreplacedbycesium.Thenewdenitionnotonlypromisesto stayconstantindenitely,butforscientistsisamoreconvenient wayofcalibratingaclockthanhavingtocarryoutastronomical measurements. self-checkC Whatisapossibleoperationaldenitionofhowstrongapersonis? Answer,p.266 Themeter TheFrenchoriginallydenedthemeteras10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(7 timesthedistancefromtheequatortothenorthpole,asmeasuredthroughParis ofcourse.Evenifthedenitionwasoperational,theoperationof travelingtothenorthpoleandlayingasurveyingchainbehindyou Section0.5BasicsoftheMetricSystem 29

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f / Theoriginaldenitionof themeter. wasnotonethatmostworkingscientistswantedtocarryout.Fairly soon,astandardwascreatedintheformofametalbarwithtwo scratchesonit.Thisdenitionpersisteduntil1960,whenthemeter wasredenedasthedistancetraveledbylightinavacuumovera periodof/299792458seconds. Thekilogram ThethirdbaseunitoftheSIisthekilogram,aunitofmass. Massisintendedtobeameasureoftheamountofasubstance, butthatisnotanoperationaldenition.Bathroomscalesworkby measuringourplanet'sgravitationalattractionfortheobjectbeing weighed,butusingthattypeofscaletodenemassoperationally wouldbeundesirablebecausegravityvariesinstrengthfromplace toplaceontheearth. There'sasurprisingamountofdisagreementamongphysicstextbooksabouthowmassshouldbedened,buthere'showit'sactually handledbythefewworkingphysicistswhospecializeinultra-highprecisionmeasurements.TheymaintainaphysicalobjectinParis, whichisthestandardkilogram,acylindermadeofplatinum-iridium alloy.Duplicatesarecheckedagainstthismotherofallkilograms byputtingtheoriginalandthecopyonthetwooppositepansofa balance.Althoughthismethodofcomparisondependsongravity, theproblemsassociatedwithdierencesingravityindierentgeographicallocationsarebypassed,becausethetwoobjectsarebeing comparedinthesameplace.Theduplicatescanthenberemoved fromtheParisiankilogramshrineandtransportedelsewhereinthe world. Combinationsofmetricunits Justaboutanythingyouwanttomeasurecanbemeasuredwith somecombinationofmeters,kilograms,andseconds.Speedcanbe measuredinm/s,volumeinm 3 ,anddensityinkg = m 3 .Partofwhat makestheSIgreatisthisbasicsimplicity.Nomorefunnyunitslike acordofwood,aboltofcloth,orajiggerofwhiskey.Nomore liquidanddrymeasure.Justasimple,consistentsetofunits.The SImeasuresputtogetherfrommeters,kilograms,andsecondsmake upthemkssystem.Forexample,themksunitofspeedism/s,not km/hr. DiscussionQuestion A IsaacNewtonwrote,...thenaturaldaysaretrulyunequal,though theyarecommonlyconsideredasequal,andusedforameasureof time...Itmaybethatthereisnosuchthingasanequablemotion,whereby timemaybeaccuratelymeasured.Allmotionsmaybeacceleratedorretarded...Newtonwasright.Eventhemoderndenitionofthesecond intermsoflightemittedbycesiumatomsissubjecttovariation.Forinstance,magneticeldscouldcausethecesiumatomstoemitlightwith aslightlydifferentrateofvibration.Whatmakesusthink,though,thata pendulumclockismoreaccuratethanasundial,orthatacesiumatom 30 Chapter0IntroductionandReview

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g / Thisisamnemonicto helpyourememberthemostimportantmetricprexes.Theword littleistoremindyouthatthe liststartswiththeprexesused forsmallquantitiesandbuilds upward.Theexponentchanges by3,exceptthatofcoursethat wedonotneedaspecialprex for10 0 ,whichequalsone. isamoreaccuratetimekeeperthanapendulumclock?Thatis,howcan onetestexperimentallyhowtheaccuraciesofdifferenttimestandards compare? 0.6TheNewton,theMetricUnitofForce Aforceisapushorapull,ormoregenerallyanythingthatcan changeanobject'sspeedordirectionofmotion.Aforceisrequired tostartacarmoving,toslowdownabaseballplayerslidinginto homebase,ortomakeanairplaneturn.Forcesmayfailtochange anobject'smotioniftheyarecanceledbyotherforces,e.g.,the forceofgravitypullingyoudownrightnowisbeingcanceledbythe forceofthechairpushinguponyou.Themetricunitofforceis theNewton,denedastheforcewhich,ifappliedforonesecond, willcausea1-kilogramobjectstartingfromresttoreachaspeedof 1m/s.Laterchapterswilldiscusstheforceconceptinmoredetail. Infact,thisentirebookisabouttherelationshipbetweenforceand motion. Insection0.5,Igaveagravitationaldenitionofmass,butby deninganumericalscaleofforce,wecanalsoturnaroundanddeneascaleofmasswithoutreferencetogravity.Forinstance,ifa forceoftwoNewtonsisrequiredtoaccelerateacertainobjectfrom restto1m/sin1s,thenthatobjectmusthaveamassof2kg. Fromthispointofview,masscharacterizesanobject'sresistance toachangeinitsmotion,whichwecallinertiaorinertialmass. Althoughthereisnofundamentalreasonwhyanobject'sresistance toachangeinitsmotionmustberelatedtohowstronglygravity aectsit,carefulandpreciseexperimentshaveshownthattheinertialdenitionandthegravitationaldenitionofmassarehighly consistentforavarietyofobjects.Itthereforedoesn'treallymatter foranypracticalpurposewhichdenitiononeadopts. DiscussionQuestion A Spendingalongtimeinweightlessnessisunhealthy.Oneofthe mostimportantnegativeeffectsexperiencedbyastronautsisalossof muscleandbonemass.Sinceanordinaryscalewon'tworkforanastronautinorbit,whatisapossiblewayofmonitoringthischangeinmass? Measuringtheastronaut'swaistorbicepswithameasuringtapeisnot goodenough,becauseitdoesn'ttellanythingaboutbonemass,orabout thereplacementofmusclewithfat. 0.7LessCommonMetricPrexes Thefollowingarethreemetricprexeswhich,whilelesscommon thantheonesdiscussedpreviously,arewellworthmemorizing. Section0.6TheNewton,theMetricUnitofForce 31

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prexmeaningexample mega-M10 6 6.4Mm=radiusoftheearth micro10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(6 10 m=sizeofawhitebloodcell nano-n10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(9 0.154nm=distancebetweencarbon nucleiinanethanemolecule NotethattheabbreviationformicroistheGreeklettermu, |acommonmistakeistoconfuseitwithmmilliorMmega. Thereareotherprexesevenlesscommon,usedforextremely largeandsmallquantities.Forinstance,1femtometer=10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(15 mis aconvenientunitofdistanceinnuclearphysics,and1gigabyte= 10 9 bytesisusedforcomputers'harddisks.TheinternationalcommitteethatmakesdecisionsabouttheSIhasrecentlyevenadded somenewprexesthatsoundlikejokes,e.g.,1yoctogram=10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(24 g isabouthalfthemassofaproton.Intheimmediatefuture,however,you'reunlikelytoseeprexeslikeyocto-"andzepto-"used exceptperhapsintriviacontestsatscience-ctionconventionsor othergeekfests. self-checkD Supposeyoucouldslowdowntimesothataccordingtoyourperception, abeamoflightwouldmoveacrossaroomatthespeedofaslowwalk. Ifyouperceivedananosecondasifitwasasecond,howwouldyou perceiveamicrosecond? Answer,p.266 0.8ScienticNotation Mostoftheinterestingphenomenainouruniversearenotonthe humanscale.Itwouldtakeabout1,000,000,000,000,000,000,000 bacteriatoequalthemassofahumanbody.Whenthephysicist ThomasYoungdiscoveredthatlightwasawave,itwasbackinthe badolddaysbeforescienticnotation,andhewasobligedtowrite thatthetimerequiredforonevibrationofthewavewas1/500of amillionthofamillionthofasecond.Scienticnotationisaless awkwardwaytowriteverylargeandverysmallnumberssuchas these.Here'saquickreview. Scienticnotationmeanswritinganumberintermsofaproduct ofsomethingfrom1to10andsomethingelsethatisapoweroften. Forinstance, 32=3.2 10 1 320=3.2 10 2 3200=3.2 10 3 ::: Eachnumberistentimesbiggerthanthepreviousone. Since10 1 istentimessmallerthan10 2 ,itmakessensetouse thenotation10 0 tostandforone,thenumberthatisinturnten timessmallerthan10 1 .Continuingon,wecanwrite10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 tostand 32 Chapter0IntroductionandReview

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for0.1,thenumbertentimessmallerthan10 0 .Negativeexponents areusedforsmallnumbers: 3.2=3.2 10 0 0.32=3.2 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 0.032=3.2 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 ::: Acommonsourceofconfusionisthenotationusedonthedisplaysofmanycalculators.Examples: 3.2 10 6 writtennotation 3.2E+6notationonsomecalculators 3.2 6 notationonsomeothercalculators Thelastexampleisparticularlyunfortunate,because3.2 6 really standsforthenumber3.2 3.2 3.2 3.2 3.2 3.2=1074,a totallydierentnumberfrom3.2 10 6 =3200000.Thecalculator notationshouldneverbeusedinwriting.It'sjustawayforthe manufacturertosavemoneybymakingasimplerdisplay. self-checkE Astudentlearnsthat10 4 bacteria,standinginlinetoregisterforclasses atParameciumCommunityCollege,wouldformaqueueofthissize: Thestudentconcludesthat10 2 bacteriawouldformalineofthislength: Whyisthestudentincorrect? Answer,p.266 0.9Conversions Isuggestyouavoidmemorizinglotsofconversionfactorsbetween SIunitsandU.S.units,buttwothatdocomeinhandyare: 1inch=2.54cm AnobjectwithaweightonEarthof2.2pounds-forcehasa massof1kg. Therstoneisthepresentdenitionoftheinch,soit'sexact.The secondoneisnotexact,butisgoodenoughformostpurposes.U.S. unitsofforceandmassareconfusing,soit'sagoodthingthey're notusedinscience.InU.S.units,theunitofforceisthepoundforce,andthebestunittouseformassistheslug,whichisabout 14.6kg. Moreimportantthanmemorizingconversionfactorsisunderstandingtherightmethodfordoingconversions.Evenwithinthe Section0.9Conversions 33

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SI,youmayneedtoconvert,say,fromgramstokilograms.Dierentpeoplehavedierentwaysofthinkingaboutconversions,but themethodI'lldescribehereissystematicandeasytounderstand. Theideaisthatif1kgand1000grepresentthesamemass,then wecanconsiderafractionlike 10 3 g 1kg tobeawayofexpressingthenumberone.Thismaybotheryou.For instance,ifyoutype1000/1intoyourcalculator,youwillget1000, notone.Again,dierentpeoplehavedierentwaysofthinking aboutit,butthejusticationisthatithelpsustodoconversions, anditworks!Nowifwewanttoconvert0.7kgtounitsofgrams, wecanmultiplykgbythenumberone: 0.7kg 10 3 g 1kg Ifyou'rewillingtotreatsymbolssuchaskg"asiftheywerevariablesasusedinalgebrawhichthey'rereallynot,youcanthen cancelthekgontopwiththekgonthebottom,resultingin 0.7 )]TJ 3.242 3.238 Td [()]TJ/F15 10.9091 Tf -2.246 0.635 Td [(kg 10 3 g 1 )]TJ 3.242 3.238 Td [()]TJ/F15 10.9091 Tf -2.245 0.635 Td [(kg =700g. Toconvertgramstokilograms,youwouldsimplyipthefraction upsidedown. Oneadvantageofthismethodisthatitcaneasilybeappliedto aseriesofconversions.Forinstance,toconvertoneyeartounitsof seconds, 1 year 365 days 1 year 24 hours 1 day 60 min 1 hour 60s 1 min = =3.15 10 7 s. Shouldthatexponentbepositive,ornegative? Acommonmistakeistowritetheconversionfractionincorrectly. Forinstancethefraction 10 3 kg 1g incorrect doesnotequalone,because10 3 kgisthemassofacar,and1gis themassofaraisin.Onecorrectwayofsettinguptheconversion factorwouldbe 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 kg 1g correct. 34 Chapter0IntroductionandReview

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Youcanusuallydetectsuchamistakeifyoutakethetimetocheck youranswerandseeifitisreasonable. Ifcommonsensedoesn'truleouteitherapositiveoranegative exponent,here'sanotherwaytomakesureyougetitright.There arebigprexesandsmallprexes: bigprexes:kM smallprexes:m n It'snothardtokeepstraightwhicharewhich,sincemega"and micro"areevocative,andit'seasytorememberthatakilometer isbiggerthanameterandamillimeterissmaller.Intheexample above,wewantthetopofthefractiontobethesameasthebottom. Since k isabigprex,weneedto compensate byputtingasmall numberlike10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 infrontofit,notabignumberlike10 3 Solvedproblem:asimpleconversionpage40,problem6 Solvedproblem:thegeometricmeanpage41,problem8 DiscussionQuestion A Eachofthefollowingconversionscontainsanerror.Ineachcase, explainwhattheerroris. a1000kg 1kg 1000g =1g b50m 1cm 100m =0.5cm cNanois10 )]TJ/F39 6.9738 Tf 6.227 0 Td [(9 ,sothereare10 )]TJ/F39 6.9738 Tf 6.227 0 Td [(9 nminameter. dMicrois10 )]TJ/F39 6.9738 Tf 6.227 0 Td [(6 ,so1kgis10 6 g. 0.10SignicantFigures Anengineerisdesigningacarengine,andhasbeentoldthatthe diameterofthepistonswhicharebeingdesignedbysomeoneelse is5cm.Heknowsthat0.02cmofclearanceisrequiredforapiston ofthissize,sohedesignsthecylindertohaveaninsidediameterof 5.04cm.Luckily,hissupervisorcatcheshismistakebeforethecar goesintoproduction.Sheexplainshiserrortohim,andmentally putshiminthedonotpromote"category. Whatwashismistake?Thepersonwhotoldhimthepistons were5cmindiameterwaswisetothewaysofsignicantgures, aswashisboss,whoexplainedtohimthatheneededtogoback andgetamoreaccuratenumberforthediameterofthepistons. Thatpersonsaidcm"ratherthan.00cm"specicallytoavoid creatingtheimpressionthatthenumberwasextremelyaccurate.In reality,thepistons'diameterwas5.13cm.Theywouldneverhave tinthe5.04-cmcylinders. Thenumberofdigitsofaccuracyinanumberisreferredtoas thenumberofsignicantgures,orsiggs"forshort.Asinthe Section0.10SignicantFigures 35

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exampleabove,siggsprovideawayofshowingtheaccuracyofa number.Inmostcases,theresultofacalculationinvolvingseveral piecesofdatacanbenomoreaccuratethantheleastaccuratepiece ofdata.Inotherwords,garbagein,garbageout."Sincethe5 cmdiameterofthepistonswasnotveryaccurate,theresultofthe engineer'scalculation,5.04cm,wasreallynotasaccurateashe thought.Ingeneral,yourresultshouldnothavemorethanthe numberofsiggsintheleastaccuratepieceofdatayoustarted with.Thecalculationaboveshouldhavebeendoneasfollows: 5cmsigg +0.04cmsigg =5cmroundedoto1sigg Thefactthatthenalresultonlyhasonesignicantgurethen alertsyoutothefactthattheresultisnotveryaccurate,andwould notbeappropriateforuseindesigningtheengine. Notethattheleadingzeroesinthenumber0.04donotcount assignicantgures,becausetheyareonlyplaceholders.Onthe otherhand,anumbersuchas50cmisambiguous|thezerocould beintendedasasignicantgure,oritmightjustbethereasa placeholder.Theambiguityinvolvingtrailingzeroescanbeavoided byusingscienticnotation,inwhich5 10 1 cmwouldimplyone siggofaccuracy,while5.0 10 1 cmwouldimplytwosiggs. 36 Chapter0IntroductionandReview

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self-checkF ThefollowingquoteistakenfromaneditorialbyNorimitsuOnishiinthe NewYorkTimes,August18,2002. ConsiderNigeria.EveryoneagreesitisAfrica'smostpopulous nation.Butwhatisitspopulation?TheUnitedNationssays 114million;theStateDepartment,120million.TheWorldBank says126.9million,whiletheCentralIntelligenceAgencyputsit at126,635,626. Whatshouldbotheryouaboutthis? Answer,p.266 Dealingcorrectlywithsignicantgurescansaveyoutime!Often,studentscopydownnumbersfromtheircalculatorswitheight signicantguresofprecision,thentypethembackinforalater calculation.That'sawasteoftime,unlessyouroriginaldatahad thatkindofincredibleprecision. Therulesaboutsignicantguresareonlyrulesofthumb,and arenotasubstituteforcarefulthinking.Forinstance,$20.00+ $0.05is$20.05.Itneednotandshouldnotberoundedoto$20. Ingeneral,thesiggrulesworkbestformultiplicationanddivision, andwealsoapplythemwhendoingacomplicatedcalculationthat involvesmanytypesofoperations.Forsimpleadditionandsubtraction,itmakesmoresensetomaintainaxednumberofdigitsafter thedecimalpoint. Whenindoubt,don'tusethesiggrulesatall.Instead,intentionallychangeonepieceofyourinitialdatabythemaximum amountbywhichyouthinkitcouldhavebeeno,andrecalculate thenalresult.Thedigitsontheendthatarecompletelyreshued aretheonesthataremeaningless,andshouldbeomitted. self-checkG Howmanysignicantguresarethereineachofthefollowingmeasurements? 9.937m 4.0s 0.0000000000000037kg Answer,p.266 Section0.10SignicantFigures 37

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Summary SelectedVocabulary matter......Anythingthatisaectedbygravity. light........Anythingthatcantravelfromoneplacetoanotherthroughemptyspaceandcaninuence matter,butisnotaectedbygravity. operationaldenition....... Adenitionthatstateswhatoperations shouldbecarriedouttomeasurethethingbeingdened. SystemeInternational....... Afancynameforthemetricsystem. mkssystem...Theuseofmetricunitsbasedonthemeter, kilogram,andsecond.Example:metersper secondisthemksunitofspeed,notcm/sor km/hr. mass.......Anumericalmeasureofhowdicultitisto changeanobject'smotion. signicantguresDigitsthatcontributetotheaccuracyofa measurement. Notation m.........meter,themetricdistanceunit kg.........kilogram,themetricunitofmass s..........second,themetricunitoftime M-.........themetricprexmega-,10 6 k-.........themetricprexkilo-,10 3 m-.........themetricprexmilli-,10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 -.........themetricprexmicro-,10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(6 n-.........themetricprexnano-,10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(9 Summary Physicsistheuseofthescienticmethodtostudythebehavior oflightandmatter.Thescienticmethodrequiresacycleoftheoryandexperiment,theorieswithbothpredictiveandexplanatory value,andreproducibleexperiments. Themetricsystemisasimple,consistentframeworkformeasurementbuiltoutofthemeter,thekilogram,andthesecondplusaset ofprexesdenotingpowersoften.Themostsystematicmethodfor doingconversionsisshowninthefollowingexample: 370ms 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 s 1ms =0.37s Massisameasureoftheamountofasubstance.Masscanbe denedgravitationally,bycomparinganobjecttoastandardmass onadouble-panbalance,orintermsofinertia,bycomparingthe eectofaforceonanobjecttotheeectofthesameforceona standardmass.Thetwodenitionsarefoundexperimentallyto beproportionaltoeachothertoahighdegreeofprecision,sowe 38 Chapter0IntroductionandReview

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usuallyrefersimplytomass,"withoutbotheringtospecifywhich type. Aforceisthatwhichcanchangethemotionofanobject.The metricunitofforceistheNewton,denedastheforcerequiredto accelerateastandard1-kgmassfromresttoaspeedof1m/sin1 s. Scienticnotationmeans,forexample,writing3.2 10 5 rather than320000. Writingnumberswiththecorrectnumberofsignicantgures correctlycommunicateshowaccuratetheyare.Asaruleofthumb, thenalresultofacalculationisnomoreaccuratethan,andshould havenomoresignicantguresthan,theleastaccuratepieceof data. ExploringFurther MicrobeHunters ,PauldeKruif.Thedramaticlife-and-death storiesinthisbookareentertaining,butalongthewaydeKruifalso presentsanexcellent,warts-and-allpictureofhowrealscienceand realscientistsreallywork|anexcellentanecdotetothesanitized pictureofthescienticmethodoftenpresentedintextbooks.Some ofthedescriptionsofeldworkinAfricaaremarredbyracism. VoodooScience:TheRoadfromFoolishnesstoFraud ,Robert L.Park.Parkhassomepenetratingpsychologicalinsightsintothe fundamentalproblemsthat Homosapiens scientistsincludedoftenhavewiththeunwelcometruthsthatsciencetossesinourlaps. UntilIreadthisbook,Ihadn'trealized,forexample,howcommon itistondpocketsofbogusscienceinsuchotherwiserespectable institutionsasNASA. Summary 39

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Correctuseofacalculator:aCalculate 74658 53222+97554 onacalculator.[Self-check:Themostcommonmistakeresultsin97555.40.] p bWhichwouldbemorelikethepriceofaTV,andwhichwould bemorelikethepriceofahouse,$3.5 10 5 or$3.5 5 ? 2 Computethefollowingthings.Iftheydon'tmakesensebecauseofunits,sayso. a3cm+5cm b1.11m+22cm c120miles+2.0hours d120miles/2.0hours 3 Yourbackyardhasbrickwallsonbothends.Youmeasurea distanceof23.4mfromtheinsideofonewalltotheinsideofthe other.Eachwallis29.4cmthick.Howfarisitfromtheoutside ofonewalltotheoutsideoftheother?Payattentiontosignicant gures. 4 Thespeedoflightis3.0 10 8 m/s.Convertthistofurlongs perfortnight.Afurlongis220yards,andafortnightis14days.An inchis2.54cm. p 5 Expresseachofthefollowingquantitiesinmicrograms: a10mg,b10 4 g,c10kg,d100 10 3 g,e1000ng. p 6 Convert134mgtounitsofkg,writingyouranswerinscientic notation. Solution,p.269 7 Inthelastcentury,theaverageageoftheonsetofpubertyfor girlshasdecreasedbyseveralyears.Urbanfolklorehasitthatthis isbecauseofhormonesfedtobeefcattle,butitismorelikelytobe becausemoderngirlshavemorebodyfatontheaverageandpossiblybecauseofestrogen-mimickingchemicalsintheenvironment fromthebreakdownofpesticides.Ahamburgerfromahormoneimplantedsteerhasabout0.2ngofestrogenaboutdoublethe amountofnaturalbeef.Aservingofpeascontainsabout300 ngofestrogen.Anadultwomanproducesabout0.5mgofestrogen perdaynotethedierentunit!.aHowmanyhamburgerswould agirlhavetoeatinonedaytoconsumeasmuchestrogenasan adultwoman'sdailyproduction?bHowmanyservingsofpeas? p 40 Chapter0IntroductionandReview

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Problem10. 8 Theusualdenitionofthemeanaverageoftwonumbers a and b is a + b = 2.Thisiscalledthearithmeticmean.Thegeometric mean,however,isdenedas ab 1 = 2 i.e.,thesquarerootof ab .For thesakeofdeniteness,let'ssaybothnumbershaveunitsofmass. aComputethearithmeticmeanoftwonumbersthathaveunits ofgrams.Thenconvertthenumberstounitsofkilogramsand recomputetheirmean.Istheanswerconsistent?bDothesame forthegeometricmean.cIf a and b bothhaveunitsofgrams, whatshouldwecalltheunitsof ab ?Doesyouranswermakesense whenyoutakethesquareroot?dSupposesomeoneproposesto youathirdkindofmean,calledthesuperdupermean,denedas ab 1 = 3 .Isthisreasonable? Solution,p.269 9 InanarticleontheSARSepidemic,theMay7,2003New YorkTimesdiscussesconictingestimatesofthedisease'sincubationperiodtheaveragetimethatelapsesfrominfectiontotherst symptoms.Thestudyestimatedittobe6.4days.Butotherstatisticalcalculations...showedthattheincubationperiodcouldbe aslongas14.22days."What'swronghere? 10 Thephotoshowsthecornerofabagofpretzels.What's wronghere? 11 Thedistancetothehorizonisgivenbytheexpression p 2 rh where r istheradiusoftheEarth,and h istheobserver'sheight abovetheEarth'ssurface.ThiscanbeprovedusingthePythagorean theorem.Showthattheunitsofthisexpressionmakesense. Problems 41

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42 Chapter0IntroductionandReview

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a / Amoebasthissizeare seldomencountered. Lifewouldbeverydifferentifyou werethesizeofaninsect. Chapter1 Scalingand Order-of-Magnitude Estimates 1.1Introduction Whycan'taninsectbethesizeofadog?Someskinnystretchedoutcellsinyourspinalcordareametertall|whydoesnature displaynosinglecellsthatarenotjustametertall,butameter wide,andameterthickaswell?Believeitornot,thesearequestions thatcanbeansweredfairlyeasilywithoutknowingmuchmoreabout physicsthanyoualreadydo.Theonlymathematicaltechniqueyou reallyneedisthehumbleconversion,appliedtoareaandvolume. Areaandvolume Areacanbedenedbysayingthatwecancopytheshapeof interestontographpaperwith1cm 1cmsquaresandcountthe numberofsquaresinside.Fractionsofsquarescanbeestimatedby eye.Wethensaytheareaequalsthenumberofsquares,inunitsof squarecm.Althoughthismightseemlesspure"thancomputing areasusingformulaelike A = r 2 foracircleor A = wh= 2fora triangle,thoseformulaearenotusefulasdenitionsofareabecause 43

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theycannotbeappliedtoirregularlyshapedareas. Unitsofsquarecmaremorecommonlywrittenascm 2 inscience. Ofcourse,theunitofmeasurementsymbolizedbycm"isnotan algebrasymbolstandingforanumberthatcanbeliterallymultiplied byitself.Butitisadvantageoustowritetheunitsofareathatway andtreattheunitsasiftheywerealgebrasymbols.Forinstance, ifyouhavearectanglewithanareaof6m 2 andawidthof2m, thencalculatingitslengthasm 2 = m=3mgivesaresult thatmakessensebothnumericallyandintermsofunits.This algebra-styletreatmentoftheunitsalsoensuresthatourmethods ofconvertingunitsworkoutcorrectly.Forinstance,ifweaccept thefraction 100cm 1m asavalidwayofwritingthenumberone,thenonetimesoneequals one,soweshouldalsosaythatonecanberepresentedby 100cm 1m 100cm 1m whichisthesameas 10000cm 2 1m 2 Thatmeanstheconversionfactorfromsquaremeterstosquarecentimetersisafactorof10 4 ,i.e.,asquaremeterhas10 4 squarecentimetersinit. Alloftheabovecanbeeasilyappliedtovolumeaswell,using one-cubic-centimeterblocksinsteadofsquaresongraphpaper. Tomanypeople,itseemshardtobelievethatasquaremeter equals10000squarecentimeters,orthatacubicmeterequalsa millioncubiccentimeters|theythinkitwouldmakemoresenseif therewere100cm 2 in1m 2 ,and100cm 3 in1m 3 ,butthatwouldbe incorrect.Theexamplesshowningurebaimtomakethecorrect answermorebelievable,usingthetraditionalU.S.unitsoffeetand yards.Onefootis12inches,andoneyardisthreefeet. b / Visualizingconversionsof areaandvolumeusingtraditional U.S.units. self-checkA Basedongureb,convinceyourselfthatthereare9ft 2 inasquareyard, 44 Chapter1ScalingandOrder-of-MagnitudeEstimates

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and27ft 3 inacubicyard,thendemonstratethesamethingsymbolically i.e.,withthemethodusingfractionsthatequalone. Answer,p. 266 Solvedproblem:converting mm 2 to cm 2 page61,problem10 Solvedproblem:scalingaliterpage62,problem19 DiscussionQuestion A Howmanysquarecentimetersarethereinasquareinch?inch= 2.54cmFirstndanapproximateanswerbymakingadrawing,thenderivetheconversionfactormoreaccuratelyusingthesymbolicmethod. c / GalileoGalilei-1642wasaRenaissanceItalianwhobroughtthe scienticmethodtobearonphysics,creatingthemodernversionofthe science.Comingfromanoblebutverypoorfamily,Galileohadtodrop outofmedicalschoolattheUniversityofPisawhenheranoutofmoney. Eventuallybecomingalecturerinmathematicsatthesameschool,he beganacareerasanotorioustroublemakerbywritingaburlesqueridiculingtheuniversity'sregulationshewasforcedtoresign,butfounda newteachingpositionatPadua.Heinventedthependulumclock,investigatedthemotionoffallingbodies,anddiscoveredthemoonsofJupiter. Thethrustofhislife'sworkwastodiscreditAristotle'sphysicsbyconfrontingitwithcontradictoryexperiments,aprogramwhichpavedtheway forNewton'sdiscoveryoftherelationshipbetweenforceandmotion.In chapter3we'llcometothestoryofGalileo'sultimatefateatthehandsof theChurch. 1.2ScalingofAreaandVolume Greateashavelessereas Upontheirbackstobite'em. Andlessereashavelesserstill, Andsoadinnitum. JonathanSwift Nowhowdotheseconversionsofareaandvolumerelatetothe questionsIposedaboutsizesoflivingthings?Well,imaginethat youareshrunklikeAliceinWonderlandtothesizeofaninsect. Onewayofthinkingaboutthechangeofscaleisthatwhatused tolooklikeacentimeternowlookslikeperhapsametertoyou, becauseyou'resomuchsmaller.Ifareaandvolumescaledaccording tomostpeople'sintuitive,incorrectexpectations,with1m 2 being thesameas100cm 2 ,thentherewouldbenoparticularreason whynatureshouldbehaveanydierentlyonyournew,reduced scale.Butnaturedoesbehavedierentlynowthatyou'resmall. Forinstance,youwillndthatyoucanwalkonwater,andjump tomanytimesyourownheight.ThephysicistGalileoGalileihad thebasicinsightthatthescalingofareaandvolumedetermines hownaturalphenomenabehavedierentlyondierentscales.He Section1.2ScalingofAreaandVolume 45

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d / Thesmallboatholdsup justne. e / Alargerboatbuiltwith thesameproportionsasthe smallonewillcollapseunderits ownweight. f / Aboatthislargeneedsto havetimbersthatarethicker comparedtoitssize. rstreasonedaboutmechanicalstructures,butlaterextendedhis insightstolivingthings,takingthethen-radicalpointofviewthatat thefundamentallevel,alivingorganismshouldfollowthesamelaws ofnatureasamachine.Wewillfollowhisleadbyrstdiscussing machinesandthenlivingthings. Galileoonthebehaviorofnatureonlargeandsmallscales Oneoftheworld'smostfamouspiecesofscienticwritingis Galileo'sDialoguesConcerningtheTwoNewSciences.Galileowas anentertainingwriterwhowantedtoexplainthingsclearlytolaypeople,andheliveneduphisworkbycastingitintheformofadialogue amongthreepeople.SalviatiisreallyGalileo'salterego.Simplicio isthestupidcharacter,andoneofthereasonsGalileogotintrouble withtheChurchwasthattherewererumorsthatSimpliciorepresentedthePope.Sagredoistheearnestandintelligentstudent,with whomthereaderissupposedtoidentify.Thefollowingexcerpts arefromthe1914translationbyCrewanddeSalvio. S AGREDO :Yes,thatiswhatImean;andIreferespeciallyto hislastassertionwhichIhavealwaysregardedasfalse...; namely,thatinspeakingoftheseandothersimilarmachines onecannotarguefromthesmalltothelarge,becausemany deviceswhichsucceedonasmallscaledonotworkona largescale.Now,sincemechanicshasitsfoundationsingeometry,wheremeresize[isunimportant],Idonotseethat thepropertiesofcircles,triangles,cylinders,conesandother solidgureswillchangewiththeirsize.If,therefore,alarge machinebeconstructedinsuchawaythatitspartsbearto oneanotherthesameratioasinasmallerone,andifthe smallerissufcientlystrongforthepurposeforwhichitis designed,Idonotseewhythelargershouldnotbeableto withstandanysevereanddestructiveteststowhichitmaybe subjected. SalviaticontradictsSagredo: S ALVIATI :...Pleaseobserve,gentlemen,howfactswhich atrstseemimprobablewill,evenonscantexplanation,drop thecloakwhichhashiddenthemandstandforthinnakedand simplebeauty.Whodoesnotknowthatahorsefallingfroma heightofthreeorfourcubitswillbreakhisbones,whileadog fallingfromthesameheightoracatfromaheightofeight ortencubitswillsuffernoinjury?Equallyharmlesswouldbe thefallofagrasshopperfromatowerorthefallofanantfrom thedistanceofthemoon. ThepointGalileoismakinghereisthatsmallthingsaresturdier inproportiontotheirsize.Therearealotofobjectionsthatcouldbe raised,however.Afterall,whatdoesitreallymeanforsomethingto bestrong",tobestronginproportiontoitssize,"ortobestrong 46 Chapter1ScalingandOrder-of-MagnitudeEstimates

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g / Galileodiscussesplanks madeofwood,buttheconcept maybeeasiertoimaginewith clay.Allthreeclayrodsinthe gurewereoriginallythesame shape.Themedium-sizedone wastwicetheheight,twicethe length,andtwicethewidthof thesmallone,andsimilarlythe largeonewastwiceasbigas themediumoneinallitslinear dimensions.Thebigonehas fourtimesthelineardimensions ofthesmallone,16timesthe cross-sectionalareawhencut perpendiculartothepage,and 64timesthevolume.Thatmeans thatthebigonehas64timesthe weighttosupport,butonly16 timesthestrengthcomparedto thesmallestone. outofproportiontoitssize?"Galileohasn'tgivenoperational denitionsofthingslikestrength,"i.e.,denitionsthatspellout howtomeasurethemnumerically. Also,acatisshapeddierentlyfromahorse|anenlarged photographofacatwouldnotbemistakenforahorse,evenifthe photo-doctoringexpertsattheNationalInquirermadeitlooklikea personwasridingonitsback.Agrasshopperisnotevenamammal, andithasanexoskeletoninsteadofaninternalskeleton.Thewhole argumentwouldbealotmoreconvincingifwecoulddosomeisolationofvariables,ascientictermthatmeanstochangeonlyone thingatatime,isolatingitfromtheothervariablesthatmighthave aneect.Ifsizeisthevariablewhoseeectwe'reinterestedinseeing,thenwedon'treallywanttocomparethingsthataredierent insizebutalsodierentinotherways. S ALVIATI :...weaskedthereasonwhy[shipbuilders]employedstocks,scaffolding,andbracingoflargerdimensions forlaunchingabigvesselthantheydoforasmallone;and [anoldman]answeredthattheydidthisinordertoavoidthe dangeroftheshippartingunderitsownheavyweight,adangertowhichsmallboatsarenotsubject? Afterthisentertainingbutnotscienticallyrigorousbeginning, Galileostartstodosomethingworthwhilebymodernstandards. Hesimplieseverythingbyconsideringthestrengthofawooden plank.Thevariablesinvolvedcanthenbenarroweddowntothe typeofwood,thewidth,thethickness,andthelength.Healso givesanoperationaldenitionofwhatitmeansfortheplankto haveacertainstrengthinproportiontoitssize,"byintroducing theconceptofaplankthatisthelongestonethatwouldnotsnap underitsownweightifsupportedatoneend.Ifyouincreased itslengthbytheslightestamount,withoutincreasingitswidthor thickness,itwouldbreak.Hesaysthatifoneplankisthesame shapeasanotherbutadierentsize,appearinglikeareducedor enlargedphotographoftheother,thentheplankswouldbestrong inproportiontotheirsizes"ifbothwerejustbarelyabletosupport theirownweight. h / 1.Thisplankisaslongasit canbewithoutcollapsingunder itsownweight.Ifitwasahundredthofaninchlonger,itwould collapse.2.Thisplankismade outofthesamekindofwood.Itis twiceasthick,twiceaslong,and twiceaswide.Itwillcollapseunderitsownweight. Section1.2ScalingofAreaandVolume 47

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Also,Galileoisdoingsomethingthatwouldbefrownedonin modernscience:heismixingexperimentswhoseresultshehasactuallyobservedbuildingboatsofdierentsizes,withexperiments thathecouldnotpossiblyhavedonedroppinganantfromthe heightofthemoon.Henowrelateshowhehasdoneactualexperimentswithsuchplanks,andfoundthat,accordingtothisoperationaldenition,theyarenotstronginproportiontotheirsizes. Thelargeronebreaks.Hemakessuretotellthereaderhowimportanttheresultis,viaSagredo'sastonishedresponse: S AGREDO :Mybrainalreadyreels.Mymind,likeacloud momentarilyilluminatedbyalightningash,isforaninstant lledwithanunusuallight,whichnowbeckonstomeand whichnowsuddenlyminglesandobscuresstrange,crude ideas.Fromwhatyouhavesaiditappearstomeimpossible tobuildtwosimilarstructuresofthesamematerial,butof differentsizesandhavethemproportionatelystrong. Inotherwords,thisspecicexperiment,usingthingslikewooden planksthathavenointrinsicscienticinterest,hasverywideimplicationsbecauseitpointsoutageneralprinciple,thatnatureacts dierentlyondierentscales. Tonishthediscussion,Galileogivesanexplanation.Hesays thatthestrengthofaplankdenedas,say,theweightoftheheaviestboulderyoucouldputontheendwithoutbreakingitisproportionaltoitscross-sectionalarea,thatis,thesurfaceareaofthe freshwoodthatwouldbeexposedifyousawedthroughitinthe middle.Itsweight,however,isproportionaltoitsvolume. 1 Howdothevolumeandcross-sectionalareaofthelongerplank comparewiththoseoftheshorterplank?Wehavealreadyseen, whilediscussingconversionsoftheunitsofareaandvolume,that thesequantitiesdon'tactthewaymostpeoplenaivelyexpect.You mightthinkthatthevolumeandareaofthelongerplankwouldboth bedoubledcomparedtotheshorterplank,sotheywouldincrease inproportiontoeachother,andthelongerplankwouldbeequally abletosupportitsweight.Youwouldbewrong,butGalileoknows thatthisisacommonmisconception,sohehasSalviatiaddressthe pointspecically: S ALVIATI :...Take,forexample,acubetwoinchesona sidesothateachfacehasanareaoffoursquareinches andthetotalarea,i.e.,thesumofthesixfaces,amounts totwenty-foursquareinches;nowimaginethiscubetobe sawedthroughthreetimes[withcutsinthreeperpendicular planes]soastodivideitintoeightsmallercubes,eachone inchontheside,eachfaceoneinchsquare,andthetotal 1 Galileomakesaslightlymorecomplicatedargument,takingintoaccount theeectofleveragetorque.TheresultI'mreferringtocomesoutthesame regardlessofthiseect. 48 Chapter1ScalingandOrder-of-MagnitudeEstimates

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i / Theareaofashapeis proportionaltothesquareofits lineardimensions,evenifthe shapeisirregular. surfaceofeachcubesixsquareinchesinsteadoftwentyfourinthecaseofthelargercube.Itisevidenttherefore, thatthesurfaceofthelittlecubeisonlyone-fourththatof thelarger,namely,theratioofsixtotwenty-four;butthevolumeofthesolidcubeitselfisonlyone-eighth;thevolume, andhencealsotheweight,diminishesthereforemuchmore rapidlythanthesurface...Yousee,therefore,Simplicio,that Iwasnotmistakenwhen...Isaidthatthesurfaceofasmall solidiscomparativelygreaterthanthatofalargeone. Thesamereasoningappliestotheplanks.Eventhoughthey arenotcubes,thelargeonecouldbesawedintoeightsmallones, eachwithhalfthelength,halfthethickness,andhalfthewidth. Thesmallplank,therefore,hasmoresurfaceareainproportionto itsweight,andisthereforeabletosupportitsownweightwhilethe largeonebreaks. Scalingofareaandvolumeforirregularlyshapedobjects YouprobablyarenotgoingtobelieveGalileo'sclaimthatthis hasdeepimplicationsforallofnatureunlessyoucanbeconvinced thatthesameistrueforanyshape.Everydrawingyou'veseenso farhasbeenofsquares,rectangles,andrectangularsolids.Clearly thereasoningaboutsawingthingsupintosmallerpieceswouldnot proveanythingabout,say,anegg,whichcannotbecutupintoeight smalleregg-shapedobjectswithhalfthelength. Isitalwaystruethatsomethinghalfthesizehasonequarter thesurfaceareaandoneeighththevolume,evenifithasanirregularshape?Taketheexampleofachild'sviolin.Violinsaremade forsmallchildreninsmallersizetoaccomodatetheirsmallbodies. Figureishowsafull-sizeviolin,alongwithtwoviolinsmadewith halfand3/4ofthenormallength. 2 Let'sstudythesurfaceareaof thefrontpanelsofthethreeviolins. Considerthesquareintheinteriorofthepanelofthefull-size violin.Inthe3/4-sizeviolin,itsheightandwidtharebothsmaller byafactorof3/4,sotheareaofthecorresponding,smallersquare becomes3 = 4 3 = 4=9 = 16oftheoriginalarea,not3/4oftheoriginal area.Similarly,thecorrespondingsquareonthesmallestviolinhas halftheheightandhalfthewidthoftheoriginalone,soitsareais 1/4theoriginalarea,nothalf. Thesamereasoningworksforpartsofthepanelneartheedge, suchasthepartthatonlypartiallyllsintheothersquare.The entiresquarescalesdownthesameasasquareintheinterior,and ineachviolinthesamefractionabout70%ofthesquareisfull,so thecontributionofthisparttothetotalareascalesdownjustthe 2 Thecustomarytermshalf-size"and/4-size"actuallydon'tdescribethe sizesinanyaccurateway.They'rereallyjuststandard,arbitrarymarketing labels. Section1.2ScalingofAreaandVolume 49

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j / Themufncomesoutof theoventoohottoeat.Breaking itupintofourpiecesincreases itssurfaceareawhilekeeping thetotalvolumethesame.It coolsfasterbecauseofthe greatersurface-to-volumeratio. Ingeneral,smallerthingshave greatersurface-to-volumeratios, butinthisexamplethereisno easywaytocomputetheeffect exactly,becausethesmallpieces aren'tthesameshapeasthe originalmufn. same. Sinceanysmallsquareregionoranysmallregioncoveringpart ofasquarescalesdownlikeasquareobject,theentiresurfacearea ofanirregularlyshapedobjectchangesinthesamemannerasthe surfaceareaofasquare:scalingitdownby3/4reducestheareaby afactorof9/16,andsoon. Ingeneral,wecanseethatanytimetherearetwoobjectswith thesameshape,butdierentlineardimensionsi.e.,onelookslikea reducedphotooftheother,theratiooftheirareasequalstheratio ofthesquaresoftheirlineardimensions: A 1 A 2 = L 1 L 2 2 Notethatitdoesn'tmatterwherewechoosetomeasurethelinear size, L ,ofanobject.Inthecaseoftheviolins,forinstance,itcould havebeenmeasuredvertically,horizontally,diagonally,orevenfrom thebottomoftheleftf-holetothemiddleoftherightf-hole.We justhavetomeasureitinaconsistentwayoneachviolin.Sinceall thepartsareassumedtoshrinkorexpandinthesamemanner,the ratio L 1 =L 2 isindependentofthechoiceofmeasurement. Itisalsoimportanttorealizethatitiscompletelyunnecessary tohaveaformulafortheareaofaviolin.Itisonlypossibleto derivesimpleformulasfortheareasofcertainshapeslikecircles, rectangles,trianglesandsoon,butthatisnoimpedimenttothe typeofreasoningweareusing. Sometimesitisinconvenienttowritealltheequationsinterms ofratios,especiallywhenmorethantwoobjectsarebeingcompared. Amorecompactwayofrewritingthepreviousequationis A / L 2 Thesymbol / "meansisproportionalto."Scientistsandengineersoftenspeakaboutsuchrelationshipsverballyusingthephrases scaleslike"orgoeslike,"forinstanceareagoeslikelengthsquared." Alloftheabovereasoningworksjustaswellinthecaseofvolume.Volumegoeslikelengthcubed: V / L 3 Ifdierentobjectsaremadeofthesamematerialwiththesame density, = m=V ,thentheirmasses, m = V ,areproportional to L 3 ,andsoaretheirweights.Thesymbolfordensityis ,the lower-caseGreekletterrho." Animportantpointisthatalloftheabovereasoningabout scalingonlyappliestoobjectsthatarethesameshape.Forinstance, apieceofpaperislargerthanapencil,buthasamuchgreater surface-to-volumeratio. 50 Chapter1ScalingandOrder-of-MagnitudeEstimates

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k / Example1.Thebigtrianglehasfourtimesmoreareathan thelittleone. l / Atrickywayofsolvingexample1,explainedinsolution#2. OneoftherstthingsIlearnedasateacherwasthatstudents werenotveryoriginalabouttheirmistakes.Everygroupofstudents tendstocomeupwiththesamegoofsasthepreviousclass.The followingaresomeexamplesofcorrectandincorrectreasoningabout proportionality. Scalingoftheareaofatriangleexample1 Ingurek,thelargertrianglehassidestwiceaslong.How manytimesgreaterisitsarea? Correctsolution#1:Areascalesinproportiontothesquareofthe lineardimensions,sothelargertrianglehasfourtimesmorearea 2 =4. Correctsolution#2:Youcouldcutthelargertriangleintofourof thesmallersize,asshowning.b,soitsareaisfourtimes greater.Thissolutioniscorrect,butitwouldnotworkforashape likeacircle,whichcan'tbecutupintosmallercircles. Correctsolution#3:Theareaofatriangleisgivenby A = bh = 2,where b isthebaseand h istheheight.Theareasof thetrianglesare A 1 = b 1 h 1 = 2 A 2 = b 2 h 2 = 2 = b 1 h 1 = 2 =2 b 1 h 1 A 2 = A 1 = b 1 h 1 = b 1 h 1 = 2 =4 Althoughthissolutioniscorrect,itisalotmoreworkthansolution #1,anditcanonlybeusedinthiscasebecauseatriangleisa simplegeometricshape,andwehappentoknowaformulaforits area. Correctsolution#4:Theareaofatriangleis A = bh = 2.The comparisonoftheareaswillcomeoutthesameaslongasthe ratiosofthelinearsizesofthetrianglesisasspecied,solet's justsay b 1 =1.00mand b 2 =2.00m.Theheightsarethenalso h 1 =1.00mand h 2 =2.00m,givingareas A 1 =0.50m 2 and A 2 =2.00m 2 ,so A 2 = A 1 =4.00. Thesolutioniscorrect,butitwouldn'tworkwithashapefor whoseareawedon'thaveaformula.Also,thenumericalcalculationmightmaketheanswerof4.00appearinexact,whereas solution#1makesitclearthatitisexactly4. Incorrectsolution:Theareaofatriangleis A = bh = 2,andifyou plugin b =2.00mand h =2.00m,youget A =2.00m 2 ,so thebiggertrianglehas2.00timesmorearea.Thissolutionis incorrectbecausenocomparisonhasbeenmadewiththesmaller triangle. Section1.2ScalingofAreaandVolume 51

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n / Example3.The48-point Shas1.78timesmorearea thanthe36-pointS. m / Example2.Thebigsphere has125timesmorevolumethan thelittleone. Scalingofthevolumeofasphereexample2 Ingurem,thelargerspherehasaradiusthatisvetimes greater.Howmanytimesgreaterisitsvolume? Correctsolution#1:Volumescaleslikethethirdpowerofthe linearsize,sothelargerspherehasavolumethatis125times greater 3 =125. Correctsolution#2:Thevolumeofasphereis V = = 3 r 3 ,so V 1 = 4 3 r 3 1 V 2 = 4 3 r 3 2 = 4 3 r 1 3 = 500 3 r 3 1 V 2 = V 1 = 500 3 r 3 1 = 4 3 r 3 1 =125 Incorrectsolution:Thevolumeofasphereis V = = 3 r 3 ,so V 1 = 4 3 r 3 1 V 2 = 4 3 r 3 2 = 4 3 5 r 3 1 = 20 3 r 3 1 V 2 = V 1 = 20 3 r 3 1 = 4 3 r 3 1 =5 Thesolutionisincorrectbecause r 1 3 isnotthesameas5 r 3 1 Scalingofamorecomplexshapeexample3 TherstletterSingurenisina36-pointfont,thesecondin 48-point.Howmanytimesmoreinkisrequiredtomakethelarger S?Pointsareaunitoflengthusedintypography. Correctsolution:Theamountofinkdependsontheareatobe coveredwithink,andareaisproportionaltothesquareofthe lineardimensions,sotheamountofinkrequiredforthesecond Sisgreaterbyafactorof = 36 2 =1.78. Incorrectsolution:ThelengthofthecurveofthesecondSis longerbyafactorof48 = 36=1.33,so1.33timesmoreinkis required. Thesolutioniswrongbecauseitassumesincorrectlythatthe widthofthecurveisthesameinbothcases.Actuallyboththe 52 Chapter1ScalingandOrder-of-MagnitudeEstimates

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widthandthelengthofthecurvearegreaterbyafactorof48/36, sotheareaisgreaterbyafactorof = 36 2 =1.78. Solvedproblem:atelescopegatherslightpage61,problem11 Solvedproblem:distancefromanearthquakepage61,problem12 DiscussionQuestions A Atoyreengineis1/30thesizeoftherealone,butisconstructed fromthesamemetalwiththesameproportions.Howmanytimessmaller isitsweight?Howmanytimeslessredpaintwouldbeneededtopaint it? B Galileospendsalotoftimeinhisdialogdiscussingwhatreally happenswhenthingsbreak.HediscusseseverythingintermsofAristotle'snow-discreditedexplanationthatthingsarehardtobreak,because ifsomethingbreaks,therehastobeagapbetweenthetwohalveswith nothinginbetween,atleastinitially.Nature,accordingtoAristotle,abhorsavacuum,i.e.,naturedoesn'tlikeemptyspacetoexist.Ofcourse, airwillrushintothegapimmediately,butattheverymomentofbreaking, Aristotleimaginedavacuuminthegap.IsAristotle'sexplanationofwhy itishardtobreakthingsanexperimentallytestablestatement?Ifso,how coulditbetestedexperimentally? 1.3 ? ScalingAppliedtoBiology Organismsofdifferentsizeswiththesameshape Theleft-handpanelingureoshowstheapproximatevalidityoftheproportionality m / L 3 forcockroachesredrawnfrom McMahonandBonner.Thescatterofthepointsaroundthecurve indicatesthatsomecockroachesareproportionedslightlydierently fromothers,butingeneralthedataseemwelldescribedby m / L 3 Thatmeansthatthelargestcockroachestheexperimentercould raiseistherea4-Hprize?hadroughlythesameshapeasthe smallestones. Anotherrelationshipthatshouldexistforanimalsofdierent sizesshapedinthesamewayisthatbetweensurfaceareaand bodymass.Ifalltheanimalshavethesameaveragedensity,then bodymassshouldbeproportionaltothecubeoftheanimal'slinearsize, m / L 3 ,whilesurfaceareashouldvaryproportionatelyto L 2 .Therefore,theanimals'surfaceareasshouldbeproportionalto m 2 = 3 .Asshownintheright-handpanelofgureo,thisrelationship appearstoholdquitewellforthedwarfsiren,atypeofsalamander. Noticehowthecurvebendsover,meaningthatthesurfaceareadoes notincreaseasquicklyasbodymass,e.g.,asalamanderwitheight timesmorebodymasswillhaveonlyfourtimesmoresurfacearea. Thisbehavioroftheratioofsurfaceareatomassor,equivSection1.3 ? ScalingAppliedtoBiology 53

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o / Geometricalscalingofanimals. alently,theratioofsurfaceareatovolumehasimportantconsequencesformammals,whichmustmaintainaconstantbodytemperature.Itwouldmakesensefortherateofheatlossthroughthe animal'sskintobeproportionaltoitssurfacearea,soweshould expectsmallanimals,havinglargeratiosofsurfaceareatovolume, toneedtoproduceagreatdealofheatincomparisontotheirsizeto avoiddyingfromlowbodytemperature.Thisexpectationisborne outbythedataoftheleft-handpanelofgurep,showingtherate ofoxygenconsumptionofguineapigsasafunctionoftheirbody mass.Neitherananimal'sheatproductionnoritssurfaceareais convenienttomeasure,butinordertoproduceheat,theanimal mustmetabolizeoxygen,sooxygenconsumptionisagoodindicator oftherateofheatproduction.Sincesurfaceareaisproportionalto m 2 = 3 ,theproportionalityoftherateofoxygenconsumptionto m 2 = 3 isconsistentwiththeideathattheanimalneedstoproduceheatata rateinproportiontoitssurfacearea.Althoughthesmalleranimals metabolizelessoxygenandproducelessheatinabsoluteterms,the amountoffoodandoxygentheymustconsumeisgreaterinproportiontotheirownmass.TheEtruscanpigmyshrew,weighinginat 54 Chapter1ScalingandOrder-of-MagnitudeEstimates

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q / Galileo'soriginaldrawing, showinghowlargeranimals' bonesmustbegreaterindiametercomparedtotheirlengths. p / Scalingofanimals'bodiesrelatedtometabolicrateandskeletalstrength. 2gramsasanadult,isataboutthelowersizelimitformammals. Itmusteatcontinually,consumingmanytimesitsbodyweighteach daytosurvive. Changesinshapetoaccommodatechangesinsize Largemammals,suchaselephants,haveasmallratioofsurface areatovolume,andhaveproblemsgettingridoftheirheatfast enough.Anelephantcannotsimplyeatsmallenoughamountsto keepfromproducingexcessiveheat,becausecellsneedtohavea certainminimummetabolicratetoruntheirinternalmachinery. Hencetheelephant'slargeears,whichaddtoitssurfaceareaand helpittocoolitself.Previously,wehaveseenseveralexamples ofdatawithinagivenspeciesthatwereconsistentwithaxed shape,scaledupanddowninthecasesofindividualspecimens.The elephant'searsareanexampleofachangeinshapenecessitatedby achangeinscale. Largeanimalsalsomustbeabletosupporttheirownweight. Returningtotheexampleofthestrengthsofplanksofdierent sizes,wecanseethatifthestrengthoftheplankdependsonarea Section1.3 ? ScalingAppliedtoBiology 55

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whileitsweightdependsonvolume,thentheratioofstrengthto weightgoesasfollows: strength = weight / A=V / 1 =L Thus,theabilityofobjectstosupporttheirownweightsdecreases inverselyinproportiontotheirlineardimensions.Ifanobjectisto bejustbarelyabletosupportitsownweight,thenalargerversion willhavetobeproportioneddierently,withadierentshape. Sincethedataonthecockroachesseemedtobeconsistentwith roughlysimilarshapeswithinthespecies,itappearsthattheabilitytosupportitsownweightwasnotthetightestdesignconstraint thatNaturewasworkingunderwhenshedesignedthem.Forlarge animals,structuralstrengthisimportant.Galileowastherstto quantifythisreasoningandtoexplainwhy,forinstance,alargeanimalmusthavebonesthatarethickerinproportiontotheirlength. Consideraroughlycylindricalbonesuchasalegboneoravertebra. Thelengthofthebone, L ,isdictatedbytheoveralllinearsizeofthe animal,sincetheanimal'sskeletonmustreachtheanimal'swhole length.Weexpecttheanimal'smasstoscaleas L 3 ,sothestrength ofthebonemustalsoscaleas L 3 .Strengthisproportionaltocrosssectionalarea,aswiththewoodenplanks,soifthediameterofthe boneis d ,then d 2 / L 3 or d / L 3 = 2 Iftheshapestayedthesameregardlessofsize,thenalllineardimensions,including d and L ,wouldbeproportionaltooneanother. Ifourreasoningholds,thenthefactthat d isproportionalto L 3 = 2 not L ,impliesachangeinproportionsofthebone.Asshowninthe right-handpanelofgurep,thevertebraeofAfricanBovidaefollow therule d / L 3 = 2 fairlywell.Thevertebraeofthegiantelandare aschunkyasacoeemug,whilethoseofaGunther'sdik-dikareas slenderasthecapofapen. DiscussionQuestions A Single-celledanimalsmustpassivelyabsorbnutrientsandoxygen fromtheirsurroundings,unlikehumanswhohavelungstopumpairinand outandahearttodistributetheoxygenatedbloodthroughouttheirbodies. Eventhecellscomposingthebodiesofmulticellularanimalsmustabsorb oxygenfromanearbycapillarythroughtheirsurfaces.Basedonthese facts,explainwhycellsarealwaysmicroscopicinsize. B Thereasoningofthepreviousquestionwouldseemtobecontradictedbythefactthathumannervecellsinthespinalcordcanbeas muchasameterlong,althoughtheirwidthsarestillverysmall.Whyis thispossible? 56 Chapter1ScalingandOrder-of-MagnitudeEstimates

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1.4Order-of-MagnitudeEstimates Itisthemarkofaninstructedmindtorestsatisedwiththe degreeofprecisionthatthenatureofthesubjectpermitsand nottoseekanexactnesswhereonlyanapproximationofthe truthispossible. Aristotle Itisacommonmisconceptionthatsciencemustbeexact.For instance,intheStarTrekTVseries,itwouldoftenhappenthat CaptainKirkwouldaskMr.Spock,Spock,we'reinaprettybad situation.Whatdoyouthinkareourchancesofgettingoutof here?"ThescienticMr.Spockwouldanswerwithsomethinglike, Captain,Iestimatetheoddsas237.345toone."Inreality,he couldnothaveestimatedtheoddswithsixsignicantguresof accuracy,butneverthelessoneofthehallmarksofapersonwitha goodeducationinscienceistheabilitytomakeestimatesthatare likelytobeatleastsomewhereintherightballpark.Inmanysuch situations,itisoftenonlynecessarytogetananswerthatisobyno morethanafactoroftenineitherdirection.Sincethingsthatdier byafactoroftenaresaidtodierbyoneorderofmagnitude,such anestimateiscalledanorder-of-magnitudeestimate.Thetilde, ,isusedtoindicatethatthingsareonlyofthesameorderof magnitude,butnotexactlyequal,asin oddsofsurvival 100toone. Thetildecanalsobeusedinfrontofanindividualnumbertoemphasizethatthenumberisonlyoftherightorderofmagnitude. Althoughmakingorder-of-magnitudeestimatesseemssimpleand naturaltoexperiencedscientists,it'samodeofreasoningthatis completelyunfamiliartomostcollegestudents.Someofthetypical mentalstepscanbeillustratedinthefollowingexample. Costoftransportingtomatoesexample4 Roughlywhatpercentageofthepriceofatomatocomesfrom thecostoftransportingitinatruck? Thefollowingincorrectsolutionillustratesoneofthemainways youcangowronginorder-of-magnitudeestimates. Incorrectsolution:Let'ssaythetruckerneedstomakea$400 protonthetrip.Takingintoaccountherbenets,thecostofgas, andmaintenanceandpaymentsonthetruck,let'ssaythetotal costismorelike$2000.I'dguessabout5000tomatoeswouldt inthebackofthetruck,sotheextracostpertomatois40cents. Thatmeansthecostoftransportingonetomatoiscomparableto thecostofthetomatoitself.Transportationreallyaddsalottothe costofproduce,Iguess. Theproblemisthatthehumanbrainisnotverygoodatestimatingareaorvolume,soitturnsouttheestimateof5000tomatoes Section1.4Order-of-MagnitudeEstimates 57

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r / Considerasphericalcow. ttinginthetruckiswayo.That'swhypeoplehaveahardtime atthosecontestswhereyouaresupposedtoestimatethenumberof jellybeansinabigjar.Anotherexampleisthatmostpeoplethink theirfamiliesuseabout10gallonsofwaterperday,butinreality theaverageisabout300gallonsperday.Whenestimatingarea orvolume,youaremuchbetteroestimatinglineardimensions, andcomputingvolumefromthelineardimensions.Here'sabetter solution: Bettersolution:Asintheprevioussolution,saythecostofthe tripis$2000.Thedimensionsofthebinareprobably4m 2m 1m,foravolumeof8m 3 .Sincethewholethingisjustanorderof-magnitudeestimate,let'sroundthatotothenearestpowerof ten,10m 3 .Theshapeofatomatoiscomplicated,andIdon'tknow anyformulaforthevolumeofatomatoshape,butsincethisisjust anestimate,let'spretendthatatomatoisacube,0.05m 0.05m 0.05m,foravolumeof1.25 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 m 3 .Sincethisisjustarough estimate,let'sroundthatto10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 m 3 .Wecanndthetotalnumber oftomatoesbydividingthevolumeofthebinbythevolumeofone tomato:10m 3 = 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 m 3 =10 5 tomatoes.Thetransportationcost pertomatois$2000 = 10 5 tomatoes=$0.02/tomato.Thatmeansthat transportationreallydoesn'tcontributeverymuchtothecostofa tomato. Approximatingtheshapeofatomatoasacubeisanexampleof anothergeneralstrategyformakingorder-of-magnitudeestimates. Asimilarsituationwouldoccurifyouweretryingtoestimatehow manym 2 ofleathercouldbeproducedfromaherdoftenthousand cattle.Thereisnopointintryingtotakeintoaccounttheshapeof thecows'bodies.Areasonableplanofattackmightbetoconsider asphericalcow.Probablyacowhasroughlythesamesurfacearea asaspherewitharadiusofabout1m,whichwouldbe4 m 2 Usingthewell-knownfactsthatpiequalsthree,andfourtimesthree equalsaboutten,wecanguessthatacowhasasurfaceareaofabout 10m 2 ,sotheherdasawholemightyield10 5 m 2 ofleather. Thefollowinglistsummarizesthestrategiesforgettingagood order-of-magnitudeestimate. 1.Don'tevenattemptmorethanonesignicantgureofprecision. 2.Don'tguessarea,volume,ormassdirectly.Guesslineardimensionsandgetarea,volume,ormassfromthem. 3.Whendealingwithareasorvolumesofobjectswithcomplex shapes,idealizethemasiftheyweresomesimplershape,a cubeorasphere,forexample. 4.Checkyournalanswertoseeifitisreasonable.Ifyouestimatethataherdoftenthousandcattlewouldyield0.01m 2 58 Chapter1ScalingandOrder-of-MagnitudeEstimates

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ofleather,thenyouhaveprobablymadeamistakewithconversionfactorssomewhere. Section1.4Order-of-MagnitudeEstimates 59

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Summary Notation / .........isproportionalto .........ontheorderof,isontheorderof Summary Naturebehavesdierentlyonlargeandsmallscales.Galileo showedthatthisresultsfundamentallyfromthewayareaandvolumescale.Areascalesasthesecondpoweroflength, A / L 2 ,while volumescalesaslengthtothethirdpower, V / L 3 Anorderofmagnitudeestimateisoneinwhichwedonotattemptorexpectanexactanswer.Themainreasonwhytheuninitiatedhavetroublewithorder-of-magnitudeestimatesisthatthe humanbraindoesnotintuitivelymakeaccurateestimatesofarea andvolume.Estimatesofareaandvolumeshouldbeapproached byrstestimatinglineardimensions,whichone'sbrainhasafeel for. 60 Chapter1ScalingandOrder-of-MagnitudeEstimates

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Howmanycubicinchesarethereinacubicfoot?Theanswer isnot12. p 2 Assumeadog'sbrainistwiceisgreatindiameterasacat's, buteachanimal'sbraincellsarethesamesizeandtheirbrainsare thesameshape.Inadditiontobeingafarbettercompanionand muchnicertocomehometo,howmanytimesmorebraincellsdoes adoghavethanacat?Theanswerisnot2. 3 ThepopulationdensityofLosAngelesisabout4000people = km 2 ThatofSanFranciscoisabout6000people = km 2 .Howmanytimes fartherawayistheaverageperson'snearestneighborinLAthanin SanFrancisco?Theanswerisnot1.5. p 4 Ahuntingdog'snosehasabout10squareinchesofactive surface.Howisthispossible,sincethedog'snoseisonlyabout1in 1in 1in=1in 3 ?Afterall,10isgreaterthan1,sohowcanit t? 5 Estimatethenumberofbladesofgrassonafootballeld. 6 Inacomputermemorychip,eachbitofinformationa0or a1isstoredinasingletinycircuitetchedontothesurfaceofa siliconchip.Thecircuitscoverthesurfaceofthechiplikelotsina housingdevelopment.Atypicalchipstores64Mbmegabytesof data,whereabyteis8bits.Estimateatheareaofeachcircuit, andbitslinearsize. 7 Supposesomeonebuiltagiganticapartmentbuilding,measuring10km 10kmatthebase.Estimatehowtallthebuilding wouldhavetobetohavespaceinitfortheentireworld'spopulation tolive. 8 Ahamburgerchainadvertisesthatithassold10billionBongo Burgers.Estimatethetotalmassoffeedrequiredtoraisethecows usedtomaketheburgers. 9 Estimatethevolumeofahumanbody,incm 3 10 Howmanycm 2 is1mm 2 ? Solution,p.269 11 Comparethelight-gatheringpowersofa3-cm-diametertelescopeanda30-cmtelescope. Solution,p.269 12 OnestepontheRichterscalecorrespondstoafactorof100 intermsoftheenergyabsorbedbysomethingonthesurfaceofthe Earth,e.g.,ahouse.Forinstance,a9.3-magnitudequakewould release100timesmoreenergythanan8.3.Theenergyspreadsout Problems 61

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AlbertEinstein,andhismoustache,problem14. Problem19. fromtheepicenterasawave,andforthesakeofthisproblemwe'll assumewe'redealingwithseismicwavesthatspreadoutinthree dimensions,sothatwecanvisualizethemashemispheresspreading outunderthesurfaceoftheearth.Ifacertain7.6-magnitudeearthquakeandacertain5.6-magnitudeearthquakeproducethesame amountofvibrationwhereIlive,comparethedistancesfrommy housetothetwoepicenters. Solution,p.269 13 InEurope,apieceofpaperofthestandardsize,calledA4, isalittlenarrowerandtallerthanitsAmericancounterpart.The ratiooftheheighttothewidthisthesquarerootof2,andthishas someusefulproperties.Forinstance,ifyoucutanA4sheetfromleft toright,yougettwosmallersheetsthathavethesameproportions. Youcanevenbuysheetsofthissmallersize,andthey'recalledA5. Thereisawholeseriesofsizesrelatedinthisway,allwiththesame proportions.aCompareanA5sheettoanA4intermsofareaand linearsize.bTheseriesofpapersizesstartsfromanA0sheet, whichhasanareaofonesquaremeter.Supposewehadaseries ofboxesdenedinasimilarway:theB0boxhasavolumeofone cubicmeter,twoB1boxestexactlyinsideanB0box,andsoon. WhatwouldbethedimensionsofaB0box? p 14 EstimatethemassofoneofthehairsinAlbertEinstein's moustache,inunitsofkg. 15 Accordingtofolklore,everytimeyoutakeabreath,youare inhalingsomeoftheatomsexhaledinCaesar'slastwords.Isthis true?Ifso,howmany? 16 TheEarth'ssurfaceisabout70%water.Mars'sdiameteris abouthalftheEarth's,butithasnosurfacewater.Comparethe landareasofthetwoplanets. p 17 ThetraditionalMartiniglassisshapedlikeaconewith thepointatthebottom.SupposeyoumakeaMartinibypouring vermouthintotheglasstoadepthof3cm,andthenaddinggin tobringthedepthto6cm.Whataretheproportionsofginand vermouth? Solution,p.269 18 ThecentralportionofaCDistakenupbytheholeandsome surroundingclearplastic,andthisareaisunavailableforstoring data.Theradiusofthecentralcircleisabout35%oftheradiusof thedata-storingarea.WhatpercentageoftheCD'sareaistherefore lost? p 19 Theone-litercubeinthephotohasbeenmarkedointo smallercubes,withlineardimensionsonetenththoseofthebig one.Whatisthevolumeofeachofthesmallcubes? Solution,p.270 62 Chapter1ScalingandOrder-of-MagnitudeEstimates

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Problem20. Problem22. 20 aBasedonthedenitionsofthesine,cosine,andtangent, whatunitsmusttheyhave?bAcuteformulafromtrigonometry letsyoundanyangleofatriangleifyouknowthelengthsof itssides.Usingthenotationshowninthegure,andletting s = a + b + c = 2behalftheperimeter,wehave tan A= 2= s s )]TJ/F20 10.9091 Tf 10.909 0 Td [(b s )]TJ/F20 10.9091 Tf 10.909 0 Td [(c s s )]TJ/F20 10.9091 Tf 10.909 0 Td [(a Showthattheunitsofthisequationmakesense.Inotherwords, checkthattheunitsoftheright-handsidearethesameasyour answertopartaofthequestion. Solution,p.270 21 Estimatethenumberofman-hoursrequiredforbuildingthe GreatWallofChina. Solution,p.270 22 aUsingthemicroscopephotointhegure,estimatethe massofaonecellofthe E.coli bacterium,whichisoneofthe mostcommononesinthehumanintestine.Notethescaleatthe lowerrightcorner,whichis1 m.Eachofthetubularobjectsin thecolumnisonecell.bThefecesinthehumanintestineare mostlybacteriasomedead,somealive,ofwhich E.coli isalarge andtypicalcomponent.Estimatethenumberofbacteriainyour intestines,andcomparewiththenumberofhumancellsinyour body,whichisbelievedtoberoughlyontheorderof10 13 .c Interpretingyourresultfrompartb,whatdoesthistellyouabout thesizeofatypicalhumancellcomparedtothesizeofatypical bacterialcell? Problems 63

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PartI MotioninOneDimension

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Ididn'tlearnuntilIwasnearlythroughwithcollegethatIcould understandabookmuchbetterifImentallyoutlineditformyself beforeIactuallybeganreading.It'satechniquethatwarnsmy braintogetlittlecerebrallefoldersreadyforthedierenttopics I'mgoingtolearn,andasI'mreadingitallowsmetosaytomyself, Oh,thereasonthey'retalkingaboutthisnowisbecausethey're preparingforthisotherthingthatcomeslater,"orIdon'tneedto sweatthedetailsofthisideanow,becausethey'regoingtoexplain itinmoredetaillateron." Atthispoint,you'reabouttodiveintothemainsubjectsof thisbook,whichareforceandmotion.Theconceptsyou'regoing tolearnbreakdownintothefollowingthreeareas: kinematics|howtodescribemotionnumerically dynamics|howforceaectsmotion vectors|amathematicalwayofhandlingthethree-dimensional natureofforceandmotion Roughlyspeaking,that'stheorderinwhichwe'llcoverthese threeareas,buttheearlierchaptersdocontainquiteabitofpreparationforthelatertopics.Forinstance,evenbeforethepresentpoint inthebookyou'velearnedabouttheNewton,aunitofforce.The discussionofforceproperlybelongstodynamics,whichwearen't tacklinghead-onforafewmorechapters,butI'vefoundthatwhen Iteachkinematicsithelpstobeabletorefertoforcesnowandthen toshowwhyitmakessensetodenecertainkinematicalconcepts. AndalthoughIdon'texplicitlyintroducevectorsuntilch.8,the groundworkisbeinglaidfortheminearlierchapters. Thegureonthenextpageisaroadmaptotherestofthebook. 66

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a / Rotation. b / Simultaneousrotationand motionthroughspace. c / Onepersonmightsaythatthe tippingchairwasonlyrotatingin acircleaboutitspointofcontact withtheoor,butanothercould describeitashavingbothrotation andmotionthroughspace. Chapter2 VelocityandRelative Motion 2.1TypesofMotion Ifyouhadtothinkconsciouslyinordertomoveyourbody,you wouldbeseverelydisabled.Evenwalking,whichweconsiderto benogreatfeat,requiresanintricateseriesofmotionsthatyour cerebrumwouldbeutterlyincapableofcoordinating.Thetaskof puttingonefootinfrontoftheotheriscontrolledbythemoreprimitivepartsofyourbrain,theonesthathavenotchangedmuchsince themammalsandreptileswenttheirseparateevolutionaryways. Thethinkingpartofyourbrainlimitsitselftogeneraldirectives suchaswalkfaster,"ordon'tsteponhertoes,"ratherthanmicromanagingeverycontractionandrelaxationofthehundredorso musclesofyourhips,legs,andfeet. Physicsisallabouttheconsciousunderstandingofmotion,but we'reobviouslynotimmediatelypreparedtounderstandthemost complicatedtypesofmotion.Instead,we'llusethedivide-andconquertechnique.We'llrstclassifythevarioustypesofmotion, andthenbeginourcampaignwithanattackonthesimplestcases. Tomakeitclearwhatweareandarenotreadytoconsider,weneed toexamineanddenecarefullywhattypesofmotioncanexist. Rigid-bodymotiondistinguishedfrommotionthatchanges anobject'sshape Nobody,withthepossibleexceptionofFredAstaire,cansimply glideforwardwithoutbendingtheirjoints.Walkingisthusanexampleinwhichthereisbothageneralmotionofthewholeobject andachangeintheshapeoftheobject.Anotherexampleisthe motionofajigglingwaterballoonasitiesthroughtheair.Weare notpresentlyattemptingamathematicaldescriptionofthewayin whichtheshapeofanobjectchanges.Motionwithoutachangein shapeiscalledrigid-bodymotion.Thewordbody"isoftenused inphysicsasasynonymforobject." Center-of-massmotionasopposedtorotation Aballerinaleapsintotheairandspinsaroundoncebeforelanding.Wefeelintuitivelythatherrigid-bodymotionwhileherfeet areothegroundconsistsoftwokindsofmotiongoingonsimul69

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e / Nomatterwhatpointyou hangthepearfrom,thestring linesupwiththepear'scenter ofmass.Thecenterofmass canthereforebedenedasthe intersectionofallthelinesmade byhangingthepearinthisway. NotethattheXinthegure shouldnotbeinterpretedas implyingthatthecenterofmass isonthesurfaceitisactually insidethepear. f / Thecircusperformershang withtheropespassingthrough theircentersofmass. taneously:arotationandamotionofherbodyasawholethrough space,alonganarc.Itisnotimmediatelyobvious,however,what isthemostusefulwaytodenethedistinctionbetweenrotation andmotionthroughspace.Imaginethatyouattempttobalancea chairanditfallsover.Onepersonmightsaythattheonlymotion wasarotationaboutthechair'spointofcontactwiththeoor,but anothermightsaythattherewasbothrotationandmotiondown andtotheside. d / Theleapingdancer'smotioniscomplicated,butthemotionof hercenterofmassissimple. Itturnsoutthatthereisoneparticularlynaturalandusefulway tomakeacleardenition,butitrequiresabriefdigression.Every objecthasabalancepoint,referredtoinphysicsasthe centerof mass .Foratwo-dimensionalobjectsuchasacardboardcutout,the centerofmassisthepointatwhichyoucouldhangtheobjectfrom astringandmakeitbalance.Inthecaseoftheballerinawhois likelytobethree-dimensionalunlessherdietisparticularlysevere, itmightbeapointeitherinsideoroutsideherbody,depending onhowsheholdsherarms.Evenifitisnotpracticaltoattacha stringtothebalancepointitself,thecenterofmasscanbedened asshowninguree. Whyisthecenterofmassconceptrelevanttothequestionof classifyingrotationalmotionasopposedtomotionthroughspace? Asillustratedinguresdandg,itturnsoutthatthemotionofan object'scenterofmassisnearlyalwaysfarsimplerthanthemotion ofanyotherpartoftheobject.Theballerina'sbodyisalargeobject withacomplexshape.Wemightexpectthathermotionwouldbe muchmorecomplicatedthanthemotionofasmall,simply-shaped 70 Chapter2VelocityandRelativeMotion

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h / Animproperlybalanced wheelhasacenterofmassthat isnotatitsgeometriccenter. Whenyougetanewtire,the mechanicclampslittleweightsto therimtobalancethewheel. i / Thistoywasintentionally designedsothatthemushroomshapedpieceofmetalontop wouldthrowoffthecenterof mass.Whenyouwinditup,the mushroomspins,butthecenter ofmassdoesn'twanttomove, sotherestofthetoytendsto counterthemushroom'smotion, causingthewholethingtojump around. object,sayamarble,thrownupatthesameangleastheangleat whichsheleapt.Butitturnsoutthatthemotionoftheballerina's centerofmassisexactlythesameasthemotionofthemarble.That is,themotionofthecenterofmassisthesameasthemotionthe ballerinawouldhaveifallhermasswasconcentratedatapoint.By restrictingourattentiontothemotionofthecenterofmass,wecan thereforesimplifythingsgreatly. g / Thesameleapingdancer,viewedfromabove.Hercenterof masstracesastraightline,butapointawayfromhercenterofmass, suchasherelbow,tracesthemuchmorecomplicatedpathshownbythe dots. Wecannowreplacetheambiguousideaofmotionasawhole throughspace"withthemoreusefulandbetterdenedconcept ofcenter-of-massmotion."Themotionofanyrigidbodycanbe cleanlysplitintorotationandcenter-of-massmotion.Bythisdenition,thetippingchairdoeshavebothrotationalandcenter-of-mass motion.Concentratingonthecenterofmassmotionallowsusto makeasimpliedmodelofthemotion,asifacomplicatedobject likeahumanbodywasjustamarbleorapoint-likeparticle.Science reallyneverdealswithreality;itdealswithmodelsofreality. Notethatthewordcenter"incenterofmass"isnotmeant toimplythatthecenterofmassmustlieatthegeometricalcenter ofanobject.Acarwheelthathasnotbeenbalancedproperlyhas acenterofmassthatdoesnotcoincidewithitsgeometricalcenter. Anobjectsuchasthehumanbodydoesnotevenhaveanobvious geometricalcenter. Itcanbehelpfultothinkofthecenterofmassastheaverage locationofallthemassintheobject.Withthisinterpretation, wecanseeforexamplethatraisingyourarmsaboveyourhead raisesyourcenterofmass,sincethehigherpositionofthearms' massraisestheaverage.Wewon'tbeconcernedrightnowwith calculatingcentersofmassmathematically;therelevantequations areinchapter4of ConservationLaws Ballerinasandprofessionalbasketballplayerscancreateanillusionofyinghorizontallythroughtheairbecauseourbrainsintuitivelyexpectthemtohaverigid-bodymotion,butthebodydoes notstayrigidwhileexecutingagrandjeteoraslamdunk.Thelegs arelowatthebeginningandendofthejump,butcomeuphigherat Section2.1TypesofMotion 71

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j / Axedpointonthedancer'sbodyfollowsatrajectorythatisatterthanwhatweexpect,creatinganillusionofight. themiddle.Regardlessofwhatthelimbsdo,thecenterofmasswill followthesamearc,butthelowpositionofthelegsatthebeginning andendmeansthatthetorsoishighercomparedtothecenterof mass,whileinthemiddleofthejumpitislowercomparedtothe centerofmass.Oureyefollowsthemotionofthetorsoandtries tointerpretitasthecenter-of-massmotionofarigidbody.But sincethetorsofollowsapaththatisatterthanweexpect,this attemptedinterpretationfails,andweexperienceanillusionthat thepersonisyinghorizontally. k / Example1. Thecenterofmassasanaverageexample1 Explainhowweknowthatthecenterofmassofeachobjectis atthelocationshowningurek. Thecenterofmassisasortofaverage,sotheheightofthe centersofmassin1and2hastobemidwaybetweenthetwo squares,becausethatheightistheaverageoftheheightsofthe twosquares.Example3isacombinationofexamples1and 2,sowecannditscenterofmassbyaveragingthehorizontal positionsoftheircentersofmass.Inexample4,eachsquare hasbeenskewedalittle,butjustasmuchmasshasbeenmoved upasdown,sotheaverageverticalpositionofthemasshasn't changed.Example5isclearlynotallthatdifferentfromexample 4,themaindifferencebeingaslightclockwiserotation,sojustas 72 Chapter2VelocityandRelativeMotion

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l / Thehigh-jumper'sbody passesoverthebar,buthis centerofmasspassesunderit. m / Self-checkB. inexample4,thecenterofmassmustbehanginginemptyspace, wherethereisn'tactuallyanymass.Horizontally,thecenterof massmustbebetweentheheelsandtoes,orelseitwouldn'tbe possibletostandwithouttippingover. Anotherinterestingexamplefromthesportsworldisthehigh jump,inwhichthejumper'scurvedbodypassesoverthebar,but thecenterofmasspassesunderthebar!Herethejumperlowershis legsandupperbodyatthepeakofthejumpinordertobringhis waisthighercomparedtothecenterofmass. Laterinthiscourse,we'llndthattherearemorefundamental reasonsbasedonNewton'slawsofmotionwhythecenterofmass behavesinsuchasimplewaycomparedtotheotherpartsofan object.We'realsopostponinganydiscussionofnumericalmethods forndinganobject'scenterofmass.Untillaterinthecourse,we willonlydealwiththemotionofobjects'centersofmass. Center-of-massmotioninonedimension Inadditiontorestrictingourstudyofmotiontocenter-of-mass motion,wewillbeginbyconsideringonlycasesinwhichthecenter ofmassmovesalongastraightline.Thiswillincludecasessuch asobjectsfallingstraightdown,oracarthatspeedsupandslows downbutdoesnotturn. Notethateventhoughwearenotexplicitlystudyingthemore complexaspectsofmotion,wecanstillanalyzethecenter-of-mass motionwhileignoringothertypesofmotionthatmightbeoccurring simultaneously.Forinstance,ifacatisfallingoutofatreeand isinitiallyupside-down,itgoesthroughaseriesofcontortionsthat bringitsfeetunderit.Thisisdenitelynotanexampleofrigidbodymotion,butwecanstillanalyzethemotionofthecat'scenter ofmassjustaswewouldforadroppingrock. self-checkA Considerapersonrunning,apersonpedalingonabicycle,aperson coastingonabicycle,andapersoncoastingoniceskates.Inwhich casesisthecenter-of-massmotionone-dimensional?Whichcasesare examplesofrigid-bodymotion? Answer,p.266 self-checkB Thegureshowsagymnastholdingontotheinsideofabigwheel. Frominsidethewheel,howcouldhemakeitrollonewayortheother? Answer,p.266 2.2DescribingDistanceandTime Center-of-massmotioninonedimensionisparticularlyeasytodeal withbecausealltheinformationaboutitcanbeencapsulatedintwo variables: x ,thepositionofthecenterofmassrelativetotheorigin, and t ,whichmeasuresapointintime.Forinstance,ifsomeone Section2.2DescribingDistanceandTime 73

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suppliedyouwithasucientlydetailedtableof x and t values,you wouldknowprettymuchalltherewastoknowaboutthemotionof theobject'scenterofmass. Apointintimeasopposedtoduration Inordinaryspeech,weusethewordtime"intwodierent senses,whicharetobedistinguishedinphysics.Itcanbeused, asinashorttime"orourtimehereonearth,"tomeanalength ordurationoftime,oritcanbeusedtoindicateaclockreading,as inIdidn'tknowwhattimeitwas,"ornow'sthetime."Insymbols, t isordinarilyusedtomeanapointintime,while t signies anintervalordurationintime.ThecapitalGreekletterdelta,, meansthechangein...,"i.e.adurationintimeisthechangeor dierencebetweenoneclockreadingandanother.Thenotation t doesnotsignifytheproductoftwonumbers,and t ,butrather onesinglenumber, t .Ifamatineebeginsatapointintime t =1 o'clockandendsat t =3o'clock,thedurationofthemoviewasthe changein t t =3hours )]TJ/F15 10.9091 Tf 10.91 0 Td [(1hour=2hours. Toavoidtheuseofnegativenumbersfor t ,wewritetheclock readingafter"totheleftoftheminussign,andtheclockreading before"totherightoftheminussign.Amorespecicdenition ofthedeltanotationisthereforethatdeltastandsforafterminus before." Eventhoughourdenitionofthedeltanotationguaranteesthat t ispositive,thereisnoreasonwhy t can'tbenegative.If t couldnotbenegative,whatwouldhavehappenedonesecondbefore t =0?Thatdoesn'tmeanthattimegoesbackward"inthesense thatadultscanshrinkintoinfantsandretreatintothewomb.It justmeansthatwehavetopickareferencepointandcallit t =0, andthentimesbeforethatarerepresentedbynegativevaluesof t Anexampleisthatayearlike2007A.D.canbethoughtofasa positive t value,whileonelike370B.C.isnegative.Similarly,when youhearacountdownforarocketlaunch,thephrasetminusten seconds"isawayofsaying t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(10s,where t =0isthetimeof blasto,and t> 0referstotimesafterlaunch. Althoughapointintimecanbethoughtofasaclockreading,it isusuallyagoodideatoavoiddoingcomputationswithexpressions suchas:35"thatarecombinationsofhoursandminutes.Times caninsteadbeexpressedentirelyintermsofasingleunit,suchas hours.Fractionsofanhourcanberepresentedbydecimalsrather thanminutes,andsimilarlyifaproblemisbeingworkedinterms ofminutes,decimalscanbeusedinsteadofseconds. self-checkC Ofthefollowingphrases,whichrefertopointsintime,whichreferto timeintervals,andwhichrefertotimeintheabstractratherthanasa measurablenumber? 74 Chapter2VelocityandRelativeMotion

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Thetimehascome. Timewaitsfornoman. Thewholetime,hehadspitonhischin. Answer,p.266 Positionasopposedtochangeinposition Aswithtime,adistinctionshouldbemadebetweenapoint inspace,symbolizedasacoordinate x ,andachangeinposition, symbolizedas x Aswith t x canbenegative.Ifatrainismovingdownthe tracks,notonlydoyouhavethefreedomtochooseanypointalong thetracksandcallit x =0,butit'salsouptoyoutodecidewhich sideofthe x =0pointispositive x andwhichsideisnegative x Sincewe'vedenedthedeltanotationtomeanafterminus before,"itispossiblethat x willbenegative,unlike t whichis guaranteedtobepositive.Supposewearedescribingthemotion ofatrainontrackslinkingTucsonandChicago.Asshowninthe gure,itisentirelyuptoyoutodecidewhichwayispositive. n / Twoequallyvalidwaysofdescribingthemotionofatrainfrom TucsontoChicago.Inexample1, thetrainhasapositive x asit goesfromEnidtoJoplin.In2, thesametraingoingforwardin thesamedirectionhasanegative x Notethatinadditionto x and x ,thereisathirdquantitywe coulddene,whichwouldbelikeanodometerreading,oractual distancetraveled.Ifyoudrive10miles,makeaU-turn,anddrive back10miles,thenyour x iszero,butyourcar'sodometerreading hasincreasedby20miles.Howeverimportanttheodometerreading istocarownersandusedcardealers,itisnotveryimportantin physics,andthereisnotevenastandardnameornotationforit. Thechangeinposition, x ,ismoreusefulbecauseitissomuch easiertocalculate:tocompute x ,weonlyneedtoknowthebeginningandendingpositionsoftheobject,notalltheinformation abouthowitgotfromonepositiontotheother. self-checkD Aballhitstheoor,bouncestoaheightofonemeter,falls,andhitsthe ooragain.Isthe x betweenthetwoimpactsequaltozero,one,or twometers? Answer,p.267 Section2.2DescribingDistanceandTime 75

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o / Motionwithconstantvelocity. p / Motionthatdecreases x isrepresentedwithnegative valuesof x and v q / Motionwithchangingvelocity. Framesofreference Theexampleaboveshowsthattherearetwoarbitrarychoices youhavetomakeinordertodeneapositionvariable, x .Youhave todecidewheretoput x =0,andalsowhichdirectionwillbepositive.Thisisreferredtoaschoosingacoordinatesystemorchoosing aframeofreference.Thetwotermsarenearlysynonymous,but therstfocusesmoreontheactual x variable,whilethesecondis moreofageneralwayofreferringtoone'spointofview.Aslongas youareconsistent,anyframeisequallyvalid.Youjustdon'twant tochangecoordinatesystemsinthemiddleofacalculation. Haveyoueverbeensittinginatraininastationwhensuddenly younoticethatthestationismovingbackward?Mostpeoplewould describethesituationbysayingthatyoujustfailedtonoticethat thetrainwasmoving|itonlyseemedlikethestationwasmoving. Butthisshowsthatthereisyetathirdarbitrarychoicethatgoes intochoosingacoordinatesystem:validframesofreferencecan dierfromeachotherbymovingrelativetooneanother.Itmight seemstrangethatanyonewouldbotherwithacoordinatesystem thatwasmovingrelativetotheearth,butforinstancetheframeof referencemovingalongwithatrainmightbefarmoreconvenient fordescribingthingshappeninginsidethetrain. 2.3GraphsofMotion;Velocity Motionwithconstantvelocity Inexampleo,anobjectismovingatconstantspeedinonedirection.Wecantellthisbecauseeverytwoseconds,itsposition changesbyvemeters. Inalgebranotation,we'dsaythatthegraphof x vs. t shows thesamechangeinposition, x =5.0m,overeachintervalof t =2.0s.Theobject'svelocityorspeedisobtainedbycalculating v = x= t =.0m = .0s=2.5m = s.Ingraphicalterms,the velocitycanbeinterpretedastheslopeoftheline.Sincethegraph isastraightline,itwouldn'thavematteredifwe'dtakenalonger timeintervalandcalculated v = x= t =.0m = .0s.The answerwouldstillhavebeenthesame,2.5m/s. Notethatwhenwedivideanumberthathasunitsofmetersby anothernumberthathasunitsofseconds,wegetunitsofmeters persecond,whichcanbewrittenm/s.Thisisanothercasewhere wetreatunitsasiftheywerealgebrasymbols,eventhoughthey're not. Inexamplep,theobjectismovingintheoppositedirection:as timeprogresses,its x coordinatedecreases.Recallingthedenition ofthenotationasafterminusbefore,"wendthat t isstill positive,but x mustbenegative.Theslopeofthelineistherefore 76 Chapter2VelocityandRelativeMotion

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r / Thevelocityatanygiven momentisdenedastheslope ofthetangentlinethroughthe relevantpointonthegraph. s / Example:ndingthevelocityatthepointindicatedwith thedot. t / Reversingthedirectionof motion. negative,andwesaythattheobjecthasanegativevelocity, v = x= t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5.0m = .0s= )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.5m = s.We'vealreadyseenthat theplusandminussignsof x valueshavetheinterpretationof tellinguswhichdirectiontheobjectmoved.Since t isalways positive,dividingby t doesn'tchangetheplusorminussign,and theplusandminussignsofvelocitiesaretobeinterpretedinthe sameway.Ingraphicalterms,apositiveslopecharacterizesaline thatgoesupaswegototheright,andanegativeslopetellsusthat thelinewentdownaswewenttotheright. Solvedproblem:light-yearspage89,problem4 Motionwithchangingvelocity Nowwhataboutagraphlikegureq?Thismightbeagraph ofacar'smotionasthedrivercruisesdownthefreeway,thenslows downtolookatacarcrashbythesideoftheroad,andthenspeeds upagain,disappointedthatthereisnothingdramaticgoingonsuch asamesorbabiestrappedintheircarseats.Notethatweare stilltalkingaboutone-dimensionalmotion.Justbecausethegraph iscurvydoesn'tmeanthatthecar'spathiscurvy.Thegraphisnot likeamap,andthehorizontaldirectionofthegraphrepresentsthe passingoftime,notdistance. Exampleqissimilartoexampleointhattheobjectmovesa totalof25.0minaperiodof10.0s,butitisnolongertruethatit makesthesameamountofprogresseverysecond.Thereisnowayto characterizetheentiregraphbyacertainvelocityorslope,because thevelocityisdierentateverymoment.Itwouldbeincorrectto saythatbecausethecarcovered25.0min10.0s,itsvelocitywas 2.5m/s.Itmovedfasterthanthatatthebeginningandend,but slowerinthemiddle.Theremayhavebeencertaininstantsatwhich thecarwasindeedgoing2.5m/s,butthespeedometersweptpast thatvaluewithoutsticking,"justasitswungthroughvariousother valuesofspeed.Idenitelywantmynextcartohaveaspeedometer calibratedinm/sandshowingbothnegativeandpositivevalues. Weassumethatourspeedometertellsuswhatishappeningto thespeedofourcarateveryinstant,buthowcanwedenespeed mathematicallyinacaselikethis?Wecan'tjustdeneitasthe slopeofthecurvygraph,becauseacurvedoesn'thaveasingle well-denedslopeasdoesaline.Amathematicaldenitionthat correspondedtothespeedometerreadingwouldhavetobeonethat attachedadierentvelocityvaluetoasinglepointonthecurve, i.e.,asingleinstantintime,ratherthantotheentiregraph.Ifwe wishtodenethespeedatoneinstantsuchastheonemarkedwith adot,thebestwaytoproceedisillustratedinr,wherewehave drawnthelinethroughthatpointcalledthetangentline,theline thathugsthecurve."Wecanthenadoptthefollowingdenition ofvelocity: Section2.3GraphsofMotion;Velocity 77

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denitionofvelocity Thevelocityofanobjectatanygivenmomentistheslopeofthe tangentlinethroughtherelevantpointonits x )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graph. Oneinterpretationofthisdenitionisthatthevelocitytellsus howmanymeterstheobjectwouldhavetraveledinonesecond,if ithadcontinuedmovingatthesamespeedforatleastonesecond. Tosomepeoplethegraphicalnatureofthisdenitionseemsinaccurate"ornotmathematical."Theequationbyitself,however, isonlyvalidifthevelocityisconstant,andsocannotserveasa generaldenition. Theslopeofthetangentlineexample2 Whatisthevelocityatthepointshownwithadotonthegraph? Firstwedrawthetangentlinethroughthatpoint.Tondthe slopeofthetangentline,weneedtopicktwopointsonit.Theoretically,theslopeshouldcomeoutthesameregardlessofwhich twopointswepick,butinpracticaltermswe'llbeabletomeasure moreaccuratelyifwepicktwopointsfairlyfarapart,suchasthe twowhitediamonds.Tosavework,wepickpointsthataredirectly abovelabeledpointsonthe t axis,sothat t =4.0siseasyto readoff.Onediamondlinesupwith x 17.5m,theotherwith x 26.5m,so x =9.0m.Thevelocityis x = t =2.2m = s. Conventionsaboutgraphing Theplacementof t onthehorizontalaxisand x ontheupright axismayseemlikeanarbitraryconvention,ormayevenhavedisturbedyou,sinceyouralgebrateacheralwaystoldyouthat x goes onthehorizontalaxisand y goesontheuprightaxis.Thereisa reasonfordoingitthisway,however.Inexamples,wehavean objectthatreversesitsdirectionofmotiontwice.Itcanonlybe inoneplaceatanygiventime,buttherecanbemorethanone timewhenitisatagivenplace.Forinstance,thisobjectpassed through x =17monthreeseparateoccasions,butthereisnoway itcouldhavebeeninmorethanoneplaceat t =5.0s.Resurrecting someterminologyyoulearnedinyourtrigonometrycourse,wesay that x isafunctionof t ,but t isnotafunctionof x .Insituations suchasthis,thereisausefulconventionthatthegraphshouldbe orientedsothatanyverticallinepassesthroughthecurveatonly onepoint.Puttingthe x axisacrossthepageand t uprightwould haveviolatedthisconvention.Topeoplewhoareusedtointerpretinggraphs,agraphthatviolatesthisconventionisasannoyingas ngernailsscratchingonachalkboard.Wesaythatthisisagraph of x versus t ."Iftheaxesweretheotherwayaround,itwould beagraphof t versus x ."Iremembertheversus"terminology byvisualizingthelabelsonthe x and t axesandrememberingthat whenyouread,yougofromlefttorightandfromtoptobottom. 78 Chapter2VelocityandRelativeMotion

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DiscussionquestionG. DiscussionQuestions A Parkisrunningslowlyingymclass,butthenhenoticesJenna watchinghim,sohespeedsuptotrytoimpressher.Whichofthegraphs couldrepresenthismotion? B Thegureshowsasequenceofpositionsfortworacingtractors. Comparethetractors'velocitiesastheraceprogresses.Whendothey havethesamevelocity?[BasedonaquestionbyLillianMcDermott.] C Ifanobjecthadastraight-linemotiongraphwith x =0and t 6 =0, whatwouldbetrueaboutitsvelocity?Whatwouldthislooklikeona graph?Whatabout t =0and x 6 =0? D Ifanobjecthasawavymotiongraphliketheoneingureton thepreviouspage,whicharethepointsatwhichtheobjectreversesits direction?Whatistrueabouttheobject'svelocityatthesepoints? E Discussanythingunusualaboutthefollowingthreegraphs. F Ihavebeenusingthetermvelocityandavoidingthemorecommon Englishwordspeed,becauseintroductoryphysicstextstypicallydene themtomeandifferentthings.Theyusethewordspeed,andthesymbol s tomeantheabsolutevalueofthevelocity, s = j v j .AlthoughI've chosennottoemphasizethisdistinctionintechnicalvocabulary,there areclearlytwodifferentconceptshere.Canyouthinkofanexampleof agraphof x -versust inwhichtheobjecthasconstantspeed,butnot constantvelocity? G Forthegraphshowninthegure,describehowtheobject'svelocity changes. H Twophysicistsduckoutofaboringscienticconferencetogo Section2.3GraphsofMotion;Velocity 79

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getbeer.Onthewaytothebar,theywitnessanaccidentinwhicha pedestrianisinjuredbyahit-and-rundriver.Acriminaltrialresults,and theymusttestify.Inhertestimony,Dr.TransverzWaivesays,Thecarwas movingalongprettyfast,I'dsaythevelocitywas+40mi/hr.Theysawthe oldladytoolate,andeventhoughtheyslammedonthebrakestheystill hitherbeforetheystopped.Thentheymadea U turnandheadedoff atavelocityofabout-20mi/hr,I'dsay.Dr.LongitudN.L.Vibrasheun says,Hewasreallygoingtoofast,maybehisvelocitywas-35or-40 mi/hr.AfterhehitMrs.Hapless,heturnedaroundandleftatavelocityof, oh,I'dguessmaybe+20or+25mi/hr.Istheirtestimonycontradictory? Explain. 2.4ThePrincipleofInertia Physicaleffectsrelateonlytoachangeinvelocity Considertwostatementsofakindthatwasatonetimemade withtheutmostseriousness: PeoplelikeGalileoandCopernicuswhosaytheearthisrotatingmustbecrazy.Weknowtheearthcan'tbemoving. Why,iftheearthwasreallyturningonceeveryday,thenour wholecitywouldhavetobemovinghundredsofleaguesin anhour.That'simpossible!Buildingswouldshakeontheir foundations.Gale-forcewindswouldknockusover.Trees wouldfalldown.TheMediterraneanwouldcomesweeping acrosstheeastcoastsofSpainandItaly.Andfurthermore, whatforcewouldbemakingtheworldturn? Allthistalkofpassengertrainsmovingatfortymilesanhour issheerhogwash!Atthatspeed,theairinapassengercompartmentwouldallbeforcedagainstthebackwall.Peoplein thefrontofthecarwouldsuffocate,andpeopleattheback woulddiebecauseinsuchconcentratedair,theywouldn'tbe abletoexpelabreath. Someoftheeectspredictedintherstquoteareclearlyjust basedonalackofexperiencewithrapidmotionthatissmoothand freeofvibration.Butthereisadeeperprincipleinvolved.Ineach case,thespeakerisassumingthatthemerefactofmotionmust havedramaticphysicaleects.Moresubtly,theyalsobelievethata forceisneededtokeepanobjectinmotion:therstpersonthinks aforcewouldbeneededtomaintaintheearth'srotation,andthe secondapparentlythinksoftherearwallaspushingontheairto keepitmoving. Commonmodernknowledgeandexperiencetellusthatthese people'spredictionsmusthavesomehowbeenbasedonincorrect reasoning,butitisnotimmediatelyobviouswherethefundamental awlies.It'soneofthosethingsafour-year-oldcouldinfuriate youbydemandingaclearexplanationof.Onewayofgettingat thefundamentalprincipleinvolvedistoconsiderhowthemodern 80 Chapter2VelocityandRelativeMotion

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v / WhydoesAristotlelook sosad?Hasherealizedthat hisentiresystemofphysicsis wrong? w / Theearthspins.People inShanghaisaythey'reatrest andpeopleinLosAngelesare moving.Angelenossaythesame abouttheShanghainese. x / Thejetsareatrest.The EmpireStateBuildingismoving. conceptoftheuniversediersfromthepopularconceptionatthe timeoftheItalianRenaissance.Tous,thewordearth"implies aplanet,oneofthenineplanetsofoursolarsystem,asmallball ofrockanddirtthatisofnosignicancetoanyoneintheuniverse exceptformembersofourspecies,whohappentoliveonit.To Galileo'scontemporaries,however,theearthwasthebiggest,most solid,mostimportantthinginallofcreation,nottobecompared withthewanderinglightsintheskyknownasplanets.Tous,the earthisjustanotherobject,andwhenwetalklooselyabouthow fast"anobjectsuchasacarisgoing,"wereallymeanthecarobject'svelocityrelativetotheearth-object. u / ThisAirForcedoctorvolunteeredtoridearocketsledasa medicalexperiment.Theobviouseffectsonhisheadandfacearenot becauseofthesled'sspeedbutbecauseofitsrapidchangesinspeed: increasingin2and3,anddecreasingin5and6.In4hisspeedis greatest,butbecausehisspeedisnotincreasingordecreasingvery muchatthismoment,thereislittleeffectonhim. Motionisrelative Accordingtoourmodernworld-view,itreallyisn'tthatreasonabletoexpectthataspecialforceshouldberequiredtomakethe airinthetrainhaveacertainvelocityrelativetoourplanet.After all,themoving"airinthemoving"trainmightjusthappento havezerovelocityrelativetosomeotherplanetwedon'tevenknow about.Aristotleclaimedthatthingsnaturally"wantedtobeat rest,lyingonthesurfaceoftheearth.ButexperimentafterexperSection2.4ThePrincipleofInertia 81

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DiscussionquestionA. DiscussionquestionB. DiscussionquestionD. imenthasshownthatthereisreallynothingsospecialaboutbeing atrestrelativetotheearth.Forinstance,ifamattressfallsoutof thebackofatruckonthefreeway,thereasonitrapidlycomesto restwithrespecttotheplanetissimplybecauseoffrictionforces exertedbytheasphalt,whichhappenstobeattachedtotheplanet. Galileo'sinsightsaresummarizedasfollows: Theprincipleofinertia Noforceisrequiredtomaintainmotionwithconstantvelocityin astraightline,andabsolutemotiondoesnotcauseanyobservable physicaleects. Therearemanyexamplesofsituationsthatseemtodisprovethe principleofinertia,buttheseallresultfromforgettingthatfriction isaforce.Forinstance,itseemsthataforceisneededtokeepa sailboatinmotion.Ifthewindstops,thesailboatstopstoo.But thewind'sforceisnottheonlyforceontheboat;thereisalsoa frictionalforcefromthewater.Ifthesailboatiscruisingandthe windsuddenlydisappears,thebackwardfrictionalforcestillexists, andsinceitisnolongerbeingcounteractedbythewind'sforward force,theboatstops.Todisprovetheprincipleofinertia,wewould havetondanexamplewhereamovingobjectsloweddowneven thoughnoforceswhatsoeverwereactingonit. self-checkE Whatisincorrectaboutthefollowingsupposedcounterexamplestothe principleofinertia? Whenastronautsblastoffinarocket,theirhugevelocitydoescause aphysicaleffectontheirbodiestheygetpressedbackintotheir seats,theeshontheirfacesgetsdistorted,andtheyhaveahardtime liftingtheirarms. Whenyou'redrivinginaconvertiblewiththetopdown,thewindin yourfaceisanobservablephysicaleffectofyourabsolutemotion. Answer,p.267 Solvedproblem:abugonawheelpage89,problem7 DiscussionQuestions A Apassengeronacruiseshipnds,whiletheshipisdocked,that hecanleapoffoftheupperdeckandjustbarelymakeitintothepool onthelowerdeck.Iftheshipleavesdockandiscruisingrapidly,willthis adrenalinejunkiestillbeabletomakeit? B Youareapassengerintheopenbaskethangingunderahelium balloon.Theballoonisbeingcarriedalongbythewindataconstant velocity.Ifyouareholdingaaginyourhand,willtheagwave?Ifso, whichway?[BasedonaquestionfromPSSCPhysics.] C Aristotlestatedthatallobjectsnaturallywantedtocometorest,with theunspokenimplicationthatrestwouldbeinterpretedrelativetothe surfaceoftheearth.SupposewegobackintimeandtransportAristotle 82 Chapter2VelocityandRelativeMotion

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tothemoon.Aristotleknew,aswedo,thatthemooncirclestheearth;he saiditdidn'tfalldownbecause,likeeverythingelseintheheavens,itwas madeoutofsomespecialsubstancewhosenaturalbehaviorwastogo incirclesaroundtheearth.Weland,puthiminaspacesuit,andkick himoutthedoor.Whatwouldheexpecthisfatetobeinthissituation?If intelligentcreaturesinhabitedthemoon,andoneofthemindependently cameupwiththeequivalentofAristotelianphysics,whatwouldtheythink aboutobjectscomingtorest? D Thebottleissittingonaleveltableinatrain'sdiningcar,butthe surfaceofthebeeristilted.Whatcanyouinferaboutthemotionofthe train? 2.5AdditionofVelocities Additionofvelocitiestodescriberelativemotion Sinceabsolutemotioncannotbeunambiguouslymeasured,the onlywaytodescribemotionunambiguouslyistodescribethemotion ofoneobjectrelativetoanother.Symbolically,wecanwrite v PQ forthevelocityofobject P relativetoobject Q Velocitiesmeasuredwithrespecttodierentreferencepointscan becomparedbyaddition.Inthegurebelow,theball'svelocity relativetothecouchequalstheball'svelocityrelativetothetruck plusthetruck'svelocityrelativetothecouch: v BC = v BT + v TC =5cm = s+10cm = s =15cm = s Thesameequationcanbeusedforanycombinationofthree objects,justbysubstitutingtherelevantsubscriptsforB,T,and C.Justremembertowritetheequationsothatthevelocitiesbeing addedhavethesamesubscripttwiceinarow.Inthisexample,if youreadothesubscriptsgoingfromlefttoright,yougetBC ::: = ::: BTTC.Thefactthatthetwoinside"subscriptsontherightare thesamemeansthattheequationhasbeensetupcorrectly.Notice howsubscriptsontheleftlookjustlikethesubscriptsontheright, butwiththetwoT'seliminated. Negativevelocitiesinrelativemotion Mydiscussionofhowtointerpretpositiveandnegativesignsof velocitymayhaveleftyouwonderingwhyweshouldbother.Why notjustmakevelocitypositivebydenition?Theoriginalreason whynegativenumberswereinventedwasthatbookkeepersdecided itwouldbeconvenienttousethenegativenumberconceptforpaymentstodistinguishthemfromreceipts.Itwasjustplaineasierthan writingreceiptsinblackandpaymentsinredink.Afteraddingup Section2.5AdditionofVelocities 83

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y / Thesetwohighlycompetentphysicistsdisagreeonabsolutevelocities,buttheywouldagreeonrelativevelocities.PurpleDino considersthecouchtobeatrest,whileGreenDinothinksofthetruckas beingatrest.Theyagree,however,thatthetruck'svelocityrelativetothe couchis v TC =10cm/s,theball'svelocityrelativetothetruckis v BT =5 cm/s,andtheball'svelocityrelativetothecouchis v BC = v BT + v TC =15 cm/s. yourmonth'spositivereceiptsandnegativepayments,youeithergot apositivenumber,indicatingprot,oranegativenumber,showing aloss.Youcouldthenshowthattotalwithahigh-tech+"or )]TJ/F15 10.9091 Tf 8.484 0 Td [(" sign,insteadoflookingaroundfortheappropriatebottleofink. Nowadaysweusepositiveandnegativenumbersforallkinds ofthings,butineverycasethepointisthatitmakessenseto addandsubtractthosethingsaccordingtotherulesyoulearned ingradeschool,suchasminusaminusmakesaplus,whythisis trueweneednotdiscuss."Addingvelocitieshasthesignicance ofcomparingrelativemotion,andwiththisinterpretationnegative andpositivevelocitiescanbeusedwithinaconsistentframework. Forexample,thetruck'svelocityrelativetothecouchequalsthe truck'svelocityrelativetotheballplustheball'svelocityrelative tothecouch: v TC = v TB + v BC = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5cm = s+15cm = s =10cm = s Ifwedidn'thavethetechnologyofnegativenumbers,wewouldhave hadtorememberacomplicatedsetofrulesforaddingvelocities: ifthetwoobjectsarebothmovingforward,youadd,ifoneis movingforwardandoneismovingbackward,yousubtract,but 84 Chapter2VelocityandRelativeMotion

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z / Graphsof x and v versus t foracaracceleratingawayfrom atrafclight,andthenstopping foranotherredlight. ifthey'rebothmovingbackward,youadd.Whatapainthatwould havebeen. Solvedproblem:twodimensionspage90,problem10 DiscussionQuestions A Interpretthegeneralrule v AB = )]TJ/F78 9.9627 Tf 7.749 0 Td [(v BA inwords. B Wa-Chuenslipsawayfromherfatheratthemallandwalksupthe downescalator,sothatshestaysinoneplace.Writethisintermsof symbols. 2.6GraphsofVelocityVersusTime Sincechangesinvelocityplaysuchaprominentroleinphysics,we needabetterwaytolookatchangesinvelocitythanbylaboriously drawingtangentlineson x -versust graphs.Agoodmethodisto drawagraphofvelocityversustime.Theexamplesontheleftshow the x )]TJ/F20 10.9091 Tf 10.361 0 Td [(t and v )]TJ/F20 10.9091 Tf 10.361 0 Td [(t graphsthatmightbeproducedbyacarstarting fromatraclight,speedingup,cruisingforawhileatconstant speed,andnallyslowingdownforastopsign.Ifyouhaveanair freshenerhangingfromyourrear-viewmirror,thenyouwillseean eectontheairfreshenerduringthebeginningandendingperiods whenthevelocityischanging,butitwillnotbetiltedduringthe periodofconstantvelocityrepresentedbytheatplateauinthe middleofthe v )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graph. Studentsoftenmixupthethingsbeingrepresentedonthesetwo typesofgraphs.Forinstance,manystudentslookingatthetop graphsaythatthecarisspeedingupthewholetime,sincethe graphisbecominggreater."Whatisgettinggreaterthroughoutthe graphis x ,not v Similarly,manystudentswouldlookatthebottomgraphand thinkitshowedthecarbackingup,becauseit'sgoingbackwards attheend."Butwhatisdecreasingattheendis v ,not x .Having boththe x )]TJ/F20 10.9091 Tf 11.63 0 Td [(t and v )]TJ/F20 10.9091 Tf 11.63 0 Td [(t graphsinfrontofyoulikethisisoften convenient,becauseonegraphmaybeeasiertointerpretthanthe otherforaparticularpurpose.Stackingthemlikethismeansthat correspondingpointsonthetwographs'timeaxesarelinedupwith eachothervertically.However,onethingthatisalittlecounterintuitiveaboutthearrangementisthatinasituationlikethisone involvingacar,oneistemptedtovisualizethelandscapestretching alongthehorizontalaxisofoneofthegraphs.Thehorizontalaxes, however,representtime,notposition.Thecorrectwaytovisualize thelandscapeisbymentallyrotatingthehorizon90degreescounterclockwiseandimaginingitstretchingalongtheuprightaxisofthe x t graph,whichistheonlyaxisthatrepresentsdierentpositions inspace. Section2.6GraphsofVelocityVersusTime 85

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2.7 R ApplicationsofCalculus Theintegralsymbol, R ,intheheadingforthissectionindicatesthat itismeanttobereadbystudentsincalculus-basedphysics.Studentsinanalgebra-basedphysicscourseshouldskipthesesections. Thecalculus-relatedsectionsinthisbookaremeanttobeusable bystudentswhoaretakingcalculusconcurrently,soatthisearly pointinthephysicscourseIdonotassumeyouknowanycalculus yet.Thissectionisthereforenotmuchmorethanaquickpreviewof calculus,tohelpyourelatewhatyou'relearninginthetwocourses. Newtonwastherstpersontogureoutthetangent-linedenitionofvelocityforcaseswherethe x )]TJ/F20 10.9091 Tf 11.165 0 Td [(t graphisnonlinear.BeforeNewton,nobodyhadconceptualizedthedescriptionofmotion intermsof x )]TJ/F20 10.9091 Tf 11.67 0 Td [(t and v )]TJ/F20 10.9091 Tf 11.67 0 Td [(t graphs.Inadditiontothegraphical techniquesdiscussedinthischapter,Newtonalsoinventedasetof symbolictechniquescalledcalculus.Ifyouhaveanequationfor x intermsof t ,calculusallowsyou,forinstance,tondanequation for v intermsof t .Incalculusterms,wesaythatthefunction v t isthederivativeofthefunction x t .Inotherwords,thederivative ofafunctionisanewfunctionthattellshowrapidlytheoriginal functionwaschanging.WenowuseneitherNewton'snameforhis techniquehecalleditthemethodofuxions"norhisnotation. ThemorecommonlyusednotationisduetoNewton'sGermancontemporaryLeibnitz,whomtheEnglishaccusedofplagiarizingthe calculusfromNewton.IntheLeibnitznotation,wewrite v = d x d t toindicatethatthefunction v t equalstheslopeofthetangentline ofthegraphof x t ateverytime t .TheLeibnitznotationismeant toevokethedeltanotation,butwithaverysmalltimeinterval. Becausethed x andd t arethoughtofasverysmall x 'sand t 's, i.e.,verysmalldierences,thepartofcalculusthathastodowith derivativesiscalleddierentialcalculus. Dierentialcalculusconsistsofthreethings: Theconceptanddenitionofthederivative,whichiscovered inthisbook,butwhichwillbediscussedmoreformallyinyour mathcourse. TheLeibnitznotationdescribedabove,whichyou'llneedto getmorecomfortablewithinyourmathcourse. Asetofrulesthatallowsyoutondanequationforthederivativeofagivenfunction.Forinstance,ifyouhappenedtohave asituationwherethepositionofanobjectwasgivenbythe equation x =2 t 7 ,youwouldbeabletousethoserulesto ndd x= d t =14 t 6 .Thisbagoftricksiscoveredinyourmath course. 86 Chapter2VelocityandRelativeMotion

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Summary SelectedVocabulary centerofmass..thebalancepointofanobject velocity......therateofchangeofposition;theslopeofthe tangentlineonan x )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graph. Notation x ..........apointinspace t ..........apointintime,aclockreading .........changein;"thevalueofavariableafterwards minusitsvaluebefore x ........adistance,ormorepreciselyachangein x whichmaybelessthanthedistancetraveled; itsplusorminussignindicatesdirection t .........adurationoftime v ..........velocity v AB ........thevelocityofobjectArelativetoobjectB OtherTerminologyandNotation displacement..anameforthesymbol x speed.......theabsolutevalueofthevelocity,i.e.,thevelocitystrippedofanyinformationaboutits direction Summary Anobject'scenterofmassisthepointatwhichitcanbebalanced.Forthetimebeing,wearestudyingthemathematicaldescriptiononlyofthemotionofanobject'scenterofmassincases restrictedtoonedimension.Themotionofanobject'scenterof massisusuallyfarsimplerthanthemotionofanyofitsotherparts. Itisimportanttodistinguishlocation, x ,fromdistance, x andclockreading, t ,fromtimeinterval t .Whenanobject's x )]TJ/F20 10.9091 Tf 10.516 0 Td [(t graphislinear,wedeneitsvelocityastheslopeoftheline, x= t Whenthegraphiscurved,wegeneralizethedenitionsothatthe velocityistheslopeofthetangentlineatagivenpointonthegraph. Galileo'sprincipleofinertiastatesthatnoforceisrequiredto maintainmotionwithconstantvelocityinastraightline,andabsolutemotiondoesnotcauseanyobservablephysicaleects.Things typicallytendtoreducetheirvelocityrelativetothesurfaceofour planetonlybecausetheyarephysicallyrubbingagainsttheplanet orsomethingattachedtotheplanet,notbecausethereisanything specialaboutbeingatrestwithrespecttotheearth'ssurface.When itseems,forinstance,thataforceisrequiredtokeepabooksliding acrossatable,infacttheforceisonlyservingtocancelthecontrary forceoffriction. Absolutemotionisnotawell-denedconcept,andiftwoobserversarenotatrestrelativetooneanothertheywilldisagree abouttheabsolutevelocitiesofobjects.Theywill,however,agree Summary 87

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aboutrelativevelocities.IfobjectAisinmotionrelativetoobject B ,and B isinmotionrelativeto C ,thenA'svelocityrelativeto C isgivenby v AC = v AB + v BC .Positiveandnegativesignsareused toindicatethedirectionofanobject'smotion. 88 Chapter2VelocityandRelativeMotion

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Problem7. Problem1. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Thegraphshowsthemotionofacarstuckinstop-and-go freewaytrac.aIfyouonlyknewhowfarthecarhadgone duringthisentiretimeperiod,whatwouldyouthinkitsvelocity was?bWhatisthecar'smaximumvelocity? p 2 aLet bethelatitudeofapointontheEarth'ssurface. Deriveanalgebraequationforthedistance, L ,traveledbythat pointduringonerotationoftheEarthaboutitsaxis,i.e.,overone day,expressedintermsof L ,and R ,theradiusoftheearth. Check:Yourequationshouldgive L =0fortheNorthPole. bAtwhatspeedisFullerton,atlatitude =34 ,movingwith therotationoftheEarthaboutitsaxis?Giveyouranswerinunits ofmi/h.[Seethetableinthebackofthebookfortherelevant data.] p 3 Apersonisparachutejumping.Duringthetimebetween whensheleapsoutoftheplaneandwhensheopensherchute,her altitudeisgivenbytheequation y =m )]TJ/F15 10.9091 Tf 10.909 0 Td [(m = s h t +.0s e )]TJ/F21 7.9701 Tf 6.586 0 Td [(t= 5.0s i Findhervelocityat t =7.0s.Thiscanbedoneonacalculator, withoutknowingcalculus.Becauseofairresistance,hervelocity doesnotincreaseatasteadyrateasitwouldforanobjectfalling invacuum. p ? 4 Alight-yearisaunitofdistanceusedinastronomy,anddened asthedistancelighttravelsinoneyear.Thespeedoflightis3.0 10 8 m/s.Findhowmanymetersthereareinonelight-year,expressing youranswerinscienticnotation. Solution,p.270 5 You'restandinginafreighttrain,andhavenowaytoseeout. Ifyouhavetoleantostayonyourfeet,what,ifanything,doesthat tellyouaboutthetrain'svelocity?Explain. Solution,p.270 6 Ahoneybee'spositionasafunctionoftimeisgivenby x = 10 t )]TJ/F20 10.9091 Tf 10.661 0 Td [(t 3 ,where t isinsecondsand x inmeters.Whatisitsvelocity at t =3.0s? R 7 Thegureshowsthemotionofapointontherimofarolling wheel.Theshapeiscalledacycloid.SupposebugAisridingon therimofthewheelonabicyclethatisrolling,whilebugBison thespinningwheelofabikethatissittingupsidedownontheoor. BugAismovingalongacycloid,whilebugBismovinginacircle. Bothwheelsaredoingthesamenumberofrevolutionsperminute. Whichbughasahardertimeholdingon,ordotheynditequally dicult? Solution,p.270 Problems 89

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Problem8. 8 Peanutplantsfolduptheirleavesatnight.Estimatethetop speedofthetipofoneoftheleavesshowninthegure,expressing yourresultinscienticnotationinSIunits. p 9 aTranslatethefollowinginformationintosymbols,using thenotationwithtwosubscriptsintroducedinsection2.5.Eowyn isridingonherhorseatavelocityof11m/s.Shetwistsaroundin hersaddleandresanarrowbackward.Herbowresarrowsat25 m/s.bFindthespeedofthearrowrelativetotheground. 10 Ourfulldiscussionoftwo-andthree-dimensionalmotionis postponeduntilthesecondhalfofthebook,buthereisachanceto usealittlemathematicalcreativityinanticipationofthatgeneralization.Supposeashipissailingeastatacertainspeed v ,anda passengeriswalkingacrossthedeckatthesamespeed v ,sothat histrackacrossthedeckisperpendiculartotheship'scenter-line. Whatishisspeedrelativetothewater,andinwhatdirectionishe movingrelativetothewater? Solution,p.270 11 FreddiFish TM hasapositionasafunctionoftimegivenby x = a= b + t 2 .Findhermaximumspeed. R 12 Drivingalonginyourcar,youtakeyourfootothegas, andyourspeedometershowsareductioninspeed.Describeaframe ofreferenceinwhichyourcarwas speedingup duringthatsame periodoftime.Theframeofreferenceshouldbedenedbyan observerwho,althoughperhapsinmotionrelativetotheearth,is notchangingherownspeedordirectionofmotion. 13 Thegureshowsthemotionofabluentuna,asmeasured byaradiotagBlocketal.,Nature,v.434,p.1121,2005,over thecourseofseveralyears.Untilthisstudy,ithadbeenbelieved thatthepopulationsoftheshintheeasternandwesternAtlantic wereseparate,butthisparticularshwasobservedtocrossthe entireAtlanticOcean,fromVirginiatoIreland.PointsA,B,andC showaperiodofonemonth,duringwhichtheshmadethemost rapidprogress.Estimateitsspeedduringthatmonth,inunitsof kilometersperhour. p Problem13. 90 Chapter2VelocityandRelativeMotion

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Galileo'scontradictionofAristotlehadseriousconsequences.Hewas interrogatedbytheChurchauthoritiesandconvictedofteachingthatthe earthwentaroundthesunasamatteroffactandnot,ashehadpromised previously,asameremathematicalhypothesis.Hewasplacedunderpermanenthousearrest,andforbiddentowriteaboutorteachhistheories. Immediatelyafterbeingforcedtorecanthisclaimthattheearthrevolved aroundthesun,theoldmanissaidtohavemuttereddeantlyandyet itdoesmove.Thestoryisdramatic,buttherearesomeomissionsin thecommonlytaughtheroicversion.TherewasarumorthattheSimpliciocharacterrepresentedthePope.Also,someoftheideasGalileo advocatedhadcontroversialreligiousovertones.Hebelievedintheexistenceofatoms,andatomismwasthoughtbysomepeopletocontradict theChurch'sdoctrineoftransubstantiation,whichsaidthatintheCatholic mass,theblessingofthebreadandwineliterallytransformedtheminto theeshandbloodofChrist.Hissupportforacosmologyinwhichthe earthcircledthesunwasalsodisreputablebecauseoneofitssupporters,GiordanoBruno,hadalsoproposedabizarresynthesisofChristianity withtheancientEgyptianreligion. Chapter3 AccelerationandFreeFall 3.1TheMotionofFallingObjects Themotionoffallingobjectsisthesimplestandmostcommon exampleofmotionwithchangingvelocity.Theearlypioneersof 91

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a / Galileodroppedacannonball andamusketballsimultaneously fromatower,andobservedthat theyhitthegroundatnearlythe sametime.Thiscontradicted Aristotle'slong-acceptedidea thatheavierobjectsfellfaster. physicshadacorrectintuitionthatthewaythingsdropwasamessagedirectlyfromNatureherselfabouthowtheuniverseworked. Otherexamplesseemlesslikelytohavedeepsignicance.Awalking personwhospeedsupismakingaconsciouschoice.Ifonestretchof ariverowsmorerapidlythananother,itmaybeonlybecausethe channelisnarrowerthere,whichisjustanaccidentofthelocalgeography.Butthereissomethingimpressivelyconsistent,universal, andinexorableaboutthewaythingsfall. Standupnowandsimultaneouslydropacoinandabitofpaper sidebyside.Thepapertakesmuchlongertohittheground.That's whyAristotlewrotethatheavyobjectsfellmorerapidly.Europeans believedhimfortwothousandyears. Nowrepeattheexperiment,butmakeitintoaracebetweenthe coinandyourshoe.Myownshoeisabout50timesheavierthan thenickelIhadhandy,butitlookstomeliketheyhitthegroundat exactlythesamemoment.SomuchforAristotle!Galileo,whohad aairforthetheatrical,didtheexperimentbydroppingabullet andaheavycannonballfromatalltower.Aristotle'sobservations hadbeenincomplete,hisinterpretationavastoversimplication. ItisinconceivablethatGalileowastherstpersontoobservea discrepancywithAristotle'spredictions.Galileowastheonewho changedthecourseofhistorybecausehewasabletoassemblethe observationsintoacoherentpattern,andalsobecausehecarried outsystematicquantitativenumericalmeasurementsratherthan justdescribingthingsqualitatively. Whyisitthatsomeobjects,likethecoinandtheshoe,havesimilarmotion,butothers,likeafeatherorabitofpaper,aredierent? Galileospeculatedthatinadditiontotheforcethatalwayspullsobjectsdown,therewasanupwardforceexertedbytheair.Anyone canspeculate,butGalileowentbeyondspeculationandcameup withtwocleverexperimentstoprobetheissue.First,heexperimentedwithobjectsfallinginwater,whichprobedthesameissues butmadethemotionslowenoughthathecouldtaketimemeasurementswithaprimitivependulumclock.Withthistechnique,he establishedthefollowingfacts: Allheavy,streamlinedobjectsforexampleasteelroddropped point-downreachthebottomofthetankinaboutthesame amountoftime,onlyslightlylongerthanthetimetheywould taketofallthesamedistanceinair. Objectsthatarelighterorlessstreamlinedtakealongertime toreachthebottom. Thissupportedhishypothesisabouttwocontraryforces.He imaginedanidealizedsituationinwhichthefallingobjectdidnot havetopushitswaythroughanysubstanceatall.Fallinginair 92 Chapter3AccelerationandFreeFall

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c / The v )]TJ/F78 9.9627 Tf 11.74 0 Td [(t graphofafalling objectisaline. d / Galileo'sexperimentsshow thatallfallingobjectshavethe samemotionifairresistanceis negligible. e / 1.Aristotlesaidthatheavier objectsfellfasterthanlighter ones.2.Iftworocksaretied together,thatmakesanextraheavyrock,whichshouldfall faster.3.ButAristotle'stheory wouldalsopredictthatthelight rockwouldholdbacktheheavy rock,resultinginaslowerfall. wouldbemorelikethisidealcasethanfallinginwater,buteven athin,sparsemediumlikeairwouldbesucienttocauseobvious eectsonfeathersandotherlightobjectsthatwerenotstreamlined. Today,wehavevacuumpumpsthatallowustosucknearlyallthe airoutofachamber,andifwedropafeatherandarocksideby sideinavacuum,thefeatherdoesnotlagbehindtherockatall. Howthespeedofafallingobjectincreaseswithtime Galileo'ssecondstrokeofgeniuswastondawaytomakequantitativemeasurementsofhowthespeedofafallingobjectincreased asitwentalong.Againitwasproblematictomakesucientlyaccuratetimemeasurementswithprimitiveclocks,andagainhefounda trickywaytoslowthingsdownwhilepreservingtheessentialphysicalphenomena:heletaballrolldownaslopeinsteadofdroppingit vertically.Thesteepertheincline,themorerapidlytheballwould gainspeed.Withoutamodernvideocamera,Galileohadinvented awaytomakeaslow-motionversionoffalling. b / Velocityincreasesmoregraduallyonthegentleslope,butthe motionisotherwisethesameasthemotionofafallingobject. AlthoughGalileo'sclockswereonlygoodenoughtodoaccurate experimentsatthesmallerangles,hewascondentaftermaking asystematicstudyatavarietyofsmallanglesthathisbasicconclusionsweregenerallyvalid.Statedinmodernlanguage,whathe foundwasthatthevelocity-versus-timegraphwasaline.Inthelanguageofalgebra,weknowthatalinehasanequationoftheform y = ax + b ,butourvariablesare v and t ,soitwouldbe v = at + b Theconstant b canbeinterpretedsimplyastheinitialvelocityof theobject,i.e.,itsvelocityatthetimewhenwestartedourclock, whichweconventionallywriteas v o self-checkA Anobjectisrollingdownanincline.Afterithasbeenrollingforashort time,itisfoundtotravel13cmduringacertainone-secondinterval. Duringthesecondafterthat,ifgoes16cm.Howmanycmwillittravel inthesecondafterthat? Answer,p.267 Section3.1TheMotionofFallingObjects 93

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Acontradictioninaristotle'sreasoning Galileo'sinclined-planeexperimentdisprovedthelong-accepted claimbyAristotlethatafallingobjecthadadenitenaturalfalling speed"proportionaltoitsweight.Galileohadfoundthatthespeed justkeptonincreasing,andweightwasirrelevantaslongasair frictionwasnegligible.NotonlydidGalileoproveexperimentally thatAristotlehadbeenwrong,buthealsopointedoutalogical contradictioninAristotle'sownreasoning.Simplicio,thestupid character,mouthstheacceptedAristotelianwisdom: S IMPLICIO :Therecanbenodoubtbutthataparticularbody ...hasaxedvelocitywhichisdeterminedbynature... S ALVIATI :Ifthenwetaketwobodieswhosenaturalspeeds aredifferent,itisclearthat,[accordingtoAristotle],onunitingthetwo,themorerapidonewillbepartlyheldbackby theslower,andtheslowerwillbesomewhathastenedbythe swifter.Doyounotagreewithmeinthisopinion? S IMPLICIO :Youareunquestionablyright. S ALVIATI :Butifthisistrue,andifalargestonemoveswitha speedof,say,eight[unspeciedunits]whileasmallermoves withaspeedoffour,thenwhentheyareunited,thesystem willmovewithaspeedlessthaneight;butthetwostones whentiedtogethermakeastonelargerthanthatwhichbefore movedwithaspeedofeight.Hencetheheavierbodymoves withlessspeedthanthelighter;aneffectwhichiscontraryto yoursupposition.Thusyouseehow,fromyourassumption thattheheavierbodymovesmorerapidlythanthelighterone, Iinferthattheheavierbodymovesmoreslowly. Whatisgravity? ThephysicistRichardFeynmanlikedtotellastoryabouthow whenhewasalittlekid,heaskedhisfather,Whydothingsfall?" Asanadult,hepraisedhisfatherforanswering,Nobodyknowswhy thingsfall.It'sadeepmystery,andthesmartestpeopleintheworld don'tknowthebasicreasonforit."Contrastthatwiththeaverage person'so-the-cuanswer,Oh,it'sbecauseofgravity."Feynman likedhisfather'sanswer,becausehisfatherrealizedthatsimply givinganametosomethingdidn'tmeanthatyouunderstoodit. TheradicalthingaboutGalileo'sandNewton'sapproachtoscience wasthattheyconcentratedrstondescribingmathematicallywhat reallydidhappen,ratherthanspendingalotoftimeonuntestable speculationsuchasAristotle'sstatementthatThingsfallbecause theyaretryingtoreachtheirnaturalplaceincontactwiththe earth."Thatdoesn'tmeanthatsciencecanneveranswerthewhy" questions.Overthenextmonthortwoasyoudelvedeeperinto physics,youwilllearnthattherearemorefundamentalreasonswhy allfallingobjectshave v )]TJ/F20 10.9091 Tf 10.813 0 Td [(t graphswiththesameslope,regardless 94 Chapter3AccelerationandFreeFall

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f / Example1. g / Example2. oftheirmass.Nevertheless,themethodsofsciencealwaysimpose limitsonhowdeepourexplanationcango. 3.2Acceleration Denitionofaccelerationforlinear v )]TJ/F78 10.9091 Tf 10.909 0 Td [(t graphs Galileo'sexperimentwithdroppingheavyandlightobjectsfrom atowershowedthatallfallingobjectshavethesamemotion,andhis inclined-planeexperimentsshowedthatthemotionwasdescribedby v = at + v o .Theinitialvelocity v o dependsonwhetheryoudropthe objectfromrestorthrowitdown,butevenifyouthrowitdown, youcannotchangetheslope, a ,ofthe v )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graph. Sincetheseexperimentsshowthatallfallingobjectshavelinear v )]TJ/F20 10.9091 Tf 11.267 0 Td [(t graphswiththesameslope,theslopeofsuchagraphis apparentlyanimportantandusefulquantity.Weusethewordacceleration,andthesymbol a ,fortheslopeofsuchagraph.Insymbols, a = v= t.Theaccelerationcanbeinterpretedastheamountof speedgainedineverysecond,andithasunitsofvelocitydividedby time,i.e.,meterspersecondpersecond,"orm/s/s.Continuingto treatunitsasiftheywerealgebrasymbols,wesimplifym/s/s"to readm = s 2 ."Accelerationcanbeausefulquantityfordescribing othertypesofmotionbesidesfalling,andthewordandthesymbol a "canbeusedinamoregeneralcontext.Wereservethemore specializedsymbol g "fortheaccelerationoffallingobjects,which onthesurfaceofourplanetequals9.8m = s 2 .Oftenwhendoing approximatecalculationsormerelyillustrativenumericalexamples itisgoodenoughtouse g =10m = s 2 ,whichisobyonly2%. Findingnalspeed,giventimeexample1 Adespondentphysicsstudentjumpsoffabridge,andfallsfor threesecondsbeforehittingthewater.Howfastishegoingwhen hehitsthewater? Approximating g as10m = s 2 ,hewillgain10m/sofspeedeach second.Afteronesecond,hisvelocityis10m/s,aftertwosecondsitis20m/s,andonimpact,afterfallingforthreeseconds, heismovingat30m/s. Extractingaccelerationfromagraphexample2 The x )]TJ/F78 10.9091 Tf 11.473 0 Td [(t and v )]TJ/F78 10.9091 Tf 11.473 0 Td [(t graphsshowthemotionofacarstarting fromastopsign.Whatisthecar'sacceleration? Accelerationisdenedastheslopeofthev-tgraph.Thegraph risesby3m/sduringatimeintervalof3s,sotheaccelerationis m = s = s=1m = s 2 Incorrectsolution#1:Thenalvelocityis3m/s,andacceleration isvelocitydividedbytime,sotheaccelerationism = s = s= 0.3m = s 2 Section3.2Acceleration 95

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Thesolutionisincorrectbecauseyoucan'tndtheslopeofa graphfromonepoint.Thispersonwasjustusingthepointatthe rightendofthev-tgraphtotrytondtheslopeofthecurve. Incorrectsolution#2:Velocityisdistancedividedbytimeso v = .5m/ s =1.5m/s.Accelerationisvelocitydividedbytime, so a =.5m/s/ s =0.5m = s 2 Thesolutionisincorrectbecausevelocityistheslopeofthetangentline.Inacaselikethiswherethevelocityischanging,you can'tjustpicktwopointsonthex-tgraphandusethemtondthe velocity. Converting g todifferentunitsexample3 Whatis g inunitsofcm = s 2 ? Theanswerisgoingtobehowmanycm/sofspeedafalling objectgainsinonesecond.Ifitgains9.8m/sinonesecond,then itgains980cm/sinonesecond,so g =980cm = s 2 .Alternatively, wecanusethemethodoffractionsthatequalone: 9.8 m s 2 100cm 1 m = 980cm s 2 Whatis g inunitsofmiles = hour 2 ? 9.8m s 2 1mile 1600m 3600s 1hour 2 =7.9 10 4 mile = hour 2 Thislargenumbercanbeinterpretedasthespeed,inmilesper hour,thatyouwouldgainbyfallingforonehour.Notethatwehad tosquaretheconversionfactorof3600s/hourinordertocancel outtheunitsofsecondssquaredinthedenominator. Whatis g inunitsofmiles/hour/s? 9.8m s 2 1mile 1600m 3600s 1hour =22mile = hour = s ThisisagurethatAmericanswillhaveanintuitivefeelfor.If yourcarhasaforwardaccelerationequaltotheaccelerationofa fallingobject,thenyouwillgain22milesperhourofspeedevery second.However,usingmixedtimeunitsofhoursandseconds likethisisusuallyinconvenientforproblem-solving.Itwouldbe likeusingunitsoffoot-inchesforareainsteadofft 2 orin 2 Theaccelerationofgravityisdifferentindifferentlocations. Everyoneknowsthatgravityisweakeronthemoon,butactuallyitisnoteventhesameeverywhereonEarth,asshownbythe samplingofnumericaldatainthefollowingtable. 96 Chapter3AccelerationandFreeFall

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locationlatitudeelevationmgm = s 2 northpole90 N09.8322 Reykjavik,Iceland64 N09.8225 Fullerton,California34 N09.7957 Guayaquil,Ecuador2 S09.7806 Mt.Cotopaxi,Ecuador1 S58969.7624 Mt.Everest28 N88489.7643 Themainvariablesthatrelatetothevalueof g onEartharelatitude andelevation.Althoughyouhavenotyetlearnedhow g would becalculatedbasedonanydeepertheoryofgravity,itisnottoo hardtoguesswhy g dependsonelevation.Gravityisanattraction betweenthingsthathavemass,andtheattractiongetsweakerwith increasingdistance.AsyouascendfromtheseaportofGuayaquil tothenearbytopofMt.Cotopaxi,youaredistancingyourselffrom themassoftheplanet.Thedependenceonlatitudeoccursbecause wearemeasuringtheaccelerationofgravityrelativetotheearth's surface,buttheearth'srotationcausestheearth'ssurfacetofall outfromunderyou.Wewilldiscussbothgravityandrotationin moredetaillaterinthecourse. h / Thisfalse-colormapshows variationsinthestrengthofthe earth'sgravity.Purpleareashave thestrongestgravity,yellowthe weakest.Theoveralltrendtoward weakergravityattheequatorand strongergravityatthepoleshas beenarticiallyremovedtoallowtheweakerlocalvariationsto showup.Themapcoversonly theoceansbecauseofthetechniqueusedtomakeit:satellites lookforbulgesanddepressions inthesurfaceoftheocean.A veryslightbulgewilloccuroveran underseamountain,forinstance, becausethemountain'sgravitationalattractionpullswatertowardit.TheUSgovernmentoriginallybegancollectingdatalike theseformilitaryuse,tocorrect forthedeviationsinthepathsof missiles.Thedatahaverecently beenreleasedforscienticand commercialusee.g.,searching forsitesforoff-shoreoilwells. Muchmorespectaculardierencesinthestrengthofgravitycan beobservedawayfromtheEarth'ssurface: Section3.2Acceleration 97

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locationgm = s 2 asteroidVestasurface0.3 Earth'smoonsurface1.6 Marssurface3.7 Earthsurface9.8 Jupitercloud-tops26 Sunvisiblesurface270 typicalneutronstarsurface10 12 blackholecenterinniteaccordingtosometheories,ontheorderof10 52 accordingtoothers Atypicalneutronstarisnotsodierentinsizefromalargeasteroid, butisordersofmagnitudemoremassive,sothemassofabody denitelycorrelateswiththe g itcreates.Ontheotherhand,a neutronstarhasaboutthesamemassasourSun,sowhyisits g billionsoftimesgreater?Ifyouhadthemisfortuneofbeingonthe surfaceofaneutronstar,you'dbewithinafewthousandmilesofall itsmass,whereasonthesurfaceoftheSun,you'dstillbemillions ofmilesfrommostofitsmass. DiscussionQuestions A Whatiswrongwiththefollowingdenitionsof g ? g isgravity. g isthespeedofafallingobject. g ishowhardgravitypullsonthings. B Whenadvertisersspecifyhowmuchaccelerationacariscapable of,theydonotgiveanaccelerationasdenedinphysics.Instead,they usuallyspecifyhowmanysecondsarerequiredforthecartogofromrest to60miles/hour.Supposeweusethenotation a fortheaccelerationas denedinphysics,and a carad forthequantityusedinadvertisementsfor cars.IntheUS'snon-metricsystemofunits,whatwouldbetheunitsof a and a carad ?Howwouldtheuseandinterpretationoflargeandsmall, positiveandnegativevaluesbedifferentforaasopposedtoacarad? C Twopeoplestandontheedgeofacliff.Astheyleanovertheedge, onepersonthrowsarockdown,whiletheotherthrowsonestraightup withanexactlyoppositeinitialvelocity.Comparethespeedsoftherocks onimpactatthebottomofthecliff. 3.3PositiveandNegativeAcceleration Gravityalwayspullsdown,butthatdoesnotmeanitalwaysspeeds thingsup.Ifyouthrowaballstraightup,gravitywillrstslow itdownto v =0andthenbeginincreasingitsspeed.WhenI tookphysicsinhighschool,Igottheimpressionthatpositivesigns ofaccelerationindicatedspeedingup,whilenegativeaccelerations representedslowingdown,i.e.,deceleration.Suchadenitionwould beinconvenient,however,becausewewouldthenhavetosaythat thesamedownwardtugofgravitycouldproduceeitherapositive 98 Chapter3AccelerationandFreeFall

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i / Theball'saccelerationstays thesameonthewayup,atthe top,andonthewaybackdown. It'salwaysnegative. oranegativeacceleration.Aswewillseeinthefollowingexample, suchadenitionalsowouldnotbethesameastheslopeofthe v )]TJ/F20 10.9091 Tf 9.808 0 Td [(t graph Let'sstudytheexampleoftherisingandfallingball.Intheexampleofthepersonfallingfromabridge,Iassumedpositivevelocity valueswithoutcallingattentiontoit,whichmeantIwasassuming acoordinatesystemwhose x axispointeddown.Inthisexample, wheretheballisreversingdirection,itisnotpossibletoavoidnegativevelocitiesbyatrickychoiceofaxis,solet'smakethemore naturalchoiceofanaxispointingup.Theball'svelocitywillinitiallybeapositivenumber,becauseitisheadingup,inthesame directionasthe x axis,butonthewaybackdown,itwillbeanegativenumber.Asshowninthegure,the v )]TJ/F20 10.9091 Tf 11.07 0 Td [(t graphdoesnotdo anythingspecialatthetopoftheball'sight,where v equals0.Its slopeisalwaysnegative.Inthelefthalfofthegraph,thereisa negativeslopebecausethepositivevelocityisgettingclosertozero. Ontherightside,thenegativeslopeisduetoanegativevelocity thatisgettingfartherfromzero,sowesaythattheballisspeeding up,butitsvelocityisdecreasing! Tosummarize,whatmakesthemostsenseistostickwiththe originaldenitionofaccelerationastheslopeofthe v )]TJ/F20 10.9091 Tf 11.677 0 Td [(t graph, v= t .Bythisdenition,itjustisn'tnecessarilytruethatthings speedinguphavepositiveaccelerationwhilethingsslowingdown havenegativeacceleration.Theworddeceleration"isnotused muchbyphysicists,andthewordacceleration"isusedunblushinglytorefertoslowingdownaswellasspeedingup:Therewasa redlight,andweacceleratedtoastop." Numericalcalculationofanegativeaccelerationexample4 Ingurei,whathappensifyoucalculatetheaccelerationbetween t =1.0and1.5s? Readingfromthegraph,itlookslikethevelocityisabout )]TJ/F39 10.9091 Tf 8.485 0 Td [(1m/s at t =1.0s,andaround )]TJ/F39 10.9091 Tf 8.485 0 Td [(6m/sat t =1.5s.Theacceleration, guredbetweenthesetwopoints,is a = v t = )]TJ/F39 10.9091 Tf 8.485 0 Td [(6m = s )]TJ/F39 10.9091 Tf 10.909 0 Td [( )]TJ/F39 10.9091 Tf 8.485 0 Td [(1m = s .5s )]TJ/F39 10.9091 Tf 10.909 0 Td [(.0s = )]TJ/F39 10.9091 Tf 8.485 0 Td [(10m = s 2 Eventhoughtheballisspeedingup,ithasanegativeacceleration. Anotherwayofconvincingyouthatthiswayofhandlingtheplus andminussignsmakessenseistothinkofadevicethatmeasures acceleration.Afterall,physicsissupposedtouseoperationaldenitions,onesthatrelatetotheresultsyougetwithactualmeasuring devices.Consideranairfreshenerhangingfromtherear-viewmirror ofyourcar.Whenyouspeedup,theairfreshenerswingsbackward. Supposewedenethisasapositivereading.Whenyouslowdown, theairfreshenerswingsforward,sowe'llcallthisanegativereading Section3.3PositiveandNegativeAcceleration 99

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onouraccelerometer.Butwhatifyouputthecarinreverseand startspeedingupbackwards?Eventhoughyou'respeedingup,the accelerometerrespondsinthesamewayasitdidwhenyouwere goingforwardandslowingdown.Therearefourpossiblecases: motionofcaraccelerometer swings slopeof v-tgraph direction offorce actingon car forward,speedingupbackward+forward forward,slowingdownforward )]TJ/F15 10.9091 Tf 57.309 0 Td [(backward backward,speedingupforward )]TJ/F15 10.9091 Tf 57.309 0 Td [(backward backward,slowingdownbackward+forward Notetheconsistencyofthethreeright-handcolumns|natureis tryingtotellusthatthisistherightsystemofclassication,not theleft-handcolumn. Becausethepositiveandnegativesignsofaccelerationdepend onthechoiceofacoordinatesystem,theaccelerationofanobject undertheinuenceofgravitycanbeeitherpositiveornegative. Ratherthanhavingtowritethingslike g =9.8m = s 2 or )]TJ/F15 10.9091 Tf 8.485 0 Td [(9.8m = s 2 everytimewewanttodiscuss g 'snumericalvalue,wesimplydene g astheabsolutevalueoftheaccelerationofobjectsmovingunder theinuenceofgravity.Weconsistentlylet g =9.8m = s 2 ,butwe mayhaveeither a = g or a = )]TJ/F20 10.9091 Tf 8.485 0 Td [(g ,dependingonourchoiceofa coordinatesystem. Accelerationwithachangeindirectionofmotionexample5 Apersonkicksaball,whichrollsupaslopingstreet,comesto ahalt,androllsbackdownagain.Theballhasconstantacceleration.Theballisinitiallymovingatavelocityof4.0m/s,and after10.0sithasreturnedtowhereitstarted.Attheend,ithas spedbackuptothesamespeedithadinitially,butintheopposite direction.Whatwasitsacceleration? Bygivingapositivenumberfortheinitialvelocity,thestatement ofthequestionimpliesacoordinateaxisthatpointsuptheslope ofthehill.Thesamespeedintheoppositedirectionshould thereforeberepresentedbyanegativenumber,-4.0m/s.The accelerationis a = v = t = v f )]TJ/F78 10.9091 Tf 10.909 0 Td [(v o = 10.0s =[ )]TJ/F39 10.9091 Tf 8.485 0 Td [(4.0m = s )]TJ/F39 10.9091 Tf 10.909 0 Td [(.0m = s] = 10.0 s = )]TJ/F39 10.9091 Tf 8.485 0 Td [(0.80m = s 2 Theaccelerationwasnodifferentduringtheupwardpartofthe rollthanonthedownwardpartoftheroll. Incorrectsolution:Accelerationis v = t,andattheendit'snot movinganyfasterorslowerthanwhenitstarted,so v=0and 100 Chapter3AccelerationandFreeFall

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DiscussionquestionC. a =0. Thevelocitydoeschange,fromapositivenumbertoanegative number. DiscussionquestionB. DiscussionQuestions A Achildrepeatedlyjumpsupanddownonatrampoline.Discussthe signandmagnitudeofhisacceleration,includingboththetimewhenheis intheairandthetimewhenhisfeetareincontactwiththetrampoline. B ThegureshowsarefugeefromaPicassopaintingblowingona rollingwaterbottle.Insomecasestheperson'sblowingisspeedingthe bottleup,butinothersitisslowingitdown.Thearrowinsidethebottle showswhichdirectionitisgoing,andacoordinatesystemisshownatthe bottomofeachgure.Ineachcase,gureouttheplusorminussignsof thevelocityandacceleration.Itmaybehelpfultodrawa v )]TJ/F78 9.9627 Tf 10.346 0 Td [(t graphin eachcase. C Sallyisonanamusementparkridewhichbeginswithherchairbeing hoistedstraightupatowerataconstantspeedof60miles/hour.Despite sternwarningsfromherfatherthathe'lltakeherhomethenexttimeshe misbehaves,shedecidesthatasascienticexperimentshereallyneeds toreleasehercorndogoverthesideasshe'sonthewayup.Shedoes notthrowit.Shesimplysticksitoutofthecar,letsitgo,andwatchesit againstthebackgroundofthesky,withnotreesorbuildingsasreference points.Whatdoesthecorndog'smotionlooklikeasobservedbySally? Doesitsspeedeverappeartohertobezero?Whataccelerationdoes sheobserveittohave:isiteverpositive?negative?zero?Whatwould herenragedfatheranswerifaskedforasimilardescriptionofitsmotion asitappearstohim,standingontheground? D Cananobjectmaintainaconstantacceleration,butmeanwhile reversethedirectionofitsvelocity? E Cananobjecthaveavelocitythatispositiveandincreasingatthe sametimethatitsaccelerationisdecreasing? Section3.3PositiveandNegativeAcceleration 101

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k / Example6. 3.4VaryingAcceleration Sofarwehaveonlybeendiscussingexamplesofmotionforwhich the v )]TJ/F20 10.9091 Tf 11.14 0 Td [(t graphislinear.Ifwewishtogeneralizeourdenitionto v-tgraphsthataremorecomplexcurves,thebestwaytoproceed issimilartohowwedenedvelocityforcurved x )]TJ/F20 10.9091 Tf 10.91 0 Td [(t graphs: denitionofacceleration Theaccelerationofanobjectatanyinstantistheslopeof thetangentlinepassingthroughits v -versust graphatthe relevantpoint. Askydiverexample6 Thegraphsingure ?? showtheresultsofafairlyrealisticcomputersimulationofthemotionofaskydiver,includingtheeffects ofairfriction.The x axishasbeenchosenpointingdown,so x isincreasingasshefalls.Findatheskydiver'saccelerationat t =3.0s,andalsobat t =7.0s. Thesolutionisshowningurel.I'veaddedtangentlinesatthe twopointsinquestion. aTondtheslopeofthetangentline,Ipicktwopointsonthe linenotnecessarilyontheactualcurve:.0s,28m = sand .0s,42m = s.Theslopeofthetangentlineism = s )]TJ/F39 10.9091 Tf 8.485 0 Td [(28m = s = .0s )]TJ/F39 10.9091 Tf -339.241 -13.549 Td [(3.0s=7.0m = s 2 bTwopointsonthistangentlineare.0s,47m = sand.0s,52m = s. Theslopeofthetangentlineism = s )]TJ/F39 10.9091 Tf 9.057 0 Td [(47m = s = .0s )]TJ/F39 10.9091 Tf 9.056 0 Td [(7.0s= 2.5m = s 2 Physically,what'shappeningisthatat t =3.0s,theskydiveris notyetgoingveryfast,soairfrictionisnotyetverystrong.She thereforehasanaccelerationalmostasgreatas g .At t =7.0s, sheismovingalmosttwiceasfastabout100milesperhour,and airfrictionisextremelystrong,resultinginasignicantdeparture fromtheidealizedcaseofnoairfriction. Inexample6,the x )]TJ/F20 10.9091 Tf 9.273 0 Td [(t graphwasnotevenusedinthesolutionof theproblem,sincethedenitionofaccelerationreferstotheslope ofthe v )]TJ/F20 10.9091 Tf 11.668 0 Td [(t graph.Itispossible,however,tointerpretan x )]TJ/F20 10.9091 Tf 11.668 0 Td [(t graphtondoutsomethingabouttheacceleration.Anobjectwith zeroacceleration,i.e.,constantvelocity,hasan x )]TJ/F20 10.9091 Tf 10.133 0 Td [(t graphthatisa straightline.Astraightlinehasnocurvature.Achangeinvelocity requiresachangeintheslopeofthe x )]TJ/F20 10.9091 Tf 10.688 0 Td [(t graph,whichmeansthat itisacurveratherthanaline.Thusaccelerationrelatestothe curvatureofthe x )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graph.Figuremshowssomeexamples. 102 Chapter3AccelerationandFreeFall

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l / Thesolutiontoexample6. Inexample6,the x )]TJ/F20 10.9091 Tf 11.11 0 Td [(t graphwasmorestronglycurvedatthe beginning,andbecamenearlystraightattheend.Ifthe x )]TJ/F20 10.9091 Tf 9.835 0 Td [(t graph isnearlystraight,thenitsslope,thevelocity,isnearlyconstant,and theaccelerationisthereforesmall.Wecanthusinterprettheaccelerationasrepresentingthecurvatureofthe x )]TJ/F20 10.9091 Tf 11.224 0 Td [(t graph,asshown ingurem.Ifthecup"ofthecurvepointsup,theaccelerationis positive,andifitpointsdown,theaccelerationisnegative. m / Accelerationrelatestothecurvatureofthe x )]TJ/F78 9.9627 Tf 9.962 0 Td [(t graph. Section3.4VaryingAcceleration 103

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o / Howposition,velocity,and accelerationarerelated. Sincetherelationshipbetween a and v isanalogoustotherelationshipbetween v and x ,wecanalsomakegraphsofacceleration asafunctionoftime,asshowninguren. n / Examplesofgraphsof x v ,and a versus t .1.Aobjectinfree fall,withnofriction.2.Acontinuationofexample6,theskydiver. Solvedproblem:Drawinga v )]TJ/F20 9.9626 Tf 9.962 0 Td [(t graph.page117,problem14 Solvedproblem:Drawing v )]TJ/F20 9.9626 Tf 9.963 0 Td [(t and a )]TJ/F20 9.9626 Tf 9.962 0 Td [(t graphs.page118,problem 20 Figureosummarizestherelationshipsamongthethreetypesof graphs. DiscussionQuestions A Describeinwordshowthechangesinthe a )]TJ/F78 9.9627 Tf 10.142 0 Td [(t graphinguren/2 relatetothebehaviorofthe v )]TJ/F78 9.9627 Tf 9.963 0 Td [(t graph. 104 Chapter3AccelerationandFreeFall

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B Explainhoweachsetofgraphscontainsinconsistencies,andx them. C Ineachcase,pickacoordinatesystemanddraw x )]TJ/F78 9.9627 Tf 10.27 0 Td [(t v )]TJ/F78 9.9627 Tf 10.269 0 Td [(t ,and a )]TJ/F78 9.9627 Tf 9.454 0 Td [(t graphs.Pickingacoordinatesystemmeanspickingwhereyouwant x =0tobe,andalsopickingadirectionforthepositive x axis. Anoceanlineriscruisinginastraightlineatconstantspeed. Youdropaball.Drawtwodifferentsetsofgraphsatotalof6,with oneset'spositive x axispointingintheoppositedirectioncomparedtothe other's. You'redrivingdownthestreetlookingforahouseyou'veneverbeen tobefore.Yourealizeyou'vepassedtheaddress,soyouslowdown,put thecarinreverse,backup,andstopinfrontofthehouse. 3.5TheAreaUndertheVelocity-TimeGraph Anaturalquestiontoaskaboutfallingobjectsishowfasttheyfall, butGalileoshowedthatthequestionhasnoanswer.Thephysical lawthathediscoveredconnectsacausetheattractionoftheplanet Earth'smasstoaneect,buttheeectispredictedintermsofan accelerationratherthanavelocity.Infact,nophysicallawpredicts adenitevelocityasaresultofaspecicphenomenon,because velocitycannotbemeasuredinabsoluteterms,andonlychangesin velocityrelatedirectlytophysicalphenomena. Theunfortunatethingaboutthissituationisthatthedenitions ofvelocityandaccelerationarestatedintermsofthetangent-line technique,whichletsyougofrom x to v to a ,butnottheother wayaround.Withoutatechniquetogobackwardsfrom a to v to x wecannotsayanythingquantitative,forinstance,aboutthe x )]TJ/F20 10.9091 Tf 11.193 0 Td [(t graphofafallingobject.Suchatechniquedoesexist,andIusedit tomakethe x )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graphsinalltheexamplesabove. Section3.5TheAreaUndertheVelocity-TimeGraph 105

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p / Theareaunderthe v )]TJ/F78 9.9627 Tf 13.05 0 Td [(t graphgives x Firstlet'sconcentrateonhowtoget x informationoutofa v )]TJ/F20 10.9091 Tf 10.015 0 Td [(t graph.Inexamplep/1,anobjectmovesataspeedof20m = sfor aperiodof4.0s.Thedistancecoveredis x = v t =m = s .0s=80m.Noticethatthequantitiesbeingmultipliedarethe widthandtheheightoftheshadedrectangle|or,strictlyspeaking, thetimerepresentedbyitswidthandthevelocityrepresentedby itsheight.Thedistanceof x =80mthuscorrespondstothearea oftheshadedpartofthegraph. Thenextstepinsophisticationisanexamplelikep/2,wherethe objectmovesataconstantspeedof10m = sfortwoseconds,then fortwosecondsatadierentconstantspeedof20m = s.Theshaded regioncanbesplitintoasmallrectangleontheleft,withanarea representing x =20m,andatalleroneontheright,corresponding toanother40 m ofmotion.Thetotaldistanceisthus60m,which correspondstothetotalareaunderthegraph. Anexamplelikep/3isnowjustatrivialgeneralization;there issimplyalargenumberofskinnyrectangularareastoaddup. Butnoticethatgraphp/3isquiteagoodapproximationtothe smoothcurvep/4.Eventhoughwehavenoformulafortheareaof afunnyshapelikep/4,wecanapproximateitsareabydividingitup intosmallerareaslikerectangles,whoseareaiseasiertocalculate. Ifsomeonehandsyouagraphlikep/4andasksyoutondthe areaunderit,thesimplestapproachisjusttocountupthelittle rectanglesontheunderlyinggraphpaper,makingroughestimates offractionalrectanglesasyougoalong. That'swhatI'vedoneingureq.Eachrectangleonthegraph paperis1.0swideand2m = stall,soitrepresents2m.Addingup allthenumbersgives x =41m.Ifyouneededbetteraccuracy, youcouldusegraphpaperwithsmallerrectangles. It'simportanttorealizethatthistechniquegivesyou x ,not x .The v )]TJ/F20 10.9091 Tf 10.938 0 Td [(t graphhasnoinformationaboutwheretheobjectwas whenitstarted. Thefollowingareimportantpointstokeepinmindwhenapplyingthistechnique: Iftherangeof v valuesonyourgraphdoesnotextenddown tozero,thenyouwillgetthewronganswerunlessyoucompensatebyaddingintheareathatisnotshown. Asintheexample,onerectangleonthegraphpaperdoesnot necessarilycorrespondtoonemeterofdistance. Negativevelocityvaluesrepresentmotionintheoppositedirection,soareaunderthe t axisshouldbesubtracted,i.e., countedasnegativearea." 106 Chapter3AccelerationandFreeFall

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q / Anexampleusingestimation offractionsofarectangle. Sincetheresultisa x value,itonlytellsyou x after )]TJ/F20 10.9091 Tf 9.931 0 Td [(x before whichmaybelessthantheactualdistancetraveled.Forinstance,theobjectcouldcomebacktoitsoriginalpositionat theend,whichwouldcorrespondto x =0,eventhoughithad actuallymovedanonzerodistance. Finally,notethatonecannd v froman a )]TJ/F20 10.9091 Tf 11.571 0 Td [(t graphusing anentirelyanalogousmethod.Eachrectangleonthe a )]TJ/F20 10.9091 Tf 11.433 0 Td [(t graph representsacertainamountofvelocitychange. DiscussionQuestion A Roughlywhatwouldapendulum's v )]TJ/F78 9.9627 Tf 9.019 0 Td [(t graphlooklike?Whatwould happenwhenyouappliedthearea-under-the-curvetechniquetondthe pendulum's x foratimeperiodcoveringmanyswings? 3.6AlgebraicResultsforConstant Acceleration Althoughthearea-under-the-curvetechniquecanbeappliedtoany graph,nomatterhowcomplicated,itmaybelaborioustocarryout, andiffractionsofrectanglesmustbeestimatedtheresultwillonly beapproximate.Inthespecialcaseofmotionwithconstantacceleration,itispossibletondaconvenientshortcutwhichproduces Section3.6AlgebraicResultsforConstantAcceleration 107

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r / Theshadedareatellsus howfaranobjectmoveswhile acceleratingataconstantrate. exactresults.Whentheaccelerationisconstant,the v )]TJ/F20 10.9091 Tf 11.507 0 Td [(t graph isastraightline,asshowninthegure.Theareaunderthecurve canbedividedintoatriangleplusarectangle,bothofwhoseareas canbecalculatedexactly: A = bh forarectangleand A = bh= 2 foratriangle.Theheightoftherectangleistheinitialvelocity, v o andtheheightofthetriangleisthechangeinvelocityfrombeginningtoend, v .Theobject's x isthereforegivenbytheequation x = v o t + v t= 2.Thiscanbesimpliedalittlebyusingthe denitionofacceleration, a = v= t ,toeliminate v ,giving x = v o t + 1 2 a t 2 .[motionwith constantacceleration] Sincethisisasecond-orderpolynomialin t ,thegraphof x versus t isaparabola,andthesameistrueofagraphof x versus t | thetwographsdieronlybyshiftingalongthetwoaxes.Although Ihavederivedtheequationusingagurethatshowsapositive v o positive a ,andsoon,itstillturnsouttobetrueregardlessofwhat plusandminussignsareinvolved. Anotherusefulequationcanbederivedifonewantstorelate thechangeinvelocitytothedistancetraveled.Thisisuseful,for instance,forndingthedistanceneededbyacartocometoastop. Forsimplicity,westartbyderivingtheequationforthespecialcase of v o =0,inwhichthenalvelocity v f isasynonymfor v .Since velocityanddistancearethevariablesofinterest,nottime,wetake theequation x = 1 2 a t 2 anduse t = v=a toeliminate t .This gives x = v 2 = 2 a ,whichcanberewrittenas v 2 f =2 a x .[motionwithconstantacceleration, v o =0] Forthemoregeneralcasewhere,weskipthetediousalgebraleading tothemoregeneralequation, v 2 f = v 2 o +2 a x .[motionwithconstantacceleration] 108 Chapter3AccelerationandFreeFall

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Tohelpgetthisallorganizedinyourhead,rstlet'scategorize thevariablesasfollows: Variablesthatchangeduringmotionwithconstantacceleration: x v t Variablethatdoesn'tchange: a Ifyouknowoneofthechangingvariablesandwanttondanother, thereisalwaysanequationthatrelatesthosetwo: Thesymmetryamongthethreevariablesisimperfectonlybecausetheequationrelating x and t includestheinitialvelocity. Therearetwomaindicultiesencounteredbystudentsinapplyingtheseequations: Theequationsapplyonlytomotionwithconstantacceleration.Youcan'tapplythemiftheaccelerationischanging. Studentsareoftenunsureofwhichequationtouse,ormay causethemselvesunnecessaryworkbytakingthelongerpath aroundthetriangleinthechartabove.Organizeyourthoughts bylistingthevariablesyouaregiven,theonesyouwantto nd,andtheonesyouaren'tgivenanddon'tcareabout. Savinganoldladyexample7 Youaretryingtopullanoldladyoutofthewayofanoncoming truck.Youareabletogiveheranaccelerationof20m = s 2 .Startingfromrest,howmuchtimeisrequiredinordertomoveher2 m? Firstweorganizeourthoughts: Variablesgiven: x a v o Variablesdesired: t Irrelevantvariables: v f Consultingthetriangularchartabove,theequationweneedis clearly x = v o t + 1 2 a t 2 ,sinceithasthefourvariablesofinterest Section3.6AlgebraicResultsforConstantAcceleration 109

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s / OnOctober4,2004,the privatelyfundedSpaceShipOne wontheten-million-dollarAnsari XPrizebyreachinganaltitude of100kmtwiceinthespaceof 14days.[CourtesyofScaled CompositesLLC.] andomitstheirrelevantone.Eliminatingthe v o termandsolving for t gives t = p 2 x = a =0.4s. Solvedproblem:Astupidcelebrationpage117,problem15 Solvedproblem:DroppingarockonMarspage117,problem16 Solvedproblem:TheDodgeViperpage118,problem18 Solvedproblem:Half-wayspeduppage118,problem22 DiscussionQuestions A Inchapter1,Igaveexamplesofcorrectandincorrectreasoning aboutproportionality,usingquestionsaboutthescalingofareaandvolume.Trytotranslatetheincorrectmodesofreasoningshownthereinto mistakesaboutthefollowingquestion:Iftheaccelerationofgravityon Marsis1/3thatonEarth,howmanytimeslongerdoesittakeforarock todropthesamedistanceonMars? B Checkthattheunitsmakesenseinthethreeequationsderivedin thissection. 3.7 ? BiologicalEffectsofWeightlessness TheusefulnessofouterspacewasbroughthometoNorthAmericansin1998bytheunexpectedfailureofthecommunicationssatellitethathadbeenhandlingalmostallofthecontinent'scellular phonetrac.Comparedtothemassiveeconomicandscienticpayosofsatellitesandspaceprobes,humanspacetravelhaslittleto boastaboutafterfourdecades.Sendingpeopleintoorbithasjust beentooexpensivetobeaneectivescienticorcommercialactivity.Thedownsizedandover-budgetInternationalSpaceStation hasproducedvirtuallynoscienticresults,andthespaceshuttle programnowhasarecordoftwocatastrophicfailuresoutof113 missions. Withinourlifetimes,weareprobablyonlylikelytoseeoneeconomicallyviablereasonforsendinghumansintospace:tourism! Nofewerthanthreeprivatecompaniesarenowwillingtotakeyour moneyforareservationonatwo-to-fourminutetripintospace, althoughnoneofthemhasarmdateonwhichtobeginservice. Withinadecade,aspacecruisemaybethenewstatussymbol amongthosesucientlyrichandbrave. Spacesickness Well,rich,brave,andpossessedofanironstomach.Travel agentswillprobablynotemphasizethecertaintyofconstantspacesickness.Forusanimalsevolvedtofunctionin g =9.8m = s 2 ,living in g =0isextremelyunpleasant.Theearlyspaceprogramfocused obsessivelyonkeepingtheastronaut-traineesinperfectphysical shape,butitsoonbecameclearthatabodylikeaGreekdemigod's wasnodefenseagainstthathorriblefeelingthatyourstomachwas 110 Chapter3AccelerationandFreeFall

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t / U.S.andRussianastronautsaboardtheInternational SpaceStation,October2000. u / TheInternationalSpace Station,September2000.The spacestationdoesnotrotateto providesimulatedgravity.The completedstationwillbemuch bigger. fallingoutfromunderyouandyouwerenevergoingtocatchup. Ourinnerear,whichnormallytellsuswhichwayisdown,tortures uswhendownisnowheretobefound.Thereiscontradictoryinformationaboutwhetheranyoneevergetsoverit;therightstu" culturecreatesastrongincentiveforastronautstodenythatthey aresick. Effectsoflongspacemissions Worsethannauseaarethehealth-threateningeectsofprolongedweightlessness.TheRussiansarethespecialistsinlong-term missions,inwhichcosmonautssuerharmtotheirblood,muscles, and,mostimportantly,theirbones. Theeectsonthemusclesandskeletonappeartobesimilarto thoseexperiencedbyoldpeopleandpeopleconnedtobedfora longtime.Everyoneknowsthatourmusclesgetstrongerorweaker dependingontheamountofexerciseweget,butthebonesarelikewiseadaptable.Normallyoldbonemassiscontinuallybeingbroken downandreplacedwithnewmaterial,butthebalancebetweenits lossandreplacementisupsetwhenpeopledonotgetenoughweightbearingexercise.Themaineectisonthebonesofthelowerbody. Moreresearchisrequiredtondoutwhetherastronauts'lossofbone massisduetofasterbreakingdownofbone,slowerreplacement,or both.Itisalsonotknownwhethertheeectcanbesuppressedvia dietordrugs. Theothersetofharmfulphysiologicaleectsappearstoderive fromtheredistributionofuids.Normally,theveinsandarteriesofthelegsaretightlyconstrictedtokeepgravityfrommaking bloodcollectthere.Itisuncomfortableforadultstostandontheir headsforverylong,becausethehead'sbloodvesselsarenotableto constrictaseectively.Weightlessastronauts'bloodtendstobeexpelledbytheconstrictedbloodvesselsofthelowerbody,andpools aroundtheirhearts,intheirthoraxes,andintheirheads.Theonly immediateresultisanuncomfortablefeelingofbloatednessinthe upperbody,butinthelongterm,aharmfulchainofeventsissetin motion.Thebody'sattemptstomaintainthecorrectbloodvolume aremostsensitivetothelevelofuidinthehead.Sinceastronauts haveextrauidintheirheads,thebodythinksthattheover-all bloodvolumehasbecometoogreat.Itrespondsbydecreasingblood volumebelownormallevels.Thisincreasestheconcentrationofred bloodcells,sothebodythendecidesthatthebloodhasbecometoo thick,andreducesthenumberofbloodcells.Inmissionslastingup toayearorso,thisisnotasharmfulasthemusculo-skeletaleects, butitisnotknownwhetherlongerperiodinspacewouldbringthe redbloodcellcountdowntoharmfullevels. Section3.7 ? BiologicalEffectsofWeightlessness 111

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Reproductioninspace Forthoseenthralledbytheromanceofactualhumancolonizationofspace,humanreproductioninweightlessnessbecomesanissue.Analready-pregnantRussiancosmonautdidspendsometime inorbitinthe1960's,andlatergavebirthtoanormalchildonthe ground.Recently,oneofNASA'spublicrelationsconcernsabout thespaceshuttleprogramhasbeentodiscouragespeculationabout spacesex,forfearofapotentialtaxpayers'backlashagainstthe spaceprogramasanexpensiveformofexoticpleasure. Scienticworkhasbeenconcentratedonstudyingplantandanimalreproductioninspace.Greenplants,fungi,insects,sh,and amphibianshaveallgonethroughatleastonegenerationinzerogravityexperimentswithoutanyseriousproblems.Inmanycases, animalembryosconceivedinorbitbeginbydevelopingabnormally, butlaterindevelopmenttheyseemtocorrectthemselves.However, chickenembryosfertilizedonearthlessthan24hoursbeforegoing intoorbithavefailedtosurvive.Sincechickensaretheorganisms mostsimilartohumansamongthespeciesinvestigatedsofar,it isnotatallcertainthathumanscouldreproducesuccessfullyina zero-gravityspacecolony. Simulatedgravity Ifhumansareevertoliveandworkinspaceformorethana yearorso,theonlysolutionisprobablytobuildspinningspacestationstoprovidetheillusionofweight,asdiscussedinsection9.2. Normalgravitycouldbesimulated,buttouristswouldprobablyenjoy g =2m = s 2 or5m = s 2 .Spaceenthusiastshaveproposedentire orbitingcitiesbuiltontherotatingcylinderplan.Althoughscience ctionhasfocusedonhumancolonizationofrelativelyearthlikebodiessuchasourmoon,Mars,andJupiter'sicymoonEuropa,there wouldprobablybenopracticalwaytobuildlargespinningstructuresontheirsurfaces.Ifthebiologicaleectsoftheir2 )]TJ/F15 10.9091 Tf 11.149 0 Td [(3m = s 2 gravitationalaccelerationsareasharmfulastheeectof g =0,then wemaybeleftwiththesurprisingresultthatinterplanetaryspace ismorehospitabletoourspeciesthanthemoonsandplanets. OptionalTopic:MoreonApparentWeightlessness Astronautsinorbitarenotreallyweightless;theyareonlyafewhundred milesup,sotheyarestillaffectedstronglybytheEarth'sgravity.Section 10.3ofthisbookdiscusseswhytheyexperienceapparentweightlessness.MoreonSimulatedGravityFormoreinformationonsimulating gravitybyspinningaspacecraft,seesection9.2ofthisbook. 3.8 R ApplicationsofCalculus IntheApplicationsofCalculussectionattheendoftheprevious chapter,Idiscussedhowtheslope-of-the-tangent-lineidearelated tothecalculusconceptofaderivative,andthebranchofcalculus 112 Chapter3AccelerationandFreeFall

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knownasdierentialcalculus.Theothermainbranchofcalculus, integralcalculus,hastodowiththearea-under-the-curveconcept discussedinsection3.5ofthischapter.Againthereisaconcept, anotation,andabagoftricksfordoingthingssymbolicallyrather thangraphically.Incalculus,theareaunderthe v )]TJ/F20 10.9091 Tf 9.28 0 Td [(t graphbetween t = t 1 and t = t 2 isnotatedlikethis: areaundercurve= x = Z t 2 t 1 v d t Theexpressionontherightiscalledanintegral,andthes-shaped symbol,theintegralsign,isreadasintegralof..." Integralcalculusanddierentialcalculusarecloselyrelated.For instance,ifyoutakethederivativeofthefunction x t ,youget thefunction v t ,andifyouintegratethefunction v t ,youget x t backagain.Inotherwords,integrationanddierentiationare inverseoperations.Thisisknownasthefundamentaltheoremof calculus. Onanunrelatedtopic,thereisaspecialnotationfortakingthe derivativeofafunctiontwice.Theacceleration,forinstance,isthe secondi.e.,doublederivativeoftheposition,becausedierentiating x oncegives v ,andthendierentiating v gives a .Thisiswritten as a = d 2 x d t 2 Theseeminglyinconsistentplacementofthetwosonthetopand bottomconfusesallbeginningcalculusstudents.Themotivation forthisfunnynotationisthataccelerationhasunitsofm = s 2 ,and thenotationcorrectlysuggeststhat:thetoplookslikeithasunitsof meters,thebottomseconds 2 .Thenotationisnotmeant,however, tosuggestthat t isreallysquared. Section3.8 R ApplicationsofCalculus 113

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Summary SelectedVocabulary gravity......Ageneraltermforthephenomenonofattractionbetweenthingshavingmass.Theattractionbetweenourplanetandahuman-sizedobjectcausestheobjecttofall. acceleration...Therateofchangeofvelocity;theslopeofthe tangentlineona v )]TJ/F20 10.9091 Tf 10.91 0 Td [(t graph. Notation v o .........initialvelocity v f .........nalvelocity a ..........acceleration g ..........theaccelerationofobjectsinfreefall;the strengthofthelocalgravitationaleld Summary Galileoshowedthatwhenairresistanceisnegligibleallfalling bodieshavethesamemotionregardlessofmass.Moreover,their v )]TJ/F20 10.9091 Tf 10.099 0 Td [(t graphsarestraightlines.Wethereforedeneaquantitycalled accelerationastheslope, v= t,ofanobject's v )]TJ/F20 10.9091 Tf 9.489 0 Td [(t graph.Incases otherthanfreefall,the v )]TJ/F20 10.9091 Tf 9.547 0 Td [(t graphmaybecurved,inwhichcasethe denitionisgeneralizedastheslopeofatangentlineonthe v )]TJ/F20 10.9091 Tf 11.206 0 Td [(t graph.Theaccelerationofobjectsinfreefallvariesslightlyacross thesurfaceoftheearth,andgreatlyonotherplanets. Positiveandnegativesignsofaccelerationaredenedaccording towhetherthe v )]TJ/F20 10.9091 Tf 11.166 0 Td [(t graphslopesupordown.Thisdenitionhas theadvantagethataforceinagivendirectionalwaysproducesthe samesignofacceleration. Theareaunderthe v )]TJ/F20 10.9091 Tf 11.238 0 Td [(t graphgives x ,andanalogouslythe areaunderthe a )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graphgives v Formotionwithconstantacceleration,thefollowingthreeequationshold: x = v o t + 1 2 a t 2 v 2 f = v 2 o +2 a x a = v t Theyarenotvalidiftheaccelerationischanging. 114 Chapter3AccelerationandFreeFall

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Problem3. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Thegraphrepresentsthevelocityofabeealongastraight line.At t =0,thebeeisatthehive.aWhenisthebeefarthest fromthehive?bHowfaristhebeeatitsfarthestpointfromthe hive?cAt t =13 s ,howfaristhebeefromthehive?[Hint:Try problem19rst.] p 2 Arockisdroppedintoapond.Drawplotsofitsposition versustime,velocityversustime,andaccelerationversustime.Includeitswholemotion,startingfromthemomentitisdropped,and continuingwhileitfallsthroughtheair,passesthroughthewater, andendsupatrestonthebottomofthepond.Doyourworkon photocopyoraprintoutofpage121. 3 Inan18th-centurynavalbattle,acannonballisshothorizontally,passesthroughthesideofanenemyship'shull,iesacrossthe galley,andlodgesinabulkhead.Drawplotsofitshorizontalposition,velocity,andaccelerationasfunctionsoftime,startingwhileit isinsidethecannonandhasnotyetbeenred,andendingwhenit comestorest.Thereisnotanysignicantamountoffrictionfrom theair.Althoughtheballmayriseandfall,youareonlyconcerned withitshorizontalmotion,asseenfromabove.Doyourworkon photocopyoraprintoutofpage121. 4 Drawgraphsofposition,velocity,andaccelerationasfunctions oftimeforapersonbunjeejumping.Inbunjeejumping,aperson hasastretchyelasticcordtiedtohis/herankles,andjumpsoofa highplatform.Atthebottomofthefall,thecordbringstheperson upshort.Presumablythepersonbouncesupalittle.Doyourwork onphotocopyoraprintoutofpage121. Problems 115

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Problem5. 5 Aballrollsdowntherampshowninthegure,consistingofa curvedknee,astraightslope,andacurvedbottom.Foreachpartof theramp,tellwhethertheball'svelocityisincreasing,decreasing, orconstant,andalsowhethertheball'saccelerationisincreasing, decreasing,orconstant.Explainyouranswers.Assumethereisno airfrictionorrollingresistance.Hint:Tryproblem20rst.[Based onaproblembyHewitt.] 6 Atoycarisreleasedononesideofapieceoftrackthatisbent intoanupright U shape.Thecargoesbackandforth.Whenthe carreachesthelimitofitsmotionononeside,itsvelocityiszero. Isitsaccelerationalsozero?Explainusinga v )]TJ/F20 10.9091 Tf 10.244 0 Td [(t graph.[Basedon aproblembySerwayandFaughn.] 7 Whatistheaccelerationofacarthatmovesatasteady velocityof100km/hfor100seconds?Explainyouranswer.[Based onaproblembyHewitt.] 8 Aphysicshomeworkquestionasks,Ifyoustartfromrestand accelerateat1.54m = s 2 for3.29s,howfardoyoutravelbytheend ofthattime?"Astudentanswersasfollows: 1.54 3.29=5.07m HisAuntWandaisgoodwithnumbers,buthasnevertakenphysics. Shedoesn'tknowtheformulaforthedistancetraveledunderconstantaccelerationoveragivenamountoftime,butshetellsher nephewhisanswercannotberight.Howdoessheknow? 9 Youarelookingintoadeepwell.Itisdark,andyoucannot seethebottom.Youwanttondouthowdeepitis,soyoudrop arockin,andyouhearasplash3.0secondslater.Howdeepisthe well? p 10 Youtakeatripinyourspaceshiptoanotherstar.Settingo, youincreaseyourspeedataconstantacceleration.Onceyouget half-waythere,youstartdecelerating,atthesamerate,sothatby thetimeyougetthere,youhavesloweddowntozerospeed.Yousee thetouristattractions,andthenheadhomebythesamemethod. aFindaformulaforthetime, T ,requiredfortheroundtrip,in termsof d ,thedistancefromoursuntothestar,and a ,themagnitudeoftheacceleration.Notethattheaccelerationisnotconstant overthewholetrip,butthetripcanbebrokenupintoconstantaccelerationparts. bTheneareststartotheEarthotherthanourownsunisProximaCentauri,atadistanceof d =4 10 16 m.Supposeyouusean accelerationof a =10m = s 2 ,justenoughtocompensateforthelack oftruegravityandmakeyoufeelcomfortable.Howlongdoesthe roundtriptake,inyears? cUsingthesamenumbersfor d and a ,ndyourmaximumspeed. Comparethistothespeedoflight,whichis3.0 10 8 m/s.Later inthiscourse,youwilllearnthattherearesomenewthingsgoing 116 Chapter3AccelerationandFreeFall

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Problem14. oninphysicswhenonegetsclosetothespeedoflight,andthatit isimpossibletoexceedthespeedoflight.Fornow,though,justuse thesimplerideasyou'velearnedsofar. p ? 11 Youclimbhalf-wayupatree,anddroparock.Thenyou climbtothetop,anddropanotherrock.Howmanytimesgreater isthevelocityofthesecondrockonimpact?Explain.Theanswer isnottwotimesgreater. 12 Alicedropsarockoacli.Bubbashootsagunstraight downfromtheedgeofthesamecli.Comparetheaccelerationsof therockandthebulletwhiletheyareintheaironthewaydown. [BasedonaproblembySerwayandFaughn.] 13 Apersonisparachutejumping.Duringthetimebetween whensheleapsoutoftheplaneandwhensheopensherchute,her altitudeisgivenbyanequationoftheform y = b )]TJ/F20 10.9091 Tf 10.909 0 Td [(c t + ke )]TJ/F21 7.9701 Tf 6.586 0 Td [(t=k where e isthebaseofnaturallogarithms,and b c ,and k areconstants.Becauseofairresistance,hervelocitydoesnotincreaseata steadyrateasitwouldforanobjectfallinginvacuum. aWhatunitswould b c ,and k havetohavefortheequationto makesense? bFindtheperson'svelocity, v ,asafunctionoftime.[Youwill needtousethechainrule,andthefactthatd e x = d x = e x .] p cUseyouranswerfrompartbtogetaninterpretationofthe constant c .[Hint: e )]TJ/F21 7.9701 Tf 6.586 0 Td [(x approacheszeroforlargevaluesof x .] dFindtheperson'sacceleration, a ,asafunctionoftime. p eUseyouranswerfrompartbtoshowthatifshewaitslong enoughtoopenherchute,heraccelerationwillbecomeverysmall. R 14 Thetoppartofthegureshowstheposition-versus-time graphforanobjectmovinginonedimension.Onthebottompart ofthegure,sketchthecorrespondingv-versus-tgraph. Solution,p.270 15 OnNewYear'sEve,astupidpersonresapistolstraightup. Thebulletleavesthegunataspeedof100m/s.Howlongdoesit takebeforethebullethitstheground? Solution,p.271 16 IftheaccelerationofgravityonMarsis1/3thatonEarth, howmanytimeslongerdoesittakeforarocktodropthesame distanceonMars?Ignoreairresistance. Solution,p.271 17 Ahoneybee'spositionasafunctionoftimeisgivenby x =10 t )]TJ/F20 10.9091 Tf 11.517 0 Td [(t 3 ,where t isinsecondsand x inmeters.Whatisits accelerationat t =3.0s? Solution,p.271 R Problems 117

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Problem19. Problem20. Problem23. 18 InJuly1999,PopularMechanicscarriedoutteststond whichcarsoldbyamajorautomakercouldcoveraquartermile metersintheshortesttime,startingfromrest.Becausethe distanceissoshort,thistypeoftestisdesignedmainlytofavorthe carwiththegreatestacceleration,notthegreatestmaximumspeed whichisirrelevanttotheaverageperson.Thewinnerwasthe DodgeViper,withatimeof12.08s.Thecar'stopandpresumably nalspeedwas118.51milesperhour.98m/s.aIfacar, startingfromrestandmovingwith constant acceleration,covers aquartermileinthistimeinterval,whatisitsacceleration?b Whatwouldbethenalspeedofacarthatcoveredaquartermile withtheconstantaccelerationyoufoundinparta?cBasedon thediscrepancybetweenyouranswerinpart b andtheactualnal speedoftheViper,whatdoyouconcludeabouthowitsacceleration changedovertime? Solution,p.271 19 Thegraphrepresentsthemotionofaballthatrollsupahill andthenbackdown.Whendoestheballreturntothelocationit hadat t =0? Solution,p.271 20 aTheballisreleasedatthetopoftherampshowninthe gure.Frictionisnegligible.Usephysicalreasoningtodraw v )]TJ/F20 10.9091 Tf 11.214 0 Td [(t and a )]TJ/F20 10.9091 Tf 10.477 0 Td [(t graphs.Assumethattheballdoesn'tbounceatthepoint wheretherampchangesslope.bDothesameforthecasewhere theballisrolleduptheslopefromtherightside,butdoesn'tquite haveenoughspeedtomakeitoverthetop. Solution,p.271 21 Youthrowarubberballup,anditfallsandbouncesseveraltimes.Drawgraphsofposition,velocity,andaccelerationas functionsoftime. Solution,p.272 22 Startingfromrest,aballrollsdownaramp,travelinga distance L andpickingupanalspeed v .Howmuchofthedistance didtheballhavetocoverbeforeachievingaspeedof v= 2?[Based onaproblembyArnoldArons.] Solution,p.273 23 ThegraphshowstheaccelerationofachipmunkinaTV cartoon.Itconsistsoftwocirculararcsandtwolinesegments. At t =0.00 s ,thechipmunk'svelocityis )]TJ/F15 10.9091 Tf 8.485 0 Td [(3.10m = s.Whatisits velocityat t =10.00s? 24 Findtheerrorinthefollowingcalculation.Astudentwants tondthedistancetraveledbyacarthatacceleratesfromrestfor 5.0swithanaccelerationof2.0m = s 2 .Firsthesolves a = v= t for v =10m = s.Thenhemultipliestondm = s.0s=50m. Donotjustrecalculatetheresultbyadierentmethod;ifthatwas allyoudid,you'dhavenowayofknowingwhichcalculationwas correct,yoursorhis. 118 Chapter3AccelerationandFreeFall

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Problem27. 25 Accelerationcouldbedenedeitheras v= t orastheslope ofthetangentlineonthe v )]TJ/F20 10.9091 Tf 11.284 0 Td [(t graph.Iseitheronesuperiorasa denition,oraretheyequivalent?Ifyousayoneisbetter,givean exampleofasituationwhereitmakesadierencewhichoneyou use. 26 Ifanobjectstartsacceleratingfromrest,wehave v 2 = 2 a x foritsspeedafterithastraveledadistance x .Explainin wordswhyitmakessensethattheequationhasvelocitysquared,but distanceonlytotherstpower.Don'trecapitulatethederivation inthebook,orgiveajusticationbasedonunits.Thepointis toexplainwhatthisfeatureoftheequationtellsusabouttheway speedincreasesasmoredistanceiscovered. 27 Thegureshowsapractical,simpleexperimentfordetermining g tohighprecision.Twosteelballsaresuspendedfromelectromagnets,andarereleasedsimultaneouslywhentheelectriccurrent isshuto.Theyfallthroughunequalheights x 1 and x 2 .A computerrecordsthesoundsthroughamicrophoneasrstoneball andthentheotherstrikestheoor.Fromthisrecording,wecan accuratelydeterminethequantity T denedas T = t 2 )]TJ/F15 10.9091 Tf 10.592 0 Td [( t 1 ,i.e., thetimelagbetweentherstandsecondimpacts.Notethatsince theballsdonotmakeanysoundwhentheyarereleased,wehave nowayofmeasuringtheindividualtimes t 2 and t 1 aFindanequationfor g intermsofthemeasuredquantities T x 1 and x 2 p bChecktheunitsofyourequation. cCheckthatyourequationgivesthecorrectresultinthecase where x 1 isveryclosetozero.However,isthiscaserealistic? dWhathappenswhen x 1 = x 2 ?Discussthisbothmathematicallyandphysically. 28 Thespeedrequiredforalow-earthorbitis7.9 10 3 m = ssee ch.10.Whenarocketislaunchedintoorbit,itgoesupalittleat rsttogetabovealmostalloftheatmosphere,butthentipsover horizontallytobuilduptoorbitalspeed.Supposethehorizontal accelerationislimitedto3 g tokeepfromdamagingthecargoor hurtingthecrew,foracrewedight.aWhatistheminimum distancetherocketmusttraveldownrangebeforeitreachesorbital speed?Howmuchdoesitmatterwhetheryoutakeintoaccountthe initialeastwardvelocityduetotherotationoftheearth?bRather thanarocketship,itmightbeadvantageoustousearailgundesign, inwhichthecraftwouldbeacceleratedtoorbitalspeedsalonga railroadtrack.Thishastheadvantagethatitisn'tnecessarytolift alargemassoffuel,sincetheenergysourceisexternal.Basedon youranswertoparta,commentonthefeasibilityofthisdesignfor crewedlaunchesfromtheearth'ssurface. Problems 119

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29 Someeascanjumpashighas30cm.Theeaonlyhasa shorttimetobuildupspeed|thetimeduringwhichitscenterof massisacceleratingupwardbutitsfeetarestillincontactwiththe ground.Makeanorder-of-magnitudeestimateoftheacceleration theeaneedstohavewhilestraighteningitslegs,andstateyour answerinunitsof g ,i.e.,howmany g 'sitpulls."Forcomparison, ghterpilotsblackoutordieiftheyexceedabout5or10 g 's. 30 ConsiderthefollowingpassagefromAliceinWonderland,in whichAlicehasbeenfallingforalongtimedownarabbithole: Down,down,down.Wouldthefall never cometoanend?I wonderhowmanymilesI'vefallenbythistime?"shesaidaloud. Imustbegettingsomewherenearthecenteroftheearth.Letme see:thatwouldbefourthousandmilesdown,Ithink"for,yousee, Alicehadlearnedseveralthingsofthissortinherlessonsinthe schoolroom,andthoughthiswasnota very goodopportunityfor showingoherknowledge,astherewasnoonetolistentoher,still itwasgoodpracticetosayitover... Alicedoesn'tknowmuchphysics,butlet'strytocalculatethe amountoftimeitwouldtaketofallfourthousandmiles,starting fromrestwithanaccelerationof10m = s 2 .Thisisreallyonlyalower limit;iftherereallywasaholethatdeep,thefallwouldactually takealongertimethantheoneyoucalculate,bothbecausethere isairfrictionandbecausegravitygetsweakerasyougetdeeperat thecenteroftheearth, g iszero,becausetheearthispullingyou equallyineverydirectionatonce. 120 Chapter3AccelerationandFreeFall

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122 Chapter3AccelerationandFreeFall

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IsaacNewton Chapter4 ForceandMotion IfIhaveseenfartherthanothers,itisbecauseIhavestood ontheshouldersofgiants. Newton,referringtoGalileo EvenasgreatandskepticalageniusasGalileowasunableto makemuchprogressonthecausesofmotion.ItwasnotuntilagenerationlaterthatIsaacNewton-1727wasabletoattackthe problemsuccessfully.Inmanyways,Newton'spersonalitywasthe oppositeofGalileo's.WhereGalileoagressivelypublicizedhisideas, 123

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a / Aristotlesaidmotionhad tobecausedbyaforce.To explainwhyanarrowkeptying afterthebowstringwasnolonger pushingonit,hesaidtheair rushedaroundbehindthearrow andpusheditforward.Weknow thisiswrong,becauseanarrow shotinavacuumchamberdoes notinstantlydroptotheoor asitleavesthebow.Galileo andNewtonrealizedthataforce wouldonlybeneededtochange thearrow'smotion,nottomake itsmotioncontinue. Newtonhadtobecoaxedbyhisfriendsintopublishingabookon hisphysicaldiscoveries.WhereGalileo'swritinghadbeenpopular anddramatic,Newtonoriginatedthestilted,impersonalstylethat mostpeoplethinkisstandardforscienticwriting.Scienticjournalstodayencouragealessponderousstyle,andpapersareoften writtenintherstperson.Galileo'stalentforarousinganimosityamongtherichandpowerfulwasmatchedbyNewton'sskillat makinghimselfapopularvisitoratcourt.Galileonarrowlyescaped beingburnedatthestake,whileNewtonhadthegoodfortuneofbeingonthewinningsideoftherevolutionthatreplacedKingJames IIwithWilliamandMaryofOrange,leadingtoalucrativepost runningtheEnglishroyalmint. Newtondiscoveredtherelationshipbetweenforceandmotion, andrevolutionizedourviewoftheuniversebyshowingthatthe samephysicallawsappliedtoallmatter,whetherlivingornonliving,onoroofourplanet'ssurface.Hisbookonforceandmotion, the MathematicalPrinciplesofNaturalPhilosophy ,wasuncontradictedbyexperimentfor200years,buthisothermainwork, Optics ,wasonthewrongtrack,assertingthatlightwascomposed ofparticlesratherthanwaves.Newtonwasalsoanavidalchemist, afactthatmodernscientistswouldliketoforget. 4.1Force Weneedonlyexplainchangesinmotion,notmotionitself. Sofaryou'vestudiedthemeasurementofmotioninsomedetail, butnotthereasonswhyacertainobjectwouldmoveinacertain way.Thischapterdealswiththewhy"questions.Aristotle'sideas aboutthecausesofmotionwerecompletelywrong,justlikeallhis otherideasaboutphysicalscience,butitwillbeinstructivetostart withthem,becausetheyamounttoaroadmapofmodernstudents' incorrectpreconceptions. Aristotlethoughtheneededtoexplainbothwhymotionoccurs andwhymotionmightchange.NewtoninheritedfromGalileothe importantcounter-Aristotelianideathatmotionneedsnoexplanation,thatitisonly changes inmotionthatrequireaphysicalcause. Aristotle'sneedlesslycomplexsystemgavethreereasonsformotion: Naturalmotion,suchasfalling,camefromthetendencyof objectstogototheirnatural"place,ontheground,and cometorest. Voluntarymotionwasthetypeofmotionexhibitedbyanimals,whichmovedbecausetheychoseto. Forcedmotionoccurredwhenanobjectwasactedonbysome otherobjectthatmadeitmove. 124 Chapter4ForceandMotion

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b / Oureyesreceiveblue lightreectedfromthispainting becauseMonetwantedtorepresentwaterwiththecolorblue. Thisisavalidstatementatone levelofexplanation,butphysics worksatthephysicallevelof explanation,inwhichbluelight getstoyoureyesbecauseitis reectedbybluepigmentsinthe paint. Motionchangesduetoaninteractionbetweentwoobjects. IntheAristoteliantheory,naturalmotionandvoluntarymotionareone-sidedphenomena:theobjectcausesitsownmotion. Forcedmotionissupposedtobeatwo-sidedphenomenon,because oneobjectimposesitscommands"onanother.WhereAristotle conceivedofsomeofthephenomenaofmotionasone-sidedand othersastwo-sided,Newtonrealizedthatachangeinmotionwas alwaysatwo-sidedrelationshipofaforceactingbetweentwophysicalobjects. Theone-sidednaturalmotion"descriptionoffallingmakesa crucialomission.Theaccelerationofafallingobjectisnotcaused byitsownnatural"tendenciesbutbyanattractiveforcebetween itandtheplanetEarth.Moonrocksbroughtbacktoourplanetdo notwant"toybackuptothemoonbecausethemoonistheir natural"place.Theyfalltotheoorwhenyoudropthem,just likeourhomegrownrocks.Aswe'lldiscussinmoredetaillaterin thiscourse,gravitationalforcesaresimplyanattractionthatoccurs betweenanytwophysicalobjects.Minutegravitationalforcescan evenbemeasuredbetweenhuman-scaleobjectsinthelaboratory. Theideaofnaturalmotionalsoexplainsincorrectlywhythings cometorest.Abasketballrollingacrossabeachslowstoastop becauseitisinteractingwiththesandviaafrictionalforce,not becauseofitsowndesiretobeatrest.Ifitwasonafrictionless surface,itwouldneverslowdown.ManyofAristotle'smistakes stemmedfromhisfailuretorecognizefrictionasaforce. Theconceptofvoluntarymotionisequallyawed.Youmay havebeenalittleuneasyaboutitfromthestart,becauseitassumes acleardistinctionbetweenlivingandnonlivingthings.Today,however,weareusedtohavingthehumanbodylikenedtoacomplex machine.Inthemodernworld-view,theborderbetweentheliving andtheinanimateisafuzzyno-man'slandinhabitedbyviruses, prions,andsiliconchips.Furthermore,Aristotle'sstatementthat youcantakeastepforwardbecauseyouchooseto"inappropriately mixestwolevelsofexplanation.Atthephysicallevelofexplanation,thereasonyourbodystepsforwardisbecauseofafrictional forceactingbetweenyourfootandtheoor.Iftheoorwascovered withapuddleofoil,noamountofchoosingto"wouldenableyou totakeagracefulstrideforward. Forcescanallbemeasuredonthesamenumericalscale. IntheAristotelian-scholastictradition,thedescriptionofmotionasnatural,voluntary,orforcedwasonlythebroadestlevelof classication,likesplittinganimalsintobirds,reptiles,mammals, andamphibians.Theremightbethousandsoftypesofmotion, eachofwhichwouldfollowitsownrules.Newton'srealizationthat allchangesinmotionwerecausedbytwo-sidedinteractionsmade itseemthatthephenomenamighthavemoreincommonthanhad Section4.1Force 125

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beenapparent.IntheNewtoniandescription,thereisonlyonecause forachangeinmotion,whichwecallforce.Forcesmaybeofdierenttypes,buttheyallproducechangesinmotionaccordingtothe samerules.Anyaccelerationthatcanbeproducedbyamagnetic forcecanequallywellbeproducedbyanappropriatelycontrolled streamofwater.Wecanspeakoftwoforcesasbeingequalifthey producethesamechangeinmotionwhenappliedinthesamesituation,whichmeansthattheypushedorpulledequallyhardinthe samedirection. Theideaofanumericalscaleofforceandthenewtonunitwere introducedinchapter0.Torecapitulatebriey,aforceiswhena pairofobjectspushorpulloneachother,andonenewtonisthe forcerequiredtoacceleratea1-kgobjectfromresttoaspeedof1 m/sin1second. Morethanoneforceonanobject Asifwehadn'tkickedpoorAristotlearoundsuciently,his theoryhasanotherimportantaw,whichisimportanttodiscuss becauseitcorrespondstoanextremelycommonstudentmisconception.Aristotleconceivedofforcedmotionasarelationshipinwhich oneobjectwasthebossandtheotherfollowedorders."Itthereforewouldonlymakesenseforanobjecttoexperienceoneforceat atime,becauseanobjectcouldn'tfollowordersfromtwosourcesat once.IntheNewtoniantheory,forcesarenumbers,notorders,and ifmorethanoneforceactsonanobjectatonce,theresultisfound byaddingupalltheforces.Itisunfortunatethattheuseofthe Englishwordforce"hasbecomestandard,becausetomanypeople itsuggeststhatyouareforcing"anobjecttodosomething.The forceoftheearth'sgravitycannotforce"aboattosink,because thereareotherforcesactingontheboat.Addingthemupgivesa totalofzero,sotheboatacceleratesneitherupnordown. Objectscanexertforcesoneachotheratadistance. Aristotledeclaredthatforcescouldonlyactbetweenobjectsthat weretouching,probablybecausehewishedtoavoidthetypeofoccultspeculationthatattributedphysicalphenomenatotheinuence ofadistantandinvisiblepantheonofgods.Hewaswrong,however, asyoucanobservewhenamagnetleapsontoyourrefrigeratoror whentheplanetearthexertsgravitationalforcesonobjectsthatare intheair.Sometypesofforces,suchasfriction,onlyoperatebetweenobjectsincontact,andarecalledcontactforces.Magnetism, ontheotherhand,isanexampleofanoncontactforce.Although themagneticforcegetsstrongerwhenthemagnetisclosertoyour refrigerator,touchingisnotrequired. Weight Inphysics,anobject'sweight, F W ,isdenedastheearth's gravitationalforceonit.TheSIunitofweightisthereforethe 126 Chapter4ForceandMotion

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c / Forcesareappliedtoa saxophone.Inthisexample, positivesignshavebeenused consistentlyforforcestothe right,andnegativesignsfor forcestotheleft.Theforces arebeingappliedtodifferent placesonthesaxophone,butthe numericalvalueofaforcecarries noinformationaboutthat. Newton.Peoplecommonlyrefertothekilogramasaunitofweight, butthekilogramisaunitofmass,notweight.Notethatanobject's weightisnotaxedpropertyofthatobject.Objectsweighmore insomeplacesthaninothers,dependingonthelocalstrengthof gravity.Itistheirmassthatalwaysstaysthesame.Abaseball pitcherwhocanthrowa90-mile-per-hourfastballonearthwould notbeabletothrowanyfasteronthemoon,becausetheball's inertiawouldstillbethesame. Positiveandnegativesignsofforce We'llstartbyconsideringonlycasesofone-dimensionalcenterof-massmotioninwhichalltheforcesareparalleltothedirectionof motion,i.e.,eitherdirectlyforwardorbackward.Inonedimension, plusandminussignscanbeusedtoindicatedirectionsofforces,as showningurec.Wecanthenrefergenericallytoadditionofforces, ratherthanhavingtospeaksometimesofadditionandsometimesof subtraction.Weaddtheforcesshowninthegureandget11N.In general,weshouldchooseaone-dimensionalcoordinatesystemwith its x axisparallelthedirectionofmotion.Forcesthatpointalong thepositive x axisarepositive,andforcesintheoppositedirection arenegative.Forcesthatarenotdirectlyalongthe x axiscannotbe immediatelyincorporatedintothisscheme,butthat'sOK,because we'reavoidingthosecasesfornow. DiscussionQuestions A Inchapter0,Idened1Nastheforcethatwouldacceleratea 1-kgmassfromrestto1m/sin1s.Anticipatingthefollowingsection,you mightguessthat2Ncouldbedenedastheforcethatwouldaccelerate thesamemasstotwicethespeed,ortwicethemasstothesamespeed. Isthereaneasierwaytodene2Nbasedonthedenitionof1N? 4.2Newton'sFirstLaw Wearenowpreparedtomakeamorepowerfulrestatementofthe principleofinertia. Newton'srstlaw Ifthetotalforceonanobjectiszero,itscenterofmasscontinues inthesamestateofmotion. Inotherwords,anobjectinitiallyatrestispredictedtoremain atrestifthetotalforceonitiszero,andanobjectinmotionremains inmotionwiththesamevelocityinthesamedirection.Theconverse ofNewton'srstlawisalsotrue:ifweobserveanobjectmoving withconstantvelocityalongastraightline,thenthetotalforceon itmustbezero. Inafuturephysicscourseorinanothertextbook,youmayencounterthetermnetforce,"whichissimplyasynonymfortotal force. Section4.2Newton'sFirstLaw 127

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Whathappensifthetotalforceonanobjectisnotzero?It accelerates.Numericalpredictionoftheresultingaccelerationisthe topicofNewton'ssecondlaw,whichwe'lldiscussinthefollowing section. ThisistherstofNewton'sthreelawsofmotion.Itisnot importanttomemorizewhichofNewton'sthreelawsarenumbers one,two,andthree.Ifafuturephysicsteacherasksyousomething like,WhichofNewton'slawsareyouthinkingof,"aperfectlyacceptableanswerisTheoneaboutconstantvelocitywhenthere's zerototalforce."Theconceptsaremoreimportantthananyspecicformulationofthem.NewtonwroteinLatin,andIamnot awareofanymoderntextbookthatusesaverbatimtranslationof hisstatementofthelawsofmotion.Clearwritingwasnotinvogue inNewton'sday,andheformulatedhisthreelawsintermsofaconceptnowcalledmomentum,onlylaterrelatingittotheconceptof force.Nearlyallmoderntexts,includingthisone,startwithforce anddomomentumlater. Anelevatorexample1 Anelevatorhasaweightof5000N.Comparetheforcesthatthe cablemustexerttoraiseitatconstantvelocity,loweritatconstant velocity,andjustkeepithanging. Inallthreecasesthecablemustpullupwithaforceofexactly 5000N.Mostpeoplethinkyou'dneedatleastalittlemorethan 5000Ntomakeitgoup,andalittlelessthan5000Ntoletitdown, butthat'sincorrect.Extraforcefromthecableisonlynecessary forspeedingthecarupwhenitstartsgoinguporslowingitdown whenitnishesgoingdown.Decreasedforceisneededtospeed thecarupwhenitgetsgoingdownandtoslowitdownwhenit nishesgoingup.Butwhentheelevatoriscruisingatconstant velocity,Newton'srstlawsaysthatyoujustneedtocancelthe forceoftheearth'sgravity. Tomanystudents,thestatementintheexamplethatthecable's upwardforcecancels"theearth'sdownwardgravitationalforceimpliesthattherehasbeenacontest,andthecable'sforcehaswon, vanquishingtheearth'sgravitationalforceandmakingitdisappear. Thatisincorrect.Bothforcescontinuetoexist,butbecausethey addupnumericallytozero,theelevatorhasnocenter-of-massacceleration.Weknowthatbothforcescontinuetoexistbecausethey bothhaveside-eectsotherthantheireectsonthecar'scenter-ofmassmotion.Theforceactingbetweenthecableandthecarcontinuestoproducetensioninthecableandkeepthecabletaut.The earth'sgravitationalforcecontinuestokeepthepassengerswhom weareconsideringaspartoftheelevator-objectstucktotheoor andtoproduceinternalstressesinthewallsofthecar,whichmust holduptheoor. 128 Chapter4ForceandMotion

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Terminalvelocityforfallingobjectsexample2 Anobjectlikeafeatherthatisnotdenseorstreamlineddoesnot fallwithconstantacceleration,becauseairresistanceisnonnegligible.Infact,itsaccelerationtapersofftonearlyzerowithina fractionofasecond,andthefeathernishesdroppingatconstant speedknownasitsterminalvelocity.Whydoesthishappen? Newton'srstlawtellsusthatthetotalforceonthefeathermust havebeenreducedtonearlyzeroafterashorttime.Thereare twoforcesactingonthefeather:adownwardgravitationalforce fromtheplanetearth,andanupwardfrictionalforcefromtheair. Asthefeatherspeedsup,theairfrictionbecomesstrongerand stronger,andeventuallyitcancelsouttheearth'sgravitational force,sothefeatherjustcontinueswithconstantvelocitywithout speedingupanymore. Thesituationforaskydiverisexactlyanalogous.It'sjustthatthe skydiverexperiencesperhapsamilliontimesmoregravitational forcethanthefeather,anditisnotuntilsheisfallingveryfast thattheforceofairfrictionbecomesasstrongasthegravitationalforce.Ittakesherseveralsecondstoreachterminalvelocity,whichisontheorderofahundredmilesperhour. Moregeneralcombinationsofforces Itistooconstrainingtorestrictourattentiontocaseswhere alltheforcesliealongthelineofthecenterofmass'smotion.For onething,wecan'tanalyzeanycaseofhorizontalmotion,since anyobjectonearthwillbesubjecttoaverticalgravitationalforce! Forinstance,whenyouaredrivingyourcardownastraightroad, therearebothhorizontalforcesandverticalforces.However,the verticalforceshavenoeectonthecenterofmassmotion,because theroad'supwardforcesimplycounteractstheearth'sdownward gravitationalforceandkeepsthecarfromsinkingintotheground. Laterinthebookwe'lldealwiththemostgeneralcaseofmany forcesactingonanobjectatanyangles,usingthemathematical techniqueofvectoraddition,butthefollowingslightgeneralization ofNewton'srstlawallowsustoanalyzeagreatmanycasesof interest: Supposethatanobjecthastwosetsofforcesactingonit,one setalongthelineoftheobject'sinitialmotionandanothersetperpendiculartotherstset.Ifbothsetsofforcescancel,thenthe object'scenterofmasscontinuesinthesamestateofmotion. Section4.2Newton'sFirstLaw 129

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d / Example4. Apassengerridingthesubwayexample3 Describetheforcesactingonapersonstandinginasubway trainthatiscruisingatconstantvelocity. Noforceisnecessarytokeepthepersonmovingrelativeto theground.Hewillnotbeswepttothebackofthetrainifthe oorisslippery.Therearetwoverticalforcesonhim,theearth's downwardgravitationalforceandtheoor'supwardforce,which cancel.Therearenohorizontalforcesonhimatall,soofcourse thetotalhorizontalforceiszero. Forcesonasailboatexample4 Ifasailboatiscruisingatconstantvelocitywiththewindcoming fromdirectlybehindit,whatmustbetrueabouttheforcesacting onit? Theforcesactingontheboatmustbecancelingeachother out.Theboatisnotsinkingorleapingintotheair,soevidently theverticalforcesarecancelingout.Theverticalforcesarethe downwardgravitationalforceexertedbytheplanetearthandan upwardforcefromthewater. Theairismakingaforwardforceonthesail,andiftheboatis notacceleratinghorizontallythenthewater'sbackwardfrictional forcemustbecancelingitout. ContrarytoAristotle,moreforceisnotneededinordertomaintain ahigherspeed.Zerototalforceisalwaysneededtomaintain constantvelocity.Considerthefollowingmade-upnumbers: boatmovingat alow,constant velocity boatmovingat ahigh,constant velocity forwardforceof thewindonthe sail... 10,000N20,000N backwardforceof thewateronthe hull... )]TJ/F39 10.9091 Tf 8.485 0 Td [(10,000N )]TJ/F39 10.9091 Tf 8.485 0 Td [(20,000N totalforceonthe boat... 0N0N Thefasterboatstillhaszerototalforceonit.Theforwardforce onitisgreater,andthebackwardforcesmallermorenegative, butthat'sirrelevantbecauseNewton'srstlawhastodowiththe totalforce,nottheindividualforces. Thisexampleisquiteanalogoustotheoneaboutterminalvelocity offallingobjects,sincethereisafrictionalforcethatincreases withspeed.Aftercastingofffromthedockandraisingthesail, theboatwillacceleratebriey,andthenreachitsterminalvelocity, atwhichthewater'sfrictionalforcehasbecomeasgreatasthe wind'sforceonthesail. 130 Chapter4ForceandMotion

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DiscussionquestionB. DiscussionquestionC. Acarcrashexample5 Ifyoudriveyourcarintoabrickwall,whatisthemysterious forcethatslamsyourfaceintothesteeringwheel? Yoursurgeonhastakenphysics,sosheisnotgoingtobelieve yourclaimthatamysteriousforceistoblame.Sheknowsthat yourfacewasjustfollowingNewton'srstlaw.Immediatelyafter yourcarhitthewall,theonlyforcesactingonyourheadwere thesamecanceling-outforcesthathadexistedpreviously:the earth'sdownwardgravitationalforceandtheupwardforcefrom yourneck.Therewerenoforwardorbackwardforcesonyour head,butthecardidexperienceabackwardforcefromthewall, sothecarsloweddownandyourfacecaughtup. DiscussionQuestions A Newtonsaidthatobjectscontinuemovingifnoforcesareacting onthem,buthispredecessorAristotlesaidthataforcewasnecessaryto keepanobjectmoving.WhydoesAristotle'stheoryseemmoreplausible, eventhoughwenowbelieveittobewrong?WhatinsightwasAristotle missingaboutthereasonwhythingsseemtoslowdownnaturally?Give anexample. B Inthegurewhatwouldhavetobetrueaboutthesaxophone'sinitial motioniftheforcesshownweretoresultincontinuedone-dimensional motionofitscenterofmass? C Thisgurerequiresaneverfurthergeneralizationofthepreceding discussion.Afterstudyingtheforces,whatdoesyourphysicalintuitiontell youwillhappen?Canyoustateinwordshowtogeneralizetheconditions forone-dimensionalmotiontoincludesituationslikethisone? 4.3Newton'sSecondLaw Whataboutcaseswherethetotalforceonanobjectisnotzero, sothatNewton'srstlawdoesn'tapply?Theobjectwillhavean acceleration.Thewaywe'vedenedpositiveandnegativesigns offorceandaccelerationguaranteesthatpositiveforcesproduce positiveaccelerations,andlikewisefornegativevalues.Howmuch accelerationwillithave?Itwillclearlydependonboththeobject's massandontheamountofforce. Experimentswithanyparticularobjectshowthatitsaccelerationisdirectlyproportionaltothetotalforceappliedtoit.This mayseemwrong,sinceweknowofmanycaseswheresmallamounts offorcefailtomoveanobjectatall,andlargerforcesgetitgoing. Thisapparentfailureofproportionalityactuallyresultsfromforgettingthatthereisafrictionalforceinadditiontotheforcewe applytomovetheobject.Theobject'saccelerationisexactlyproportionaltothetotalforceonit,nottoanyindividualforceonit. Intheabsenceoffriction,evenaverytinyforcecanslowlychange thevelocityofaverymassiveobject. ExperimentsalsoshowthattheaccelerationisinverselyproporSection4.3Newton'sSecondLaw 131

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tionaltotheobject'smass,andcombiningthesetwoproportionalitiesgivesthefollowingwayofpredictingtheaccelerationofany object: Newton'ssecondlaw a = F total =m where m isanobject'smass F total isthesumoftheforcesactingonit,and a istheaccelerationoftheobject'scenterofmass. Wearepresentlyrestrictedtothecasewheretheforcesofinterest areparalleltothedirectionofmotion. Anacceleratingbusexample6 AVWbuswithamassof2000kgacceleratesfrom0to25m/s freewayspeedin34s.Assumingtheaccelerationisconstant, whatisthetotalforceonthebus? WesolveNewton'ssecondlawfor F total = ma ,andsubstitute v = t for a ,giving F total = m v = t =kgm = s )]TJ/F39 10.9091 Tf 10.909 0 Td [(0m = s = s =1.5kN. Ageneralization Aswiththerstlaw,thesecondlawcanbeeasilygeneralized toincludeamuchlargerclassofinterestingsituations: Supposeanobjectisbeingactedonbytwosetsofforces, onesetlyingalongtheobject'sinitialdirectionofmotionand anothersetactingalongaperpendicularline.Iftheforces perpendiculartotheinitialdirectionofmotioncancelout, thentheobjectacceleratesalongitsoriginallineofmotion accordingto a = F total =m Therelationshipbetweenmassandweight Massisdierentfromweight,butthey'rerelated.Anapple's masstellsushowharditistochangeitsmotion.Itsweightmeasures thestrengthofthegravitationalattractionbetweentheappleand theplanetearth.Theapple'sweightislessonthemoon,butits 132 Chapter4ForceandMotion

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e / Asimpledouble-panbalanceworksbycomparingthe weightforcesexertedbythe earthonthecontentsofthetwo pans.Sincethetwopansare atalmostthesamelocationon theearth'ssurface,thevalue of g isessentiallythesamefor eachone,andequalityofweight thereforealsoimpliesequalityof mass. f / Example7. massisthesame.AstronautsassemblingtheInternationalSpace Stationinzerogravitycannotjustpitchmassivemodulesbackand forthwiththeirbarehands;themodulesareweightless,butnot massless. Wehavealreadyseentheexperimentalevidencethatwhenweight theforceoftheearth'sgravityistheonlyforceactingonanobject,itsaccelerationequalstheconstant g ,and g dependsonwhere youareonthesurfaceoftheearth,butnotonthemassoftheobject.ApplyingNewton'ssecondlawthenallowsustocalculatethe magnitudeofthegravitationalforceonanyobjectintermsofits mass: j F W j = mg Theequationonlygivesthemagnitude,i.e.theabsolutevalue,of F W ,becausewe'redening g asapositivenumber,soitequalsthe absolutevalueofafallingobject'sacceleration. Solvedproblem:Deceleratingacarpage142,problem7 Weightandmassexample7 Figurefshowsmassesofoneandtwokilogramshungfroma springscale,whichmeasuresforceinunitsofnewtons.Explain thereadings. Let'sstartwiththesinglekilogram.It'snotaccelerating,so evidentlythetotalforceonitiszero:thespringscale'supward forceonitiscancelingouttheearth'sdownwardgravitational force.Thespringscaletellsushowmuchforceitisbeingobliged tosupply,butsincethetwoforcesareequalinstrength,the springscale'sreadingcanalsobeinterpretedasmeasuringthe strengthofthegravitationalforce,i.e.,theweightoftheonekilogrammass.Theweightofaone-kilogrammassshouldbe F W = mg =.0kg.8m = s 2 =9.8N, andthat'sindeedthereadingonthespringscale. Similarlyforthetwo-kilogrammass,wehave F W = mg =.0kg.8m = s 2 =19.6N. Calculatingterminalvelocityexample8 Experimentsshowthattheforceofairfrictiononafallingobject suchasaskydiverorafeathercanbeapproximatedfairlywell withtheequation j F air j = c Av 2 ,where c isaconstant, isthe densityoftheair, A isthecross-sectionalareaoftheobjectas seenfrombelow,and v istheobject'svelocity.Predicttheobject's terminalvelocity,i.e.,thenalvelocityitreachesafteralongtime. Section4.3Newton'sSecondLaw 133

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x m t s 101.84 202.86 303.80 404.67 505.53 606.38 707.23 808.10 908.96 1009.83 g / DiscussionquestionD. Astheobjectaccelerates,itsgreater v causestheupwardforce oftheairtoincreaseuntilnallythegravitationalforceandthe forceofairfrictioncancelout,afterwhichtheobjectcontinues atconstantvelocity.Wechooseacoordinatesysteminwhich positiveisup,sothatthegravitationalforceisnegativeandthe forceofairfrictionispositive.Wewanttondthevelocityatwhich F air + F W =0, i e ., c Av 2 )]TJ/F78 10.9091 Tf 10.909 0 Td [(mg =0. Solvingfor v gives v terminal = r mg c A self-checkA Itisimportanttogetintothehabitofinterpretingequations.Thismaybe difcultatrst,buteventuallyyouwillgetusedtothiskindofreasoning. Interprettheequation v terminal = p mg = c A inthecaseof =0. Howwouldtheterminalvelocityofa4-cmsteelballcomparetothat ofa1-cmball? Inadditiontoteasingoutthe mathematical meaningofanequation, wealsohavetobeabletoplaceitinits physical context.Howgenerally importantisthisequation? Answer,p.267 DiscussionQuestions A ShowthattheNewtoncanbereexpressedintermsofthethree basicmksunitsasthecombinationkg m = s 2 B Whatiswrongwiththefollowingstatements? gistheforceofgravity. Massisameasureofhowmuchspacesomethingtakesup. C Criticizethefollowingincorrectstatement: Ifanobjectisatrestandthetotalforceonitiszero,itstaysatrest. Therecanalsobecaseswhereanobjectismovingandkeepsonmoving withouthavinganytotalforceonit,butthatcanonlyhappenwhenthere's nofriction,likeinouterspace. D TableggiveslasertimingdataforBenJohnson's100mdashatthe 1987WorldChampionshipinRome.Hisworldrecordwaslaterrevoked becausehetestedpositiveforsteroids.Howdoesthetotalforceonhim changeoverthedurationoftherace? 134 Chapter4ForceandMotion

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4.4WhatForceIsNot Violinteachershavetoenduretheirbeginningstudents'screeching. Afrownappearsonthewoodwindteacher'sfaceasshewatchesher studenttakeabreathwithanexpansionofhisribcagebutnone inhisbelly.Whatmakesphysicsteacherscringeistheirstudents' verbalstatementsaboutforces.BelowIhavelistedseveraldicta aboutwhatforceisnot. Forceisnotapropertyofoneobject. Agreatmanyofstudents'incorrectdescriptionsofforcescould becuredbykeepinginmindthataforceisaninteractionoftwo objects,notapropertyofoneobject. Incorrectstatement: Thatmagnethasalotofforce. Ifthemagnetisonemillimeterawayfromasteelballbearing,they mayexertaverystrongattractiononeachother,butiftheywerea meterapart,theforcewouldbevirtuallyundetectable.Themagnet's strengthcanberatedusingcertainelectricalunitsampere )]TJ/F39 9.9626 Tf 9.767 0 Td [(meters 2 butnotinunitsofforce. Forceisnotameasureofanobject'smotion. Ifforceisnotapropertyofasingleobject,thenitcannotbe usedasameasureoftheobject'smotion. Incorrectstatement: Thefreighttrainrumbleddownthetrackswith awesomeforce. Forceisnotameasureofmotion.Ifthefreighttraincollideswitha stalledcementtruck,thensomeawesomeforceswilloccur,butifithits aytheforcewillbesmall. Forceisnotenergy. Therearetwomainapproachestounderstandingthemotionof objects,onebasedonforceandoneonadierentconcept,calledenergy.TheSIunitofenergyistheJoule,butyouareprobablymore familiarwiththecalorie,usedformeasuringfood'senergy,andthe kilowatt-hour,theunittheelectriccompanyusesforbillingyou. Physicsstudents'previousfamiliaritywithcaloriesandkilowatthoursismatchedbytheiruniversalunfamiliaritywithmeasuring forcesinunitsofNewtons,butthepreciseoperationaldenitionsof theenergyconceptsaremorecomplexthanthoseoftheforceconcepts,andtextbooks,includingthisone,almostuniversallyplacethe forcedescriptionofphysicsbeforetheenergydescription.During thelongperiodaftertheintroductionofforceandbeforethecareful denitionofenergy,studentsarethereforevulnerabletosituations inwhich,withoutrealizingit,theyareimputingthepropertiesof energytophenomenaofforce. Incorrectstatement: Howcanmychairbemakinganupwardforceon myrearend?Ithasnopower! Powerisaconceptrelatedtoenergy,e.g.,a100-wattlightbulbuses Section4.4WhatForceIsNot 135

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up100joulespersecondofenergy.Whenyousitinachair,noenergy isusedup,soforcescanexistbetweenyouandthechairwithoutany needforasourceofpower. Forceisnotstoredorusedup. Becauseenergycanbestoredandusedup,peoplethinkforce alsocanbestoredorusedup. Incorrectstatement: Ifyoudon'tllupyourtankwithgas,you'llrun outofforce. Energyiswhatyou'llrunoutof,notforce. Forcesneednotbeexertedbylivingthingsormachines. Transformingenergyfromoneformintoanotherusuallyrequires somekindoflivingormechanicalmechanism.Theconceptisnot applicabletoforces,whichareaninteractionbetweenobjects,not athingtobetransferredortransformed. Incorrectstatement: Howcanawoodenbenchbemakinganupward forceonmyrearend?Itdoesn'thaveanyspringsoranythinginsideit. Nospringsorotherinternalmechanismsarerequired.Ifthebench didn'tmakeanyforceonyou,youwouldobeyNewton'ssecondlawand fallthroughit.Evidentlyitdoesmakeaforceonyou! Aforceisthedirectcauseofachangeinmotion. Icanclickaremotecontroltomakemygaragedoorchangefrom beingatresttobeinginmotion.Mynger'sforceonthebutton, however,wasnottheforcethatactedonthedoor.Whenwespeak ofaforceonanobjectinphysics,wearetalkingaboutaforcethat actsdirectly.Similarly,whenyoupullareluctantdogalongbyits leash,theleashandthedogaremakingforcesoneachother,not yourhandandthedog.Thedogisnoteventouchingyourhand. self-checkB Whichofthefollowingthingscanbecorrectlydescribedintermsof force? Anuclearsubmarineischargingaheadatfullsteam. Anuclearsubmarine'spropellersspininthewater. Anuclearsubmarineneedstorefuelitsreactorperiodically. Answer,p.267 DiscussionQuestions A Criticizethefollowingincorrectstatement:Ifyoushoveabook acrossatable,frictiontakesawaymoreandmoreofitsforce,untilnally itstops. B Youhitatennisballagainstawall.Explainanyandallincorrect ideasinthefollowingdescriptionofthephysicsinvolved:Theballgets someforcefromyouwhenyouhitit,andwhenithitsthewall,itlosespart ofthatforce,soitdoesn'tbouncebackasfast.Themusclesinyourarm aretheonlythingsthataforcecancomefrom. 136 Chapter4ForceandMotion

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4.5InertialandNoninertialFramesof Reference Oneday,you'redrivingdownthestreetinyourpickuptruck,on yourwaytodeliverabowlingball.Theballisinthebackofthe truck,enjoyingitslittlejauntandtakinginthefreshairandsunshine.Thenyouhavetoslowdownbecauseastopsigniscoming up.Asyoubrake,youglanceinyourrearviewmirror,andseeyour trustycompanionacceleratingtowardyou.Didsomemysterious forcepushitforward?No,itonlyseemsthatwaybecauseyouand thecarareslowingdown.TheballisfaithfullyobeyingNewton's rstlaw,andasitcontinuesatconstantvelocityitgetsaheadrelativetotheslowingtruck.Noforcesareactingonitotherthanthe samecanceling-outverticalforcesthatwerealwaysactingonit. 1 TheballonlyappearedtoviolateNewton'srstlawbecausethere wassomethingwrongwithyourframeofreference,whichwasbased onthetruck. h / 1.Inaframeofreferencethat moveswiththetruck,thebowlingballappearstoviolateNewton'srstlawbyacceleratingdespitehavingnohorizontalforces onit.2.Inaninertialframeofreference,whichthesurfaceofthe earthapproximatelyis,thebowlingballobeysNewton'srstlaw. Itmovesequaldistancesinequal timeintervals,i.e.,maintainsconstantvelocity.Inthisframeof reference,itisthetruckthatappearstohaveachangeinvelocity,whichmakessense,sincethe roadismakingahorizontalforce onit. How,then,arewetotellinwhichframesofreferenceNewton's lawsarevalid?It'snogoodtosaythatweshouldavoidmoving framesofreference,becausethereisnosuchthingasabsoluterest orabsolutemotion.Allframescanbeconsideredasbeingeitherat restorinmotion.AccordingtoanobserverinIndia,thestripmall thatconstitutedtheframeofreferenceinpanelbofthegure wasmovingalongwiththeearth'srotationathundredsofmilesper hour. 1 Let'sassumeforsimplicitythatthereisnofriction. Section4.5InertialandNoninertialFramesofReference 137

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ThereasonwhyNewton'slawsfailinthetruck'sframeofreferenceisnotbecausethetruckis moving butbecauseitis accelerating Recallthatphysicistsusethewordtorefereithertospeedingupor slowingdown.Newton'slawswereworkingjustneinthemoving truck'sframeofreferenceaslongasthetruckwasmovingatconstantvelocity.Itwasonlywhenitsspeedchangedthattherewas aproblem.How,then,arewetotellwhichframesareaccelerating andwhicharenot?Whatifyouclaimthatyourtruckisnotaccelerating,andthesidewalk,theasphalt,andtheBurgerKingare accelerating?Thewaytosettlesuchadisputeistoexaminethe motionofsomeobject,suchasthebowlingball,whichweknow haszerototalforceonit.Anyframeofreferenceinwhichtheball appearstoobeyNewton'srstlawisthenavalidframeofreference, andtoanobserverinthatframe,Mr.Newtonassuresusthatall theotherobjectsintheuniversewillobeyhislawsofmotion,not justtheball. Validframesofreference,inwhichNewton'slawsareobeyed, arecalledinertialframesofreference.Framesofreferencethatare notinertialarecallednoninertialframes.Inthoseframes,objects violatetheprincipleofinertiaandNewton'srstlaw.Whilethe truckwasmovingatconstantvelocity,bothitandthesidewalk werevalidinertialframes.Thetruckbecameaninvalidframeof referencewhenitbeganchangingitsvelocity. Youusuallyassumethegroundunderyourfeetisaperfectly inertialframeofreference,andwemadethatassumptionabove.It isn'tperfectlyinertial,however.Itsmotionthroughspaceisquite complicated,beingcomposedofapartduetotheearth'sdailyrotationarounditsownaxis,themonthlywobbleoftheplanetcaused bythemoon'sgravity,andtherotationoftheeartharoundthesun. Sincetheaccelerationsinvolvedarenumericallysmall,theearthis approximatelyavalidinertialframe. Noninertialframesareavoidedwheneverpossible,andwewill seldom,ifever,haveoccasiontousetheminthiscourse.Sometimes, however,anoninertialframecanbeconvenient.Navalgunners,for instance,getalltheirdatafromradars,humaneyeballs,andother detectionsystemsthataremovingalongwiththeearth'ssurface. Sincetheirgunshaverangesofmanymiles,thesmalldiscrepanciesbetweentheirshells'actualaccelerationsandtheaccelerations predictedbyNewton'ssecondlawcanhaveeectsthataccumulate andbecomesignicant.Inordertokillthepeopletheywanttokill, theyhavetoaddsmallcorrectionsontotheequation a = F total =m Doingtheircalculationsinaninertialframewouldallowthemto usetheusualformofNewton'ssecondlaw,buttheywouldhave toconvertalltheirdataintoadierentframeofreference,which wouldrequirecumbersomecalculations. 138 Chapter4ForceandMotion

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DiscussionQuestion A Ifanobjecthasalinear x )]TJ/F78 9.9627 Tf 10.855 0 Td [(t graphinacertaininertialframe, whatistheeffectonthegraphifwechangetoacoordinatesystemwith adifferentorigin?Whatistheeffectifwekeepthesameoriginbutreversethepositivedirectionofthe x axis?Howaboutaninertialframe movingalongsidetheobject?Whatifwedescribetheobject'smotionin anoninertialframe? Section4.5InertialandNoninertialFramesofReference 139

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Summary SelectedVocabulary weight.......theforceofgravityonanobject,equalto mg inertialframe..aframeofreferencethatisnotaccelerating, oneinwhichNewton'srstlawistrue noninertialframeanacceleratingframeofreference,inwhich Newton'srstlawisviolated Notation F W ........weight OtherTerminologyandNotation netforce.....anotherwayofsayingtotalforce" Summary Newton'srstlawofmotionstatesthatifalltheforcesonan objectcanceleachotherout,thentheobjectcontinuesinthesame stateofmotion.ThisisessentiallyamorerenedversionofGalileo's principleofinertia,whichdidnotrefertoanumericalscaleofforce. Newton'ssecondlawofmotionallowsthepredictionofanobject'saccelerationgivenitsmassandthetotalforceonit, a cm = F total =m .Thisisonlytheone-dimensionalversionofthelaw;the full-threedimensionaltreatmentwillcomeinchapter8,Vectors. Withoutthevectortechniques,wecanstillsaythatthesituation remainsunchangedbyincludinganadditionalsetofvectorsthat cancelamongthemselves,eveniftheyarenotinthedirectionof motion. Newton'slawsofmotionareonlytrueinframesofreferencethat arenotaccelerating,knownasinertialframes. ExploringFurther IsaacNewton:TheLastSorcerer ,MichaelWhite.AnexcellentbiographyofNewtonthatbringsusclosertotherealman. 140 Chapter4ForceandMotion

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Anobjectisobservedtobemovingatconstantspeedina certaindirection.Canyouconcludethatnoforcesareactingonit? Explain.[BasedonaproblembySerwayandFaughn.] 2 Acarisnormallycapableofanaccelerationof3m = s 2 .Ifit istowingatrailerwithhalfasmuchmassasthecaritself,whataccelerationcanitachieve?[BasedonaproblemfromPSSCPhysics.] 3 aLet T bethemaximumtensionthatanelevator'scablecan withstandwithoutbreaking,i.e.,themaximumforceitcanexert. Ifthemotorisprogrammedtogivethecaranacceleration a ,what isthemaximummassthatthecarcanhave,includingpassengers, ifthecableisnottobreak? p bInterprettheequationyouderivedinthespecialcasesof a =0 andofadownwardaccelerationofmagnitude g Interpret"meanstoanalyzethebehavioroftheequation,and connectthattoreality,asintheself-checkonpage134. 4 Ahelicopterofmass m istakingovertically.Theonlyforces actingonitaretheearth'sgravitationalforceandtheforce, F air oftheairpushinguponthepropellerblades. aIfthehelicopterliftsoat t =0,whatisitsverticalspeedat time t ? bPlugnumbersintoyourequationfromparta,using m =2300 kg, F air =27000N,and t =4.0s. 5 Inthe1964OlympicsinTokyo,thebestmen'shighjumpwas 2.18m.FouryearslaterinMexicoCity,thegoldmedalinthesame eventwasforajumpof2.24m.BecauseofMexicoCity'saltitude m,theaccelerationofgravitythereislowerthanthatin Tokyobyabout0.01m = s 2 .Supposeahigh-jumperhasamassof 72kg. aComparehismassandweightinthetwolocations. bAssumethatheisabletojumpwiththesameinitialvertical velocityinbothlocations,andthatallotherconditionsarethesame exceptforgravity.Howmuchhighershouldhebeabletojumpin MexicoCity? p Actually,thereasonforthebigchangebetween'64and'68wasthe introductionoftheFosburyop." ? Problems 141

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Problem6. 6 Ablimpisinitiallyatrest,hovering,whenat t =0thepilot turnsonthemotorofthepropeller.Themotorcannotinstantly getthepropellergoing,butthepropellerspeedsupsteadily.The steadilyincreasingforcebetweentheairandthepropellerisgiven bytheequation F = kt ,where k isaconstant.Ifthemassofthe blimpis m ,nditspositionasafunctionoftime.Assumethat duringtheperiodoftimeyou'redealingwith,theblimpisnotyet movingfastenoughtocauseasignicantbackwardforceduetoair resistance. p R 7 Acarisacceleratingforwardalongastraightroad.Iftheforce oftheroadonthecar'swheels,pushingitforward,isaconstant3.0 kN,andthecar'smassis1000kg,thenhowlongwillthecartake togofrom20m/sto50m/s? Solution,p.273 8 Somegardenshearsarelikeapairofscissors:onesharpblade slicespastanother.Intheanvil"type,however,asharpblade pressesagainstaatoneratherthangoingpastit.Agardening booksaysthatforpeoplewhoarenotveryphysicallystrong,the anviltypecanmakeiteasiertocuttoughbranches,becauseit concentratestheforceononeside.Evaluatethisclaimbasedon Newton'slaws.[Hint:Considertheforcesactingonthebranch, andthemotionofthebranch.] 9 Auraniumatomdeepintheearthspitsoutanalphaparticle. Analphaparticleisafragmentofanatom.Thisalphaparticlehas initialspeed v ,andtravelsadistance d beforestoppingintheearth. aFindtheforce, F ,thatactedontheparticle,intermsof v d anditsmass, m .Don'tpluginanynumbersyet.Assumethatthe forcewasconstant. p bShowthatyouranswerhastherightunits. cDiscusshowyouranswertopartadependsonallthreevariables, andshowthatitmakessense.Thatis,foreachvariable,discuss whatwouldhappentotheresultifyouchangeditwhilekeepingthe othertwovariablesconstant.Wouldabiggervaluegiveasmaller result,orabiggerresult?Onceyou'veguredoutthis mathematical relationship,showthatitmakessense physically dEvaluateyourresultfor m =6.7 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(27 kg, v =2.0 10 4 km/s, and d =0.71mm. p 142 Chapter4ForceandMotion

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Problem10,partc. 10 Youaregivenalargesealedbox,andarenotallowedtoopen it.Whichofthefollowingexperimentsmeasureitsmass,andwhich measureitsweight?[Hint:Whichexperimentswouldgivedierent resultsonthemoon?] aPutitonafrozenlake,throwarockatit,andseehowfastit scootsawayafterbeinghit. bDropitfromathird-oorbalcony,andmeasurehowloudthe soundiswhenithitstheground. cAsshowninthegure,connectitwithaspringtothewall,and watchitvibrate. Solution,p.273 11 WhileescapingfromthepalaceoftheevilMartianemperor,SallySpacehoundjumpsfromatowerofheight h downto theground.Ordinarilythefallwouldbefatal,butsheresher blasterriestraightdown,producinganupwardforceofmagnitude F B .Thisforceisinsucienttolevitateher,butitdoescancelout someoftheforceofgravity.Duringthetime t thatsheisfalling, Sallyisunfortunatelyexposedtorefromtheemperor'sminions, andcan'tdodgetheirshots.Let m behermass,and g thestrength ofgravityonMars. aFindthetime t intermsoftheothervariables. bChecktheunitsofyouranswertoparta. bForsucientlylargevaluesof F B ,youranswertopartabecomesnonsense|explainwhat'sgoingon. p 12 WhenIcookrice,someofthedrygrainsalwayssticktothe measuringcup.Togetthemout,Iturnthemeasuringcupupsidedown,andhitthebackofthecupwithmyhand.Explainwhythis works,andwhyitssuccessdependsonhittingthecuphardenough. Problems 143

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Whatforcesactonthegirl? Chapter5 AnalysisofForces 5.1Newton'sThirdLaw Newtoncreatedthemodernconceptofforcestartingfromhisinsight thatalltheeectsthatgovernmotionareinteractionsbetweentwo objects:unliketheAristoteliantheory,Newtonianphysicshasno phenomenainwhichanobjectchangesitsownmotion. Isoneobjectalwaystheorder-giver"andtheothertheorder145

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c / Rocketsworkbypushing exhaustgasesouttheback. Newton'sthirdlawsaysthatifthe rocketexertsabackwardforce onthegases,thegasesmust makeanequalforwardforceon therocket.Rocketenginescan functionabovetheatmosphere, unlikepropellersandjets,which workbypushingagainstthe surroundingair. a / Twomagnetsexertforces oneachother. b / Twopeople'shandsexert forcesoneachother. follower"?Asanexample,considerabatterhittingabaseball.The batdenitelyexertsalargeforceontheball,becausetheballacceleratesdrastically.Butifyouhaveeverhitabaseball,youalso knowthattheballmakesaforceonthebat|oftenwithpainful resultsifyourtechniqueisasbadasmine! Howdoestheball'sforceonthebatcomparewiththebat's forceontheball?Thebat'saccelerationisnotasspectacularas theball's,butmaybeweshouldn'texpectittobe,sincethebat's massismuchgreater.Infact,carefulmeasurementsofbothobjects' massesandaccelerationswouldshowthat m ball a ball isverynearly equalto )]TJ/F20 10.9091 Tf 8.485 0 Td [(m bat a bat ,whichsuggeststhattheball'sforceonthebat isofthesamemagnitudeasthebat'sforceontheball,butinthe oppositedirection. Figuresaandbshowtwosomewhatmorepracticallaboratory experimentsforinvestigatingthisissueaccuratelyandwithouttoo muchinterferencefromextraneousforces. Inexperimenta,alargemagnetandasmallmagnetareweighed separately,andthenonemagnetishungfromthepanofthetop balancesothatitisdirectlyabovetheothermagnet.Thereisan attractionbetweenthetwomagnets,causingthereadingonthetop scaletoincreaseandthereadingonthebottomscaletodecrease. Thelargemagnetismorepowerful"inthesensethatitcanpick upaheavierpaperclipfromthesamedistance,somanypeoplehave astrongexpectationthatonescale'sreadingwillchangebyafar dierentamountthantheother.Instead,wendthatthetwo changesareequalinmagnitudebutoppositeindirection:theforce ofthebottommagnetpullingdownonthetoponehasthesame strengthastheforceofthetoponepullinguponthebottomone. Inexperimentb,twopeoplepullontwospringscales.Regardless ofwhotriestopullharder,thetwoforcesasmeasuredonthespring scalesareequal.Interposingthetwospringscalesisnecessaryin ordertomeasuretheforces,buttheoutcomeisnotsomearticial resultofthescales'interactionswitheachother.Ifonepersonslaps anotherhardonthehand,theslapper'shandhurtsjustasmuch astheslappee's,anditdoesn'tmatteriftherecipientoftheslap triestobeinactive.Punchingsomeoneinthemouthcausesjust asmuchforceonthestasonthelips.It'sjustthatthelipsare moredelicate.Theforcesareequal,butnotthelevelsofpainand injury. Newton,afterobservingaseriesofresultssuchasthese,decided thattheremustbeafundamentallawofnatureatwork: Newton'sthirdlaw Forcesoccurinequalandoppositepairs:wheneverobjectAexerts aforceonobjectB,objectBmustalsobeexertingaforceonobject A.Thetwoforcesareequalinmagnitudeandoppositeindirection. 146 Chapter5AnalysisofForces

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d / Aswimmerdoingthebreast strokepushesbackwardagainst thewater.ByNewton'sthirdlaw, thewaterpushesforwardonher. e / Newton'sthirdlawdoes notmeanthatforcesalways canceloutsothatnothingcan evermove.Ifthesetwogure skaters,initiallyatrest,push againsteachother,theywillboth move. Inone-dimensionalsituations,wecanuseplusandminussignsto indicatethedirectionsofforces,andNewton'sthirdlawcanbe writtensuccinctlyas F AonB = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F BonA self-checkA FiguredanalyzesswimmingusingNewton'sthirdlaw.Doasimilar analysisforasprinterleavingthestartingline. Answer,p.267 Thereisnocauseandeectrelationshipbetweenthetwoforces inNewton'sthirdlaw.Thereisnooriginal"force,andneitherone isaresponsetotheother.Thepairofforcesisarelationship,like marriage,notaback-and-forthprocesslikeatennismatch.Newton cameupwiththethirdlawasageneralizationaboutallthetypesof forceswithwhichhewasfamiliar,suchasfrictionalandgravitational forces.Whenlaterphysicistsdiscoveredanewtypeforce,such astheforcethatholdsatomicnucleitogether,theyhadtocheck whetheritobeyedNewton'sthirdlaw.Sofar,noviolationofthe thirdlawhaseverbeendiscovered,whereastherstandsecond lawswereshowntohavelimitationsbyEinsteinandthepioneersof atomicphysics. TheEnglishvocabularyfordescribingforcesisunfortunately rootedinAristotelianism,andoftenimpliesincorrectlythatforces areone-wayrelationships.Itisunfortunatethatahalf-truthsuchas thetableexertsanupwardforceonthebook"issoeasilyexpressed, whileamorecompleteandcorrectdescriptionendsupsounding awkwardorstrange:thetableandthebookinteractviaaforce," orthetableandbookparticipateinaforce." Tostudents,itoftensoundsasthoughNewton'sthirdlawimpliesnothingcouldeverchangeitsmotion,sincethetwoequaland oppositeforceswouldalwayscancel.Thetwoforces,however,are alwaysontwodierentobjects,soitdoesn'tmakesensetoadd themintherstplace|weonlyaddforcesthatareactingonthe sameobject.Iftwoobjectsareinteractingviaaforceandnoother forcesareinvolved,then both objectswillaccelerate|inopposite directions! Amnemonicforusingnewton'sthirdlawcorrectly Mnemonicsaretricksformemorizingthings.Forinstance,the musicalnotesthatliebetweenthelinesonthetrebleclefspellthe wordFACE,whichiseasytoremember.Manypeopleusethe mnemonicSOHCAHTOA"torememberthedenitionsofthesine, cosine,andtangentintrigonometry.Ihavemyownmodestoering, POFOSTITO,whichIhopewillmakeitintothemnemonicshallof fame.It'sawaytoavoidsomeofthemostcommonproblemswith applyingNewton'sthirdlawcorrectly: Section5.1Newton'sThirdLaw 147

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f / Itdoesn'tmakesenseforthe mantotalkaboutusingthe woman'smoneytocancelouthis bartab,becausethereisnogood reasontocombinehisdebtsand herassets.Similarly,itdoesn't makesensetorefertotheequal andoppositeforcesofNewton's thirdlawascanceling.Itonly makessensetoaddupforces thatareactingonthe same object,whereastwoforcesrelated toeachotherbyNewton'sthird lawarealwaysactingontwo different objects. Abooklyingonatableexample1 Abookislyingonatable.WhatforceistheNewton's-third-law partneroftheearth'sgravitationalforceonthebook? Answer:Newton'sthirdlawworkslikeBonA,Aon B ,sothe partnermustbethebook'sgravitationalforcepullingupwardon theplanetearth.Yes,thereissuchaforce!No,itdoesnotcause theearthtodoanythingnoticeable. Incorrectanswer:Thetable'supwardforceonthebookisthe Newton's-third-lawpartneroftheearth'sgravitationalforceonthe book. Thisanswerviolatestwooutofthreeofthecommandmentsof POFOSTITO.Theforcesarenotofthesametype,becausethe table'supwardforceonthebookisnotgravitational.Also,three objectsareinvolvedinsteadoftwo:thebook,thetable,andthe planetearth. Pushingaboxupahillexample2 Apersonispushingaboxupahill.Whatforceisrelatedby Newton'sthirdlawtotheperson'sforceonthebox? Thebox'sforceontheperson. Incorrectanswer:Theperson'sforceontheboxisopposedby friction,andalsobygravity. 148 Chapter5AnalysisofForces

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ThisanswerfailsallthreepartsofthePOFOSTITOtest,themost obviousofwhichisthatthreeforcesarereferredtoinsteadofa pair. Solvedproblem:Moreaboutexample2page171,problem20 Solvedproblem:Whydiditaccelerate?page171,problem18 OptionalTopic:Newton'sThirdLawandActionataDistance Newton'sthirdlawiscompletelysymmetricinthesensethatneither forceconstitutesadelayedresponsetotheother.Newton'sthirdlaw doesnotevenmentiontime,andtheforcesaresupposedtoagreeat anygiveninstant.Thiscreatesaninterestingsituationwhenitcomes tononcontactforces.Supposetwopeopleareholdingmagnets,and whenonepersonwavesorwiggleshermagnet,theotherpersonfeels aneffectonhis.Inthiswaytheycansendsignalstoeachotherfrom oppositesidesofawall,andifNewton'sthirdlawiscorrect,itwould seemthatthesignalsaretransmittedinstantly,withnotimelag.The signalsareindeedtransmittedquitequickly,butexperimentswithelectronicallycontrolledmagnetsshowthatthesignalsdonotleapthegap instantly:theytravelatthesamespeedaslight,whichisanextremely highspeedbutnotaninniteone. IsthisacontradictiontoNewton'sthirdlaw?Notreally.Accordingtocurrenttheories,therearenotruenoncontactforces.Actionat adistancedoesnotexist.Althoughitappearsthatthewigglingofone magnetaffectstheotherwithnoneedforanythingtobeincontactwith anything,whatreallyhappensisthatwigglingamagnetunleashesa showeroftinyparticlescalledphotons.Themagnetshovesthephotonsoutwithakick,andreceivesakickinreturn,instrictobedienceto Newton'sthirdlaw.Thephotonsyoutinalldirections,andtheones thathittheothermagnettheninteractwithit,againobeyingNewton's thirdlaw. Photonsarenothingexotic,really.Lightismadeofphotons,butour eyesreceivesuchhugenumbersofphotonsthatwedonotperceive themindividually.Thephotonsyouwouldmakebywigglingamagnet withyourhandwouldbeofacolorthatyoucannotsee,faroffthered endoftherainbow.Book6inthisseriesdescribestheevidenceforthe photonmodeloflight. DiscussionQuestions A Whenyoureagun,theexplodinggasespushoutwardinall directions,causingthebullettoacceleratedownthebarrel.Whatthirdlawpairsareinvolved?[Hint:Rememberthatthegasesthemselvesare anobject.] B TamAnhgrabsSarahbythehandandtriestopullher.Shetries toremainstandingwithoutmoving.Astudentanalyzesthesituationas follows.IfTamAnh'sforceonSarahisgreaterthanherforceonhim, hecangethertomove.Otherwise,she'llbeabletostaywheresheis. What'swrongwiththisanalysis? C Youhitatennisballagainstawall.Explainanyandallincorrect ideasinthefollowingdescriptionofthephysicsinvolved:Accordingto Newton'sthirdlaw,therehastobeaforceoppositetoyourforceonthe ball.Theoppositeforceistheball'smass,whichresistsacceleration,and Section5.1Newton'sThirdLaw 149

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g / Ascienticclassication system. alsoairresistance. 5.2ClassicationandBehaviorofForces Oneofthemostbasicandimportanttasksofphysicsistoclassify theforcesofnature.Ihavealreadyreferredinformallytotypes"of forcessuchasfriction,magnetism,gravitationalforces,andsoon. Classicationsystemsarecreationsofthehumanmind,sothereis alwayssomedegreeofarbitrarinessinthem.Foronething,thelevel ofdetailthatisappropriateforaclassicationsystemdependson whatyou'retryingtondout.Somelinguists,thelumpers,"liketo emphasizethesimilaritiesamonglanguages,andafewextremists haveeventriedtondsignsofsimilaritiesbetweenwordsinlanguagesasdierentasEnglishandChinese,lumpingtheworld'slanguagesintoonlyafewlargegroups.Otherlinguists,thesplitters," mightbemoreinterestedinstudyingthedierencesinpronunciationbetweenEnglishspeakersinNewYorkandConnecticut.The splitterscallthelumperssloppy,butthelumperssaythatscience isn'tworthwhileunlessitcanndbroad,simplepatternswithinthe seeminglycomplexuniverse. Scienticclassicationsystemsarealsousuallycompromisesbetweenpracticalityandnaturalness.Anexampleisthequestionof howtoclassifyoweringplants.Mostpeoplethinkthatbiological classicationisaboutdiscoveringnewspecies,namingthem,and classifyingthemintheclass-order-family-genus-speciessystemaccordingtoguidelinessetlongago.Inreality,thewholesystemisin aconstantstateofuxandcontroversy.Oneverypracticalwayof classifyingoweringplantsisaccordingtowhethertheirpetalsare separateorjoinedintoatubeorcone|thecriterionissoclearthat itcanbeappliedtoaplantseenfromacrossthestreet.Buthere practicalityconictswithnaturalness.Forinstance,thebegoniahas separatepetalsandthepumpkinhasjoinedpetals,buttheyareso similarinsomanyotherwaysthattheyareusuallyplacedwithin thesameorder.Sometaxonomistshavecomeupwithclassication criteriathattheyclaimcorrespondmorenaturallytotheapparent relationshipsamongplants,withouthavingtomakespecialexceptions,butthesemaybefarlesspractical,requiringforinstancethe examinationofpollengrainsunderanelectronmicroscope. Inphysics,therearetwomainsystemsofclassicationforforces. Atthispointinthecourse,youaregoingtolearnonethatisvery practicalandeasytouse,andthatsplitstheforcesupintoarelativelylargenumberoftypes:sevenverycommononesthatwe'll discussexplicitlyinthischapter,plusperhapstenlessimportant onessuchassurfacetension,whichwewillnotbotherwithright now. Professionalphysicists,however,areobsessedwithndingsimplepatterns,sorecognizingasmanyasfteenortwentytypesof 150 Chapter5AnalysisofForces

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forcesstrikesthemasdistastefulandoverlycomplex.Sinceabout theyear1900,physicshasbeenonanaggressiveprogramtodiscover waysinwhichthesemanyseeminglydierenttypesofforcesarise fromasmallernumberoffundamentalones.Forinstance,whenyou pressyourhandstogether,theforcethatkeepsthemfrompassing througheachothermayseemtohavenothingtodowithelectricity,butattheatomiclevel,itactuallydoesarisefromelectrical repulsionbetweenatoms.Byabout1950,alltheforcesofnature hadbeenexplainedasarisingfromfourfundamentaltypesofforces attheatomicandnuclearlevel,andthelumping-togetherprocess didn'tstopthere.Bythe1960'sthelengthofthelisthadbeen reducedtothree,andsometheoristsevenbelievethattheymaybe abletoreduceittotwoorone.Althoughtheunicationoftheforces ofnatureisoneofthemostbeautifulandimportantachievements ofphysics,itmakesmuchmoresensetostartthiscoursewiththe morepracticalandeasysystemofclassication.Theuniedsystemoffourforceswillbeoneofthehighlightsoftheendofyour introductoryphysicssequence. Thepracticalclassicationschemewhichconcernsusnowcan belaidoutintheformofthetreeshowningureh.Themost specictypesofforcesareshownatthetipsofthebranches,and itisthesetypesofforcesthatarereferredtointhePOFOSTITO mnemonic.Forexample,electricalandmagneticforcesbelongto thesamegeneralgroup,butNewton'sthirdlawwouldneverrelate anelectricalforcetoamagneticforce. Thebroadestdistinctionisthatbetweencontactandnoncontact forces,whichhasbeendiscussedinthepreviouschapter.Among thecontactforces,wedistinguishbetweenthosethatinvolvesolids onlyandthosethathavetodowithuids,atermusedinphysicsto includebothgasesandliquids.Thetermsrepulsive,"attractive," andoblique"refertothedirectionsoftheforces. Repulsiveforcesarethosethattendtopushthetwoparticipatingobjectsawayfromeachother.Morespecically,a repulsivecontactforceactsperpendiculartothesurfacesat whichthetwoobjectstouch,andarepulsivenoncontactforce actsalongthelinebetweenthetwoobjects. Attractiveforcespullthetwoobjectstowardoneanother,i.e., theyactalongthesamelineasrepulsiveforces,butinthe oppositedirection. Obliqueforcesarethosethatactatsomeotherangle. Itshouldnotbenecessarytomemorizethisdiagrambyrote. Itisbettertoreinforceyourmemoryofthissystembycallingto Section5.2ClassicationandBehaviorofForces 151

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h / Apracticalclassicationschemeforforces. mindyourcommonsenseknowledgeofcertainordinaryphenomena. Forinstance,weknowthatthegravitationalattractionbetweenus andtheplanetearthwillactevenifourfeetmomentarilyleavethe ground,andthatalthoughmagnetshavemassandareaectedby gravity,mostobjectsthathavemassarenonmagnetic. Thisdiagramismeanttobeassimpleaspossiblewhileincluding mostoftheforceswedealwithineverydaylife.Ifyouwereaninsect, youwouldbemuchmoreinterestedintheforceofsurfacetension, whichallowedyoutowalkonwater.Ihavenotincludedthenuclear forces,whichareresponsibleforholdingthenucleiofatoms,because theyarenotevidentineverydaylife. Youshouldnotbeafraidtoinventyourownnamesfortypesof forcesthatdonottintothediagram.Forinstance,theforcethat holdsapieceoftapetothewallhasbeenleftoofthetree,andif youwereanalyzingasituationinvolvingscotchtape,youwouldbe absolutelyrighttorefertoitbysomecommonsensenamesuchas stickyforce." 152 Chapter5AnalysisofForces

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i / Amodelthatcorrectlyexplainsmanypropertiesoffriction. Themicroscopicbumpsand holesintwosurfacesdiginto eachother,causingafrictional force. Ontheotherhand,ifyouarehavingtroubleclassifyingacertain force,youshouldalsoconsiderwhetheritisaforceatall.For instance,ifsomeoneasksyoutoclassifytheforcethattheearthhas becauseofitsrotation,youwouldhavegreatdicultycreatinga placeforitonthediagram.That'sbecauseit'satypeofmotion, notatypeofforce! Normalforces Anormalforce, F N ,isaforcethatkeepsonesolidobjectfrom passingthroughanother.Normal"issimplyafancywordforperpendicular,"meaningthattheforceisperpendiculartothesurface ofcontact.Intuitively,itseemsthenormalforcemagicallyadjusts itselftoprovidewhateverforceisneededtokeeptheobjectsfrom occupyingthesamespace.Ifyourmusclespressyourhandstogether gently,thereisagentlenormalforce.Pressharder,andthenormal forcegetsstronger.Howdoesthenormalforceknowhowstrongto be?Theansweristhattheharderyoujamyourhandstogether, themorecompressedyoureshbecomes.Youreshisactinglike aspring:moreforceisrequiredtocompressitmore.Thesameis truewhenyoupushonawall.Thewallexesimperceptiblyinproportiontoyourforceonit.Ifyouexertedenoughforce,woulditbe possiblefortwoobjectstopassthrougheachother?No,typically theresultissimplytostraintheobjectssomuchthatoneofthem breaks. Gravitationalforces Aswe'lldiscussinmoredetaillaterinthecourse,agravitational forceexistsbetweenanytwothingsthathavemass.Ineverydaylife, thegravitationalforcebetweentwocarsortwopeopleisnegligible, sotheonlynoticeablegravitationalforcesaretheonesbetweenthe earthandvarioushuman-scaleobjects.Werefertotheseplanetearth-inducedgravitationalforcesasweightforces,andaswehave alreadyseen,theirmagnitudeisgivenby j F W j = mg Solvedproblem:Weightandmasspage172,problem26 Staticandkineticfriction Ifyouhavepushedarefrigeratoracrossakitchenoor,youhave feltacertainseriesofsensations.Atrst,yougraduallyincreased yourforceontherefrigerator,butitdidn'tmove.Finally,yousuppliedenoughforcetounstickthefridge,andtherewasasuddenjerk asthefridgestartedmoving.Oncethefridgeisunstuck,youcan reduceyourforcesignicantlyandstillkeepitmoving. Whileyouweregraduallyincreasingyourforce,theoor'sfrictionalforceonthefridgeincreasedinresponse.Thetwoforceson thefridgecanceled,andthefridgedidn'taccelerate.Howdidthe oorknowhowtorespondwithjusttherightamountofforce?Figureishowsonepossible model offrictionthatexplainsthisbehavior. Section5.2ClassicationandBehaviorofForces 153

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j / Staticfriction:thetraydoesn't sliponthewaiter'sngers. k / Kineticfriction:thecar skids. Ascienticmodelisadescriptionthatweexpecttobeincomplete, approximate,orunrealisticinsomeways,butthatneverthelesssucceedsinexplainingavarietyofphenomena.Figurei/1showsa microscopicviewofthetinybumpsandholesinthesurfacesofthe oorandtherefrigerator.Theweightofthefridgepressesthetwo surfacestogether,andsomeofthebumpsinonesurfacewillsettle asdeeplyaspossibleintosomeoftheholesintheothersurface.In i/2,yourleftwardforceonthefridgehascausedittorideupalittle higheronthebumpintheoorlabeledwithasmallarrow.Still moreforceisneededtogetthefridgeoverthebumpandallowitto startmoving.Ofcourse,thisisoccurringsimultaneouslyatmillions ofplacesonthetwosurfaces. Onceyouhadgottenthefridgemovingatconstantspeed,you foundthatyouneededtoexertlessforceonit.Sincezerototalforce isneededtomakeanobjectmovewithconstantvelocity,theoor's rightwardfrictionalforceonthefridgehasapparentlydecreased somewhat,makingiteasierforyoutocancelitout.Ourmodelalso givesaplausibleexplanationforthisfact:asthesurfacesslidepast eachother,theydon'thavetimetosettledownandmeshwithone another,sothereislessfriction. Eventhoughthismodelisintuitivelyappealingandfairlysuccessful,itshouldnotbetakentooseriously,andinsomesituations itismisleading.Forinstance,fancyracingbikesthesedaysare madewithsmoothtiresthathavenotread|contrarytowhat we'dexpectfromourmodel,thisdoesnotcauseanydecreasein friction.Machinistsknowthattwoverysmoothandcleanmetal surfacesmaysticktoeachotherrmlyandbeverydiculttoslide apart.Thiscannotbeexplainedinourmodel,butmakesmore senseintermsofamodelinwhichfrictionisdescribedasarising fromchemicalbondsbetweentheatomsofthetwosurfacesattheir pointsofcontact:veryatsurfacesallowmoreatomstocomein contact. Sincefrictionchangesitsbehaviordramaticallyoncethesurfacescomeunstuck,wedenetwoseparatetypesoffrictionalforces. Staticfriction isfrictionthatoccursbetweensurfacesthatarenot slippingovereachother.Slippingsurfacesexperience kineticfriction .Kinetic"meanshavingtodowithmotion.Theforcesof staticandkineticfriction,notated F s and F k ,arealwaysparallelto thesurfaceofcontactbetweenthetwoobjects. self-checkB 1.Whenabaseballplayerslidesintoabase,isthefrictionstatic,or kinetic? 2.Amattressstaysontheroofofaslowlyacceleratingcar.Isthe frictionstatic,orkinetic? 3.Doesstaticfrictioncreateheat?Kineticfriction? Answer,p.267 Themaximumpossibleforceofstaticfrictiondependsonwhat 154 Chapter5AnalysisofForces

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kindsofsurfacestheyare,andalsoonhowhardtheyarebeing pressedtogether.Theapproximatemathematicalrelationshipscan beexpressedasfollows: F s = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F applied ,when j F applied j < s j F N j where s isaunitlessnumber,calledthecoecientofstaticfriction, whichdependsonwhatkindsofsurfacestheyare.Themaximum forcethatstaticfrictioncansupply, s j F N j ,representstheboundary betweenstaticandkineticfriction.Itdependsonthenormalforce, whichisnumericallyequaltowhateverforceispressingthetwo surfacestogether.Intermsofourmodel,ifthetwosurfacesare beingpressedtogethermorermly,agreatersidewaysforcewillbe requiredinordertomaketheirregularitiesinthesurfacesrideup andovereachother. Notethatjustbecauseweuseanadjectivesuchasapplied"to refertoaforce,thatdoesn'tmeanthatthereissomespecialtype offorcecalledtheappliedforce."Theappliedforcecouldbeany typeofforce,oritcouldbethesumofmorethanoneforcetrying tomakeanobjectmove. Theforceofkineticfrictiononeachofthetwoobjectsisinthe directionthatresiststheslippageofthesurfaces.Itsmagnitudeis usuallywellapproximatedas j F k j = k j F N j where k isthecoecientofkineticfriction.Kineticfrictionis usuallymoreorlessindependentofvelocity. l / Wechooseacoordinatesysteminwhichtheappliedforce, i.e.,theforcetryingtomovethe objects,ispositive.Thefriction forceisthennegative,sinceitis intheoppositedirection.Asyou increasetheappliedforce,the forceofstaticfrictionincreasesto matchitandcancelitout,untilthe maximumforceofstaticfrictionis surpassed.Thesurfacesthenbeginslippingpasteachother,and thefrictionforcebecomessmaller inabsolutevalue. self-checkC Canafrictionlesssurfaceexertanormalforce?Canafrictionalforce existwithoutanormalforce? Answer,p.267 Ifyoutrytoaccelerateordecelerateyourcartooquickly,the forcesbetweenyourwheelsandtheroadbecometoogreat,andthey Section5.2ClassicationandBehaviorofForces 155

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beginslipping.Thisisnotgood,becausekineticfrictionisweaker thanstaticfriction,resultinginlesscontrol.Also,ifthisoccurs whileyouareturning,thecar'shandlingchangesabruptlybecause thekineticfrictionforceisinadierentdirectionthanthestatic frictionforcehadbeen:contrarytothecar'sdirectionofmotion, ratherthancontrarytotheforcesappliedtothetire. Mostpeoplerespondwithdisbeliefwhentoldoftheexperimentalevidencethatbothstaticandkineticfrictionareapproximately independentoftheamountofsurfaceareaincontact.Evenafter doingahands-onexercisewithspringscalestoshowthatitistrue, manystudentsareunwillingtobelievetheirownobservations,and insistthatbiggertiresgivemoretraction."Infact,themainreasonwhyyouwouldnotwanttoputsmalltiresonabigheavycar isthatthetireswouldburst! Althoughmanypeopleexpectthatfrictionwouldbeproportionaltosurfacearea,suchaproportionalitywouldmakepredictions contrarytomanyeverydayobservations.Adog'sfeet,forexample, haveverylittlesurfaceareaincontactwiththegroundcompared toahuman'sfeet,andyetweknowthatadogcanoftenwina tug-of-warwithaperson. Thereasonasmallersurfaceareadoesnotleadtolessfriction isthattheforcebetweenthetwosurfacesismoreconcentrated, causingtheirbumpsandholestodigintoeachothermoredeeply. self-checkD Findthedirectionofeachoftheforcesingurem. Answer,p.267 m / 1.Thecliff'snormalforceon theclimber'sfeet.2.Thetrack's staticfrictionalforceonthewheel oftheacceleratingdragster.3. Theball'snormalforceonthe bat. Locomotivesexample3 Lookingatapictureofalocomotive,n,wenoticetwoobvious thingsthataredifferentfromanautomobile.Whereacartypicallyhastwodrivewheels,alocomotivenormallyhasmany teninthisexample.Somealsohavesmaller,unpoweredwheels infrontofandbehindthedrivewheels,butthisexampledoesn't. Also,carsthesedaysaregenerallybuilttobeaslightaspossiblefortheirsize,whereaslocomotivesareverymassive,andno effortseemstobemadetokeeptheirweightlow.Thesteam locomotiveinthephotoisfromabout1900,butthisistrueeven 156 Chapter5AnalysisofForces

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formoderndieselandelectrictrains. n / Example3. Thereasonlocomotivesarebuilttobesoheavyisfortraction. Theupwardnormalforceoftherailsonthewheels, F N ,cancels thedownwardforceofgravity, F W ,soignoringplusandminus signs,thesetwoforcesareequalinabsolutevalue, F N = F W Giventhisamountofnormalforce,themaximumforceofstatic frictionis F s = s F N = s F W .Thisstaticfrictionalforce,ofthe railspushingforwardonthewheels,istheonlyforcethatcan acceleratethetrain,pullituphill,orcancelouttheforceofair resistancewhilecruisingatconstantspeed.Thecoefcientof staticfrictionforsteelonsteelisabout1/4,sonolocomotivecan pullwithaforcegreaterthanabout1/4ofitsownweight.Ifthe engineiscapableofsupplyingmorethanthatamountofforce,the resultwillbesimplytobreakstaticfrictionandspinthewheels. Thereasonthisisallsodifferentfromthesituationwithacaris thatacarisn'tpullingsomethingelse.Ifyouputextraweightin acar,youimprovethetraction,butyoualsoincreasetheinertia ofthecar,andmakeitjustashardtoaccelerate.Inatrain,the inertiaisalmostallinthecarsbeingpulled,notinthelocomotive. Theotherfactwehavetoexplainisthelargenumberofdrivingwheels.First,wehavetorealizethatincreasingthenumberofdrivingwheelsneitherincreasesnordecreasesthetotal amountofstaticfriction,becausestaticfrictionisindependentof theamountofsurfaceareaincontact.Thereasonfour-wheeldriveisgoodinacaristhatifoneormoreofthewheelsisslippingoniceorinmud,theotherwheelsmaystillhavetraction. Thisisn'ttypicallyanissueforatrain,sinceallthewheelsexperiencethesameconditions.Theadvantageofhavingmoredriving wheelsonatrainisthatitallowsustoincreasetheweightofthe locomotivewithoutcrushingtherails,ordamagingbridges. Fluidfriction Trytodriveanailintoawaterfallandyouwillbeconfronted withthemaindierencebetweensolidfrictionanduidfriction. Fluidfrictionispurelykinetic;thereisnostaticuidfriction.The nailinthewaterfallmaytendtogetdraggedalongbythewater owingpastit,butitdoesnotstickinthewater.Thesameistrue forgasessuchasair:recallthatweareusingtheworduid"to includebothgasesandliquids. Section5.2ClassicationandBehaviorofForces 157

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o / Thewheelbasesofthe HummerH3andtheToyotaPrius aresurprisinglysimilar,differing byonly10%.Themaindifference inshapeisthattheHummeris muchtallerandwider.Itpresents amuchgreatercross-sectional areatothewind,andthisisthe mainreasonthatitusesabout2.5 timesmoregasonthefreeway. Unlikekineticfrictionbetweensolids,uidfrictionincreases rapidlywithvelocity.Italsodependsontheshapeoftheobject, whichiswhyaghterjetismorestreamlinedthanaModelT.For objectsofthesameshapebutdierentsizes,uidfrictiontypically scalesupwiththecross-sectionalareaoftheobject,whichisone ofthemainreasonsthatanSUVgetsworsemileageonthefreeway thanacompactcar. DiscussionQuestions A Astudentstatesthatwhenhetriestopushhisrefrigerator,the reasonitwon'tmoveisbecauseNewton'sthirdlawsaysthere'sanequal andoppositefrictionalforcepushingback.Afterall,thestaticfrictionforce isequalandoppositetotheappliedforce.Howwouldyouconvincehim heiswrong? B Kineticfrictionisusuallymoreorlessindependentofvelocity.However,inexperienceddriverstendtoproduceajerkatthelastmomentof decelerationwhentheystopatastoplight.Whatdoesthistellyouabout thekineticfrictionbetweenthebrakeshoesandthebrakedrums? C Someofthefollowingarecorrectdescriptionsoftypesofforcesthat couldbeaddedonasnewbranchesoftheclassicationtree.Othersare notreallytypesofforces,andstillothersarenotforcephenomenaatall. Ineachcase,decidewhat'sgoingon,andifappropriate,gureouthow youwouldincorporatethemintothetree. stickyforcemakestapesticktothings oppositeforcetheforcethatNewton'sthirdlawsaysrelatestoeveryforceyoumake owingforcetheforcethatwatercarrieswithitasitowsoutofa hose surfacetensionletsinsectswalkonwater horizontalforceaforcethatishorizontal motorforcetheforcethatamotormakesonthethingitisturning canceledforceaforcethatisbeingcanceledoutbysomeother force 5.3AnalysisofForces Newton'srstandsecondlawsdealwiththetotalofalltheforces exertedonaspecicobject,soitisveryimportanttobeableto gureoutwhatforcesthereare.Onceyouhavefocusedyourattentionononeobjectandlistedtheforcesonit,itisalsohelpfulto describeallthecorrespondingforcesthatmustexistaccordingto Newton'sthirdlaw.Werefertothisasanalyzingtheforces"in whichtheobjectparticipates. 158 Chapter5AnalysisofForces

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Abarge example4 Abargeisbeingpulledalongacanalbyteamsofhorsesontheshores.Analyzealltheforcesinwhichthe bargeparticipates. forceactingonbarge forcerelatedtoitbyNewton'sthirdlaw ropes'forwardnormalforcesonbarge barge'sbackwardnormalforceonropes water'sbackwarduidfrictionforceonbarge barge'sforwarduidfrictionforceonwater planetearth'sdownwardgravitationalforce onbarge barge'supwardgravitationalforceonearth water'supwardoatingforceonbarge barge'sdownwardoatingforceonwater HereI'veusedthewordoatingforceasanexampleofasensibleinventedtermforatypeofforcenot classiedonthetreeintheprevioussection.Amoreformaltechnicaltermwouldbehydrostaticforce. NotehowthepairsofforcesareallstructuredasA'sforceonB,B'sforceonA:ropesonbargeandbarge onropes;wateronbargeandbargeonwater.Becausealltheforcesintheleftcolumnareforcesactingon thebarge,alltheforcesintherightcolumnareforcesbeingexertedbythebarge,whichiswhyeachentryin thecolumnbeginswithbarge. Oftenyoumaybeunsurewhetheryouhaveforgottenoneofthe forces.Herearethreestrategiesforcheckingyourlist: Seewhatphysicalresultwouldcomefromtheforcesyou've foundsofar.Suppose,forinstance,thatyou'dforgottenthe oating"forceonthebargeintheexampleabove.Looking attheforcesyou'dfound,youwouldhavefoundthatthere wasadownwardgravitationalforceonthebargewhichwas notcanceledbyanyupwardforce.Thebargeisn'tsupposed tosink,soyouknowyouneedtondafourth,upwardforce. Anothertechniqueforndingmissingforcesissimplytogo throughthelistofallthecommontypesofforcesandseeif anyofthemapply. Makeadrawingoftheobject,anddrawadashedboundary linearounditthatseparatesitfromitsenvironment.Lookfor pointsontheboundarywhereotherobjectscomeincontact withyourobject.Thisstrategyguaranteesthatyou'llnd everycontactforcethatactsontheobject,althoughitwon't helpyoutondnon-contactforces. Thefollowingisanotherexampleinwhichwecanprotbycheckingagainstourphysicalintuitionforwhatshouldbehappening. Section5.3AnalysisofForces 159

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Rappelling example5 Asshowninthegurebelow,Cindyisrappellingdownacliff.Herdownwardmotionisatconstantspeed,and shetakeslittlehopsoffofthecliff,asshownbythedashedline.Analyzetheforcesinwhichsheparticipates atamomentwhenherfeetareonthecliffandsheispushingoff. forceactingonCindy forcerelatedtoitbyNewton'sthirdlaw planetearth'sdownwardgravitationalforce onCindy Cindy'supwardgravitationalforceonearth ropesupwardfrictionalforceonCindyher hand Cindy'sdownwardfrictionalforceontherope cliff'srightwardnormalforceonCindy Cindy'sleftwardnormalforceonthecliff Thetwoverticalforcescancel,whichiswhattheyshouldbedoingifsheistogodownataconstantrate.The onlyhorizontalforceonheristhecliff'sforce,whichisnotcanceledbyanyotherforce,andwhichtherefore willproduceanaccelerationofCindytotheright.Thismakessense,sincesheishoppingoff.Thissolution isalittleoversimplied,becausetheropeisslanting,soitalsoappliesasmallleftwardforcetoCindy.Asshe iesouttotheright,theslantoftheropewillincrease,pullingherbackinmorestrongly. Ibelievethatconstructingthetypeoftabledescribedinthis sectionisthebestmethodforbeginningstudents.Mosttextbooks, however,prescribeapictorialwayofshowingalltheforcesactingon anobject.Suchapictureiscalledafree-bodydiagram.Itshould notbeabigproblemifafuturephysicsprofessorexpectsyouto beabletodrawsuchdiagrams,becausetheconceptualreasoning isthesame.Yousimplydrawapictureoftheobject,witharrows representingtheforcesthatareactingonit.Arrowsrepresenting contactforcesaredrawnfromthepointofcontact,noncontactforces fromthecenterofmass.Free-bodydiagramsdonotshowtheequal andoppositeforcesexertedbytheobjectitself. Oftenyoumaybeunsurewhetheryouhavemissedoneofthe forces.Herearethreestrategiesforcheckingyourlist: Seewhatphysicalresultwouldcomefromtheforcesyou've foundsofar.Suppose,forinstance,thatyou'dforgottenthe oating"forceonthebargeintheexampleabove.Looking attheforcesyou'dfound,youwouldhavefoundthatthere wasadownwardgravitationalforceonthebargewhichwas notcanceledbyanyupwardforce.Thebargeisn'tsupposed tosink,soyouknowyouneedtondafourth,upwardforce. Wheneveronesolidobjecttouchesanother,therewillbea normalforce,andpossiblyalsoafrictionalforce;checkfor both. 160 Chapter5AnalysisofForces

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DiscussionquestionC. Makeadrawingoftheobject,anddrawadashedboundary linearounditthatseparatesitfromitsenvironment.Lookfor pointsontheboundarywhereotherobjectscomeincontact withyourobject.Thisstrategyguaranteesthatyou'llnd everycontactforcethatactsontheobject,althoughitwon't helpyoutondnon-contactforces. DiscussionQuestions A Intheexampleofthebargegoingdownthecanal,Ireferredto aoatingorhydrostaticforcethatkeepstheboatfromsinking.Ifyou wereaddinganewbranchontheforce-classicationtreetorepresentthis force,wherewoulditgo? B Apoolballisreboundingfromthesideofthepooltable.Analyze theforcesinwhichtheballparticipatesduringtheshorttimewhenitisin contactwiththesideofthetable. C Theearth'sgravitationalforceonyou,i.e.,yourweight,isalways equalto mg ,where m isyourmass.Sowhycanyougetashoveltogo deeperintothegroundbyjumpingontoit?Justbecauseyou'rejumping, thatdoesn'tmeanyourmassorweightisanygreater,doesit? 5.4TransmissionofForcesbyLow-Mass Objects You'rewalkingyourdog.Thedogwantstogofasterthanyoudo, andtheleashistaut.DoesNewton'sthirdlawguaranteethatyour forceonyourendoftheleashisequalandoppositetothedog's forceonitsend?Ifthey'renotexactlyequal,isthereanyreason whytheyshouldbeapproximatelyequal? Iftherewasnoleashbetweenyou,andyouwereindirectcontact withthedog,thenNewton'sthirdlawwouldapply,butNewton's thirdlawcannotrelateyourforceontheleashtothedog'sforce ontheleash,becausethatwouldinvolvethreeseparateobjects. Newton'sthirdlawonlysaysthatyourforceontheleashisequal andoppositetotheleash'sforceonyou, F yL = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F Ly andthatthedog'sforceontheleashisequalandoppositetoits forceonthedog F dL = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F Ld Still,wehaveastrongintuitiveexpectationthatwhateverforcewe makeonourendoftheleashistransmittedtothedog,andviceversa.Wecananalyzethesituationbyconcentratingontheforces thatactontheleash, F dL and F yL .AccordingtoNewton'ssecond law,theserelatetotheleash'smassandacceleration: F dL + F yL = m L a L Theleashisfarlessmassivethenanyoftheotherobjectsinvolved, andif m L isverysmall,thenapparentlythetotalforceontheleash Section5.4TransmissionofForcesbyLow-MassObjects 161

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q / TheGoldenGateBridge's roadwayisheldupbythetension intheverticalcables. isalsoverysmall, F dL + F yL 0,andtherefore F dL )]TJ/F20 10.9091 Tf 20 0 Td [(F yL ThuseventhoughNewton'sthirdlawdoesnotapplydirectlyto thesetwoforces,wecanapproximatethelow-massleashasifitwas notinterveningbetweenyouandthedog.It'satleastapproximately asifyouandthedogwereactingdirectlyoneachother,inwhich caseNewton'sthirdlawwouldhaveapplied. Ingeneral,low-massobjectscanbetreatedapproximatelyasif theysimplytransmittedforcesfromoneobjecttoanother.Thiscan betrueforstrings,ropes,andcords,andalsoforrigidobjectssuch asrodsandsticks. p / Ifweimaginedividingatautropeupintosmallsegments,then anysegmenthasforcespullingoutwardonitateachend.Iftherope isofnegligiblemass,thenalltheforcesequal+ T or )]TJ/F78 9.9627 Tf 7.749 0 Td [(T ,where T ,the tension,isasinglenumber. Ifyoulookatapieceofstringunderamagnifyingglassasyou pullontheendsmoreandmorestrongly,youwillseethebers straighteningandbecomingtaut.Dierentpartsofthestringare apparentlyexertingforcesoneachother.Forinstance,ifwethinkof thetwohalvesofthestringastwoobjects,theneachhalfisexerting aforceontheotherhalf.Ifweimaginethestringasconsisting ofmanysmallparts,theneachsegmentistransmittingaforceto thenextsegment,andifthestringhasverylittlemass,thenall theforcesareequalinmagnitude.Werefertothemagnitudeof theforcesasthetensioninthestring, T .Althoughthetension ismeasuredinunitsofNewtons,itisnotitselfaforce.Thereare manyforceswithinthestring,someinonedirectionandsomeinthe otherdirection,andtheirmagnitudesareonlyapproximatelyequal. Theconceptoftensiononlymakessenseasageneral,approximate statementofhowbigalltheforcesare. Ifaropegoesoverapulleyoraroundsomeotherobject,thenthe tensionthroughouttheropeisapproximatelyequalsolongasthere isnottoomuchfriction.Arodorstickcanbetreatedinmuchthe samewayasastring,butitispossibletohaveeithercompression ortension. Sincetensionisnotatypeofforce,theforceexertedbyarope onsomeotherobjectmustbeofsomedenitetypesuchasstatic friction,kineticfriction,oranormalforce.Ifyouholdyourdog's 162 Chapter5AnalysisofForces

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leashwithyourhandthroughtheloop,thentheforceexertedbythe leashonyourhandisanormalforce:itistheforcethatkeepsthe leashfromoccupyingthesamespaceasyourhand.Ifyougraspa plainendofarope,thentheforcebetweentheropeandyourhand isafrictionalforce. Amorecomplexexampleoftransmissionofforcesistheway acaraccelerates.Manypeoplewoulddescribethecar'sengineas makingtheforcethatacceleratesthecar,buttheengineispartof thecar,sothat'simpossible:objectscan'tmakeforcesonthemselves.Whatreallyhappensisthattheengine'sforceistransmitted throughthetransmissiontotheaxles,thenthroughthetirestothe road.ByNewton'sthirdlaw,therewillthusbeaforwardforcefrom theroadonthetires,whichacceleratesthecar. DiscussionQuestion A Whenyousteponthegaspedal,isyourfoot'sforcebeingtransmitted inthesenseofthewordusedinthissection? 5.5ObjectsUnderStrain Astringlengthensslightlywhenyoustretchit.Similarly,wehave alreadydiscussedhowanapparentlyrigidobjectsuchasawallis actuallyexingwhenitparticipatesinanormalforce.Inother cases,theeectismoreobvious.Aspringorarubberbandvisibly elongateswhenstretched. Commontoalltheseexamplesisachangeinshapeofsomekind: lengthening,bending,compressing,etc.Thechangeinshapecan bemeasuredbypickingsomepartoftheobjectandmeasuringits position, x .Forconcreteness,let'simagineaspringwithoneend attachedtoawall.Whennoforceisexerted,theunxedendofthe springisatsomeposition x o .Ifaforceactsattheunxedend,its positionwillchangetosomenewvalueof x .Themoreforce,the greaterthedepartureof x from x o BackinNewton'stime,experimentslikethiswereconsidered cutting-edgeresearch,andhiscontemporaryHookeisremembered todayfordoingthemandforcomingupwithasimplemathematical generalizationcalledHooke'slaw: F k x )]TJ/F20 10.9091 Tf 10.909 0 Td [(x o .[forcerequiredtostretchaspring;valid forsmallforcesonly] Here k isaconstant,calledthespringconstant,thatdependson howstitheobjectis.Iftoomuchforceisapplied,thespring exhibitsmorecomplicatedbehavior,sotheequationisonlyagood approximationiftheforceissucientlysmall.Usuallywhenthe forceissolargethatHooke'slawisabadapproximation,theforce endsuppermanentlybendingorbreakingthespring. Section5.5ObjectsUnderStrain 163

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r / Deningthequantities F x and x o inHooke'slaw. AlthoughHooke'slawmayseemlikeapieceoftriviaabout springs,itisactuallyfarmoreimportantthanthat,becauseall solidobjectsexertHooke's-lawbehavioroversomerangeofsucientlysmallforces.Forexample,ifyoupushdownonthehoodof acar,itdipsbyanamountthatisdirectlyproportionaltotheforce. Butthecar'sbehaviorwouldnotbeasmathematicallysimpleif youdroppedaboulderonthehood! Solvedproblem:Combiningspringspage170,problem14 Solvedproblem:Young'smoduluspage170,problem16 DiscussionQuestion A Acarisconnectedtoitsaxlesthroughbig,stiffspringscalledshock absorbers,orshocks.Althoughwe'vediscussedHooke'slawaboveonly inthecaseofstretchingaspring,acar'sshocksarecontinuallygoing throughbothstretchingandcompression.Inthissituation,howwould youinterpretthepositiveandnegativesignsinHooke'slaw? 5.6SimpleMachines:ThePulley Eventhemostcomplexmachines,suchascarsorpianos,arebuilt outofcertainbasicunitscalled simplemachines .Thefollowingare someofthemainfunctionsofsimplemachines: transmittingaforce:Thechainonabicycletransmitsaforce fromthecranksettotherearwheel. changingthedirectionofaforce:Ifyoupushdownonaseesaw,theotherendgoesup. changingthespeedandprecisionofmotion:Whenyoumake thecomehere"motion,yourbicepsonlymovesacoupleof centimeterswhereitattachestoyourforearm,butyourarm movesmuchfartherandmorerapidly. changingtheamountofforce:Aleverorpulleycanbeused 164 Chapter5AnalysisofForces

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toincreaseordecreasetheamountofforce. Youarenowpreparedtounderstandone-dimensionalsimplemachines,ofwhichthepulleyisthemainexample. s / Example6. Apulleyexample6 FarmerBillsaysthispulleyarrangementdoublestheforceof histractor.Ishejustadumbhayseed,ordoesheknowwhathe's doing? TouseNewton'srstlaw,weneedtopickanobjectandconsiderthesumoftheforcesonit.Sinceourgoalistorelatethe tensioninthepartofthecableattachedtothestumptothetensioninthepartattachedtothetractor,weshouldpickanobject towhichboththosecablesareattached,i.e.,thepulleyitself.As discussedinsection5.4,thetensioninastringorcableremains approximatelyconstantasitpassesaroundapulley,providedthat thereisnottoomuchfriction.Therearethereforetwoleftward forcesactingonthepulley,eachequaltotheforceexertedbythe tractor.Sincetheaccelerationofthepulleyisessentiallyzero,the forcesonitmustbecancelingout,sotherightwardforceofthe pulley-stumpcableonthepulleymustbedoubletheforceexerted bythetractor.Yes,FarmerBillknowswhathe'stalkingabout. Section5.6SimpleMachines:ThePulley 165

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Summary SelectedVocabulary repulsive.....describesaforcethattendstopushthetwo participatingobjectsapart attractive....describesaforcethattendstopullthetwo participatingobjectstogether oblique......describesaforcethatactsatsomeotherangle, onethatisnotadirectrepulsionorattraction normalforce...theforcethatkeepstwoobjectsfromoccupyingthesamespace staticfriction..africtionforcebetweensurfacesthatarenot slippingpasteachother kineticfriction.africtionforcebetweensurfacesthatareslippingpasteachother uid........agasoraliquid uidfriction...africtionforceinwhichatleastoneofthe objectisisauid springconstant.theconstantofproportionalitybetweenforce andelongationofaspringorotherobjectunderstrain Notation F N .........anormalforce F s .........astaticfrictionalforce F k .........akineticfrictionalforce s .........thecoecientofstaticfriction;theconstantof proportionalitybetweenthemaximumstatic frictionalforceandthenormalforce;depends onwhattypesofsurfacesareinvolved k .........thecoecientofkineticfriction;theconstant ofproportionalitybetweenthekineticfrictionalforceandthenormalforce;dependson whattypesofsurfacesareinvolved k..........thespringconstant;theconstantofproportionalitybetweentheforceexertedonanobjectandtheamountbywhichtheobjectis lengthenedorcompressed Summary Newton'sthirdlawstatesthatforcesoccurinequalandopposite pairs.IfobjectAexertsaforceonobjectB,thenobjectBmust simultaneouslybeexertinganequalandoppositeforceonobjectA. EachinstanceofNewton'sthirdlawinvolvesexactlytwoobjects, andexactlytwoforces,whichareofthesametype. Therearetwosystemsforclassifyingforces.Wearepresently usingthemorepracticalbutlessfundamentalone.Inthissystem, forcesareclassiedbywhethertheyarerepulsive,attractive,or oblique;whethertheyarecontactornoncontactforces;andwhether 166 Chapter5AnalysisofForces

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thetwoobjectsinvolvedaresolidsoruids. Staticfrictionadjustsitselftomatchtheforcethatistryingto makethesurfacesslidepasteachother,untilthemaximumvalueis reached, j F s j < s j F N j Oncethisforceisexceeded,thesurfacesslippastoneanother,and kineticfrictionapplies, j F k j = k j F N j Bothtypesoffrictionalforcearenearlyindependentofsurfacearea, andkineticfrictionisusuallyapproximatelyindependentofthe speedatwhichthesurfacesareslipping. AgoodrststepinapplyingNewton'slawsofmotiontoany physicalsituationistopickanobjectofinterest,andthentolist alltheforcesactingonthatobject.Weclassifyeachforcebyits type,andnditsNewton's-third-lawpartner,whichisexertedby theobjectonsomeotherobject. Whentwoobjectsareconnectedbyathirdlow-massobject, theirforcesaretransmittedtoeachothernearlyunchanged. ObjectsunderstrainalwaysobeyHooke'slawtoagoodapproximation,aslongastheforceissmall.Hooke'slawstatesthatthe stretchingorcompressionoftheobjectisproportionaltotheforce exertedonit, F k x )]TJ/F20 10.9091 Tf 10.909 0 Td [(x o Summary 167

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Problem1. Problem6. Problem9. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Alittleoldladyandaprofootballplayercollidehead-on. Comparetheirforcesoneachother,andcomparetheiraccelerations. Explain. 2 Theearthisattractedtoanobjectwithaforceequaland oppositetotheforceoftheearthontheobject.Ifthisistrue, whyisitthatwhenyoudropanobject,theearthdoesnothavean accelerationequalandoppositetothatoftheobject? 3 Whenyoustandstill,therearetwoforcesactingonyou, theforceofgravityyourweightandthenormalforceoftheoor pushinguponyourfeet.Aretheseforcesequalandopposite?Does Newton'sthirdlawrelatethemtoeachother?Explain. Inproblems4-8,analyzetheforcesusingatableintheformatshown insection5.3.Analyzetheforcesinwhichtheitalicizedobjectparticipates. 4 A magnet isstuckunderneathaparkedcar.Seeinstructions above. 5 Analyzetwoexamplesof objects atrestrelativetotheearth thatarebeingkeptfromfallingbyforcesotherthanthenormal force.Donotuseobjectsinouterspace,anddonotduplicate problem4or8.Seeinstructionsabove. 6 A person isrowingaboat,withherfeetbraced.Sheisdoing thepartofthestrokethatpropelstheboat,withtheendsofthe oarsinthewaternotthepartwheretheoarsareoutofthewater. Seeinstructionsabove. 7 A farmer isinastallwithacowwhenthecowdecidestopress himagainstthewall,pinninghimwithhisfeetotheground.Analyzetheforcesinwhichthefarmerparticipates.Seeinstructions above. 8 Apropeller plane iscruisingeastatconstantspeedandaltitude.Seeinstructionsabove. 9 Today'stallestbuildingsarereallynotthatmuchtallerthan thetallestbuildingsofthe1940 s .Onebigproblemwithmakingan eventallerskyscraperisthateveryelevatorneedsitsownshaftrunningthewholeheightofthebuilding.Somanyelevatorsareneeded toservethebuilding'sthousandsofoccupantsthattheelevator shaftsstarttakinguptoomuchofthespacewithinthebuilding. Analternativeistohaveelevatorsthatcanmovebothhorizontally andvertically:withsuchadesign,manyelevatorcarscansharea 168 Chapter5AnalysisofForces

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Problem10. fewshafts,andtheydon'tgetineachother'swaytoomuchbecause theycandetouraroundeachother.Inthisdesign,itbecomesimpossibletohangthecarsfromcables,sotheywouldinsteadhaveto rideonrailswhichtheygrabontowithwheels.Frictionwouldkeep themfromslipping.Thegureshowssuchafrictionalelevatorin itsverticaltravelmode.Thewheelsonthebottomareforwhenit needstoswitchtohorizontalmotion. aIfthecoecientofstaticfrictionbetweenrubberandsteelis s ,andthemaximummassofthecarplusitspassengersis M howmuchforcemusttherebepressingeachwheelagainsttherail inordertokeepthecarfromslipping?Assumethecarisnot accelerating. p bShowthatyourresulthasphysicallyreasonablebehaviorwith respectto s .Inotherwords,iftherewaslessfriction,wouldthe wheelsneedtobepressedmorermlyorlessrmly?Doesyour equationbehavethatway? 10 Unequalmasses M and m aresuspendedfromapulleyas showninthegure. aAnalyzetheforcesinwhichmass m participates,usingatable theformatshowninsection5.3.[Theforcesinwhichtheothermass participateswillofcoursebesimilar,butnotnumericallythesame.] bFindthemagnitudeoftheaccelerationsofthetwomasses. [Hints:Pickacoordinatesystem,andusepositiveandnegativesignsconsistentlytoindicatethedirectionsoftheforcesand accelerations.Thetwoaccelerationsofthetwomasseshaveto beequalinmagnitudebutofoppositesigns,sinceonesideeatsup ropeatthesamerateatwhichtheothersidepaysitout.You needtoapplyNewton'ssecondlawtwice,oncetoeachmass,and thensolvethetwoequationsfortheunknowns:theacceleration, a andthetensionintherope, T .] cManypeopleexpectthatinthespecialcaseof M = m ,thetwo masseswillnaturallysettledowntoanequilibriumpositionsideby side.Basedonyouranswerfrompartb,isthiscorrect? dFindthetensionintherope, T eInterpretyourequationfrompartdinthespecialcasewhereone ofthemassesiszero.Hereinterpret"meanstogureoutwhathappensmathematically,gureoutwhatshouldhappenphysically,and connectthetwo. 11 Atugboatofmass m pullsashipofmass M ,acceleratingit. Thespeedsarelowenoughthatyoucanignoreuidfrictionacting ontheirhulls,althoughtherewillofcourseneedtobeuidfriction actingonthetug'spropellers. aAnalyzetheforcesinwhichthetugboatparticipates,usinga tableintheformatshowninsection5.3.Don'tworryaboutvertical forces. bDothesamefortheship. Problems 169

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Problem17. Problem13. Problem14. cAssumenowthatwaterfrictiononthetwovessels'hullsisnegligible.Iftheforceactingonthetug'spropelleris F ,whatisthe tension, T ,inthecableconnectingthetwoships?[Hint:Write downtwoequations,oneforNewton'ssecondlawappliedtoeach object.Solvetheseforthetwounknowns T and a .] p dInterpretyouranswerinthespecialcasesof M =0and M = 1 12 SomeonetellsyousheknowsofacertaintypeofCentral Americanearthwormwhoseskin,whenrubbedonpolisheddiamond,has k > s .Whyisthisnotjustempiricallyunlikelybut logicallysuspect? 13 Inthesystemshowninthegure,thepulleysontheleftand rightarexed,butthepulleyinthecentercanmovetotheleftor right.Thetwomassesareidentical.Showthatthemassontheleft willhaveanupwardaccelerationequalto g= 5.Assumealltheropes andpulleysaremasslessandrictionless. 14 Thegureshowstwodierentwaysofcombiningapairof identicalsprings,eachwithspringconstant k .Werefertothetop setupasparallel,andthebottomoneasaseriesarrangement. aFortheparallelarrangement,analyzetheforcesactingonthe connectorpieceontheleft,andthenusethisanalysistodetermine theequivalentspringconstantofthewholesetup.Explainwhether thecombinedspringconstantshouldbeinterpretedasbeingstier orlesssti. bFortheseriesarrangement,analyzetheforcesactingoneach springandgureoutthesamethings. Solution,p.273 15 Generalizetheresultsofproblem14tothecasewherethe twospringconstantsareunequal. 16 aUsingthesolutionofproblem14,whichisgiveninthe backofthebook,predicthowthespringconstantofaberwill dependonitslengthandcross-sectionalarea. bTheconstantofproportionalityiscalledtheYoung'smodulus, E ,andtypicalvaluesoftheYoung'smodulusareabout10 10 to 10 11 .WhatunitswouldtheYoung'smodulushaveintheSImeterkilogram-secondsystem? Solution,p.274 17 Thisproblemdependsontheresultsofproblems14and 16,whosesolutionsareinthebackofthebook.Whenatomsform chemicalbonds,itmakessensetotalkaboutthespringconstantof thebondasameasureofhowsti"itis.Ofcourse,therearen't reallylittlesprings|thisisjustamechanicalmodel.Thepurpose ofthisproblemistoestimatethespringconstant, k ,forasingle bondinatypicalpieceofsolidmatter.Supposewehaveaber, likeahairorapieceofshingline,andimagineforsimplicitythat itismadeofatomsofasingleelementstackedinacubicalmanner, asshowninthegure,withacenter-to-centerspacing b .Atypical valuefor b wouldbeabout10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(10 m. 170 Chapter5AnalysisofForces

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Problem19. aFindanequationfor k intermsof b ,andintermsoftheYoung's modulus, E ,denedinproblem16anditssolution. bEstimate k usingthenumericaldatagiveninproblem16. cSupposeyoucouldgraboneoftheatomsinadiatomicmolecule likeH 2 orO 2 ,andlettheotheratomhangverticallybelowit.Does thebondstretchbyanyappreciablefractionduetogravity? 18 Ineachcase,identifytheforcethatcausestheacceleration, andgiveitsNewton's-third-lawpartner.Describetheeectofthe partnerforce.aAswimmerspeedsup.bAgolferhitstheball oofthetee.cAnarcherresanarrow.dAlocomotiveslows down. Solution,p.274 19 Ginnyhasaplan.Sheisgoingtoridehersledwhileher dogFoopullsher.However,Ginnyhasn'ttakenphysics,sothere maybeaproblem:shemaysliderightothesledwhenFoostarts pulling. aAnalyzealltheforcesinwhichGinnyparticipates,makinga tableasinsection5.3. bAnalyzealltheforcesinwhichthesledparticipates. cThesledhasmass m ,andGinnyhasmass M .Thecoecient ofstaticfrictionbetweenthesledandthesnowis 1 ,and 2 is thecorrespondingquantityforstaticfrictionbetweenthesledand hersnowpants.Ginnymusthaveacertainminimummasssothat shewillnotslipothesled.Findthisintermsoftheotherthree variables. p dInterpretingyourequationfrompartc,underwhatconditions willtherebenophysicallyrealisticsolutionfor M ?Discusswhat thismeansphysically. 20 Example2onpage148involvesapersonpushingaboxupa hill.Theincorrectanswerdescribesthreeforces.Foreachofthese threeforces,givetheforcethatitisrelatedtobyNewton'sthird law,andstatethetypeofforce. Solution,p.274 21 Example6onpage165describesaforce-doublingsetup involvingapulley.Makeupamorecomplicatedarrangement,using morethanonepulley,thatwouldmultiplytheforcebyafactor greaterthantwo. 22 Pickupaheavyobjectsuchasabackpackorachair,and standonabathroomscale.Shaketheobjectupanddown.What doyouobserve?InterpretyourobservationsintermsofNewton's thirdlaw. 23 Acopinvestigatingthesceneofanaccidentmeasuresthe length L ofacar'sskidmarksinordertondoutitsspeed v at thebeginningoftheskid.Express v intermsof L andanyother relevantvariables. p 24 Thefollowingreasoningleadstoanapparentparadox;explain what'swrongwiththelogic.Abaseballplayerhitsaball.Theball Problems 171

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andthebatspendafractionofasecondincontact.Duringthat timethey'removingtogether,sotheiraccelerationsmustbeequal. Newton'sthirdlawsaysthattheirforcesoneachotherarealso equal.But a = F=m ,sohowcanthisbe,sincetheirmassesare unequal?Notethattheparadoxisn'tresolvedbyconsideringthe forceofthebatter'shandsonthebat.Notonlyisthisforcevery smallcomparedtotheball-batforce,butthebattercouldhavejust thrownthebatattheball. 25 Thisproblemhasbeendeleted. 26 aComparethemassofaone-literwaterbottleonearth, onthemoon,andininterstellarspace. Solution,p.274 bDothesameforitsweight. 27 Aniceskaterbuildsupsomespeed,andthencoastsacross theicepassivelyinastraightline.aAnalyzetheforces.bIf hisinitialspeedis v ,andthecoecientofkineticfrictionis k ndthemaximumtheoreticaldistancehecanglidebeforecoming toastop.Ignoreairresistance.cShowthatyouranswerto partbhastherightunits.dShowthatyouranswertopartb dependsonthevariablesinawaythatmakessensephysically.e Evaluateyouranswernumericallyfor k =0.0046,andaworldrecordspeedof14.58m/s.Thecoecientoffrictionwasmeasured byDeKoningetal.,usingspecialskateswornbyrealspeedskaters. fCommentonwhetheryouranswerinpartdseemsrealistic.If itdoesn't,suggestpossiblereasonswhy. 172 Chapter5AnalysisofForces

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PartII MotioninThree Dimensions

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Chapter6 Newton'sLawsinThree Dimensions 6.1ForcesHaveNoPerpendicularEffects Supposeyoucouldshootarieandarrangeforasecondbulletto bedroppedfromthesameheightattheexactmomentwhenthe rstleftthebarrel.Whichwouldhitthegroundrst?Nearly everyoneexpectsthatthedroppedbulletwillreachthedirtrst, 175

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andAristotlewouldhaveagreed.Aristotlewouldhavedescribedit likethis.Theshotbulletreceivessomeforcedmotionfromthegun. Ittravelsforwardforasplitsecond,slowingdownrapidlybecause thereisnolongeranyforcetomakeitcontinueinmotion.Once itisdonewithitsforcedmotion,itchangestonaturalmotion,i.e. fallingstraightdown.Whiletheshotbulletisslowingdown,the droppedbulletgetsonwiththebusinessoffalling,soaccordingto Aristotleitwillhitthegroundrst. a / Abulletisshotfromagun,andanotherbulletissimultaneouslydroppedfromthesameheight.1. Aristotelianphysicssaysthatthehorizontalmotionoftheshotbulletdelaystheonsetoffalling,sothedropped bullethitsthegroundrst.2.Newtonianphysicssaysthetwobulletshavethesameverticalmotion,regardless oftheirdifferenthorizontalmotions. Luckily,natureisn'tascomplicatedasAristotlethought!To convinceyourselfthatAristotle'sideaswerewrongandneedlessly complex,standupnowandtrythisexperiment.Takeyourkeys outofyourpocket,andbeginwalkingbrisklyforward.Without speedinguporslowingdown,releaseyourkeysandletthemfall whileyoucontinuewalkingatthesamepace. Youhavefoundthatyourkeyshitthegroundrightnexttoyour feet.Theirhorizontalmotionneversloweddownatall,andthe wholetimetheyweredropping,theywererightnexttoyou.The horizontalmotionandtheverticalmotionhappenatthesametime, andtheyareindependentofeachother.Yourexperimentproves thatthehorizontalmotionisunaectedbytheverticalmotion,but it'salsotruethattheverticalmotionisnotchangedinanywayby thehorizontalmotion.Thekeystakeexactlythesameamountof timetogettothegroundastheywouldhaveifyousimplydropped them,andthesameistrueofthebullets:bothbulletshittheground 176 Chapter6Newton'sLawsinThreeDimensions

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simultaneously. Thesehavebeenourrstexamplesofmotioninmorethanone dimension,andtheyillustratethemostimportantnewideathat isrequiredtounderstandthethree-dimensionalgeneralizationof Newtonianphysics: Forceshavenoperpendiculareects. Whenaforceactsonanobject,ithasnoeectonthepartofthe object'smotionthatisperpendiculartotheforce. Intheexamplesabove,theverticalforceofgravityhadnoeect onthehorizontalmotionsoftheobjects.Thesewereexamplesof projectilemotion,whichinterestedpeoplelikeGalileobecauseof itsmilitaryapplications.Theprincipleismoregeneralthanthat, however.Forinstance,ifarollingballisinitiallyheadingstraight forawall,butasteadywindbeginsblowingfromtheside,theball doesnottakeanylongertogettothewall.Inthecaseofprojectile motion,theforceinvolvedisgravity,sowecansaymorespecically thattheverticalaccelerationis9.8m = s 2 ,regardlessofthehorizontal motion. self-checkA Intheexampleoftheballbeingblownsideways,whydoesn'ttheball takelongertogetthere,sinceithastotravelagreaterdistance? Answer,p.268 Relationshiptorelativemotion Theseconceptsaredirectlyrelatedtotheideathatmotionisrelative.Galileo'sopponentsarguedthattheearthcouldnotpossibly berotatingasheclaimed,becausethenifyoujumpedstraightupin theairyouwouldn'tbeabletocomedowninthesameplace.Their argumentwasbasedontheirincorrectAristotelianassumptionthat oncetheforceofgravitybegantoactonyouandbringyouback down,yourhorizontalmotionwouldstop.InthecorrectNewtonian theory,theearth'sdownwardgravitationalforceisactingbefore, during,andafteryourjump,buthasnoeectonyourmotionin theperpendicularhorizontaldirection. IfAristotlehadbeencorrect,thenwewouldhaveahandyway todetermineabsolutemotionandabsoluterest:jumpstraightup intheair,andifyoulandbackwhereyoustarted,thesurfacefrom whichyoujumpedmusthavebeeninastateofrest.Inreality,this testgivesthesameresultaslongasthesurfaceunderyouisan inertialframe.Ifyoutrythisinajetplane,youlandbackonthe samespotonthedeckfromwhichyoustarted,regardlessofwhether theplaneisyingat500milesperhourorparkedontherunway. Themethodwouldinfactonlybegoodfordetectingwhetherthe planewasaccelerating. Section6.1ForcesHaveNoPerpendicularEffects 177

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DiscussionQuestions A Thefollowingisanincorrectexplanationofafactabouttarget shooting: Shootingahigh-poweredriewithahighmuzzlevelocityisdifferentfrom shootingalesspowerfulgun.Withalesspowerfulgun,youhavetoaim quiteabitaboveyourtarget,butwithamorepowerfuloneyoudon'thave toaimsohighbecausethebulletdoesn'tdropasfast. Whatisthecorrectexplanation? B Youhavethrownarock,anditisyingthroughtheairinanarc.If theearth'sgravitationalforceonitisalwaysstraightdown,whydoesn'tit justgostraightdownonceitleavesyourhand? C Considertheexampleofthebulletthatisdroppedatthesame momentanotherbulletisredfromagun.Whatwouldthemotionofthe twobulletslookliketoajetpilotyingalongsideinthesamedirectionas theshotbulletandatthesamehorizontalspeed? b / Thisobjectexperiencesaforcethatpullsitdowntowardthe bottomofthepage.Ineachequaltimeinterval,itmovesthreeunitsto theright.Atthesametime,itsverticalmotionismakingasimplepattern of+1,0, )]TJ/F39 9.9626 Tf 7.748 0 Td [(1, )]TJ/F39 9.9626 Tf 7.749 0 Td [(2, )]TJ/F39 9.9626 Tf 7.748 0 Td [(3, )]TJ/F39 9.9626 Tf 7.749 0 Td [(4,...units.Itsmotioncanbedescribedbyan x coordinatethathaszeroaccelerationanda y coordinatewithconstant acceleration.Thearrowslabeled x and y servetoexplainthatweare deningincreas-ing x totherightandincreasing y asupward. 178 Chapter6Newton'sLawsinThreeDimensions

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c / Theshadowonthewall showstheball's y motion,the shadowontheoorits x motion. 6.2CoordinatesandComponents 'Causewe'reall Boldaslove, Justasktheaxis. JimiHendrix Howdoweconverttheseideasintomathematics?Figurebshows agoodwayofconnectingtheintuitiveideastothenumbers.Inone dimension,weimposeanumberlinewithan x coordinateona certainstretchofspace.Intwodimensions,weimagineagridof squareswhichwelabelwith x and y values,asshowningureb. Butofcoursemotiondoesn'treallyoccurinaseriesofdiscrete hopslikeinchessorcheckers.Thegureontheleftshowsaway ofconceptualizingthesmoothvariationofthe x and y coordinates. Theball'sshadowonthewallmovesalongaline,andwedescribeits positionwithasinglecoordinate, y ,itsheightabovetheoor.The wallshadowhasaconstantaccelerationof-9.8 m=s 2 .Ashadowon theoor,madebyasecondlightsource,alsomovesalongaline, andwedescribeitsmotionwithan x coordinate,measuredfromthe wall. Thevelocityoftheoorshadowisreferredtoasthe x component ofthevelocity,written v x .Similarlywecannotatetheacceleration oftheoorshadowas a x .Since v x isconstant, a x iszero. Similarly,thevelocityofthewallshadowiscalled v y ,itsacceleration a y .Thisexamplehas a y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(9.8m = s 2 Becausetheearth'sgravitationalforceontheballisactingalong the y axis,wesaythattheforcehasanegative y component, F y but F x = F z =0. Thegeneralideaisthatweimaginetwoobservers,eachofwhom perceivestheentireuniverseasifitwasatteneddowntoasingle line.The y -observer,forinstance,perceives y v y ,and a y ,andwill inferthatthereisaforce, F y ,actingdownwardontheball.That is,a y componentmeanstheaspectofaphysicalphenomenon,such asvelocity,acceleration,orforce,thatisobservabletosomeonewho canonlyseemotionalongthe y axis. Allofthiscaneasilybegeneralizedtothreedimensions.Inthe exampleabove,therecouldbea z -observerwhoonlyseesmotion towardorawayfromthebackwalloftheroom. Section6.2CoordinatesandComponents 179

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d / Example1. Acargoingoveracliffexample1 Thepolicendacaratadistance w =20mfromthebaseofa cliffofheight h =100m.Howfastwasthecargoingwhenitwent overtheedge?Solvetheproblemsymbolicallyrst,thenplugin thenumbers. Let'schoose y pointingupand x pointingawayfromthecliff. Thecar'sverticalmotionwasindependentofitshorizontalmotion,soweknowithadaconstantverticalaccelerationof a = )]TJ/F78 10.9091 Tf 8.485 0 Td [(g = )]TJ/F39 10.9091 Tf 8.485 0 Td [(9.8m = s 2 .Thetimeitspentintheairisthereforerelated totheverticaldistanceitfellbytheconstant-accelerationequation y = 1 2 a y t 2 or )]TJ/F78 10.9091 Tf 8.485 0 Td [(h = 1 2 )]TJ/F78 10.9091 Tf 8.485 0 Td [(g t 2 Solvingfor t gives t = s 2 h g Sincetheverticalforcehadnoeffectonthecar'shorizontalmotion,ithad a x =0,i.e.,constanthorizontalvelocity.Wecanapply theconstant-velocityequation v x = x t i.e., v x = w t Wenowsubstitutefor t tond v x = w = s 2 h g whichsimpliesto v x = w r g 2 h Plugginginnumbers,wendthatthecar'sspeedwhenitwent overtheedgewas4m/s,orabout10mi/hr. 180 Chapter6Newton'sLawsinThreeDimensions

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e / Aparabolacanbedenedas theshapemadebycuttingacone paralleltoitsside.Aparabolais alsothegraphofanequationof theform y / x 2 f / Eachwaterdropletfollows aparabola.Thefasterdrops' parabolasarebigger. Projectilesmovealongparabolas. Whattypeofmathematicalcurvedoesaprojectilefollowthrough space?Tondout,wemustrelate x to y ,eliminating t .Thereasoningisverysimilartothatusedintheexampleabove.Arbitrarily choosing x = y = t =0tobeatthetopofthearc,weconveniently have x = x y = y ,and t = t ,so y = 1 2 a y t 2 a y < 0 x = v x t Wesolvethesecondequationfor t = x=v x andeliminate t inthe rstequation: y = 1 2 a y x v x 2 Sinceeverythinginthisequationisaconstantexceptfor x and y weconcludethat y isproportionaltothesquareof x .Asyoumay ormaynotrecallfromamathclass, y / x 2 describesaparabola. Solvedproblem:Acannonpage184,problem5 DiscussionQuestion A AtthebeginningofthissectionIrepresentedthemotionofaprojectileongraphpaper,breakingitsmotionintoequaltimeintervals.Suppose insteadthatthereisnoforceontheobjectatall.ItobeysNewton'srstlaw andcontinueswithoutchangingitsstateofmotion.Whatwouldthecorrespondinggraph-paperdiagramlooklike?Ifthetimeintervalrepresented byeacharrowwas1second,howwouldyourelatethegraph-paperdiagramtothevelocitycomponents v x and v y ? B Makeupseveraldifferentcoordinatesystemsorientedindifferent ways,anddescribethe a x and a y ofafallingobjectineachone. 6.3Newton'sLawsinThreeDimensions ItisnowfairlystraightforwardtoextendNewton'slawstothree dimensions: Newton'srstlaw Ifallthreecomponentsofthetotalforceonanobjectarezero, thenitwillcontinueinthesamestateofmotion. Newton'ssecondlaw Thecomponentsofanobject'saccelerationarepredictedby theequations a x = F x total =m a y = F y total =m ,and a z = F z total =m Newton'sthirdlaw Section6.3Newton'sLawsinThreeDimensions 181

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g / Example2. IftwoobjectsAandBinteractviaforces,thenthecomponentsoftheirforcesoneachotherareequalandopposite: F AonB, x = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F BonA, x F AonB, y = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F BonA, y ,and F AonB, z = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F BonA, z Forcesinperpendiculardirectionsonthesameobjectexample2 Anobjectisinitiallyatrest.Twoconstantforcesbeginactingon it,andcontinueactingonitforawhile.Assuggestedbythetwo arrows,theforcesareperpendicular,andtherightwardforceis stronger.Whathappens? Aristotlebelieved,andmanystudentsstilldo,thatonlyoneforce cangiveorderstoanobjectatonetime.Theythereforethink thattheobjectwillbeginspeedingupandmovinginthedirection ofthestrongerforce.Infacttheobjectwillmovealongadiagonal. Intheexampleshowninthegure,theobjectwillrespondtothe largerightwardforcewithalargeaccelerationcomponenttothe right,andthesmallupwardforcewillgiveitasmallacceleration componentupward.Thestrongerforcedoesnotoverwhelmthe weakerforce,orhaveanyeffectontheupwardmotionatall.The forcecomponentssimplyaddtogether: F x total = F 1, x + 0 F 2, x F y total = > 0 F 1, y + F 2, y DiscussionQuestion A Thegureshowstwotrajectories,madebysplicingtogetherlines andcirculararcs,whichareunphysicalforanobjectthatisonlybeing actedonbygravity.ProvethattheyareimpossiblebasedonNewton's laws. 182 Chapter6Newton'sLawsinThreeDimensions

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Summary SelectedVocabulary component....thepartofavelocity,acceleration,orforce thatwouldbeperceptibletoanobserverwho couldonlyseetheuniverseprojectedalonga certainone-dimensionalaxis parabola.....themathematicalcurvewhosegraphhas y proportionalto x 2 Notation x y z ......anobject'spositionsalongthe x y ,and z axes v x v y v z .....the x y ,and z componentsofanobject'svelocity;theratesofchangeoftheobject's x y and z coordinates a x a y a z .....the x y ,and z componentsofanobject'sacceleration;theratesofchangeof v x v y ,and v z Summary Aforcedoesnotproduceanyeectonthemotionofanobject inaperpendiculardirection.Themostimportantapplicationof thisprincipleisthatthehorizontalmotionofaprojectilehaszero acceleration,whiletheverticalmotionhasanaccelerationequalto g Thatis,anobject'shorizontalandverticalmotionsareindependent. Thearcofaprojectileisaparabola. Motioninthreedimensionsismeasuredusingthreecoordinates, x y ,and z .Eachofthesecoordinateshasitsowncorresponding velocityandacceleration.Wesaythatthevelocityandacceleration bothhave x y ,and z components Newton'ssecondlawisreadilyextendedtothreedimensionsby rewritingitasthreeequationspredictingthethreecomponentsof theacceleration, a x = F x total =m a y = F y total =m a z = F z total =m andlikewisefortherstandthirdlaws. Summary 183

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 aAballisthrownstraightupwithvelocity v .Findan equationfortheheighttowhichitrises. bGeneralizeyourequationforaballthrownatanangle above horizontal,inwhichcaseitsinitialvelocitycomponentsare v x = v cos and v y = v sin 2 AttheSalinasLettuceFestivalParade,MissLettuceof1996 dropsherbouquetwhileridingonaoatmovingtowardtheright. Comparetheshapeofitstrajectoryasseenbyhertotheshapeseen byoneofheradmirersstandingonthesidewalk. 3 Twodaredevils,WendyandBill,gooverNiagaraFalls.Wendy sitsinaninnertube,andletsthe30km/hrvelocityoftheriverthrow herouthorizontallyoverthefalls.Billpaddlesakayak,addingan extra10km/hrtohisvelocity.Theygoovertheedgeofthefalls atthesamemoment,sidebyside.Ignoreairfriction.Explainyour reasoning. aWhohitsthebottomrst? bWhatisthehorizontalcomponentofWendy'svelocityonimpact? cWhatisthehorizontalcomponentofBill'svelocityonimpact? dWhoisgoingfasteronimpact? 4 Abaseballpitcherthrowsapitchclockedat v x =73.3mi/h. Hethrowshorizontally.Bywhatamount, d ,doestheballdropby thetimeitreacheshomeplate, L =60.0ftaway? aFirstndasymbolicanswerintermsof L v x ,and g bPluginandndanumericalanswer.Expressyouranswerin unitsofft.[Note:1ft=12in,1mi=5280ft,and1in=2.54cm] p Problem4. 5 Acannonstandingonaateldresacannonballwitha muzzlevelocity v ,atanangle abovehorizontal.Thecannonball thusinitiallyhasvelocitycomponents v x = v cos and v y = v sin aShowthatthecannon'srangehorizontaldistancetowherethe 184 Chapter6Newton'sLawsinThreeDimensions

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cannonballfallsisgivenbytheequation R = v 2 =g sin cos bInterpretyourequationinthecasesof =0and =90 Solution,p.275 6 Assumingtheresultofproblem5fortherangeofaprojectile, R = v 2 =g sin cos ,showthatthemaximumrangeisfor =45 R 7 Twocarsgooverthesamebumpintheroad,Maria'sMaserati at25milesperhourandPark'sPorscheat37.Howmanytimes greateristheverticalaccelerationofthePorsche?Hint:Remember thataccelerationdependsbothonhowmuchthevelocitychanges andonhowmuchtimeittakestochange. p Problems 185

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a / Vectorsareusedinaerialnavigation. Chapter7 Vectors 7.1VectorNotation Theideaofcomponentsfreedusfromtheconnesofone-dimensional physics,butthecomponentnotationcanbeunwieldy,sinceevery one-dimensionalequationhastobewrittenasasetofthreeseparate equationsinthethree-dimensionalcase.Newtonwasstuckwiththe componentnotationuntilthedayhedied,buteventuallysomeone sucientlylazyandcleverguredoutawayofabbreviatingthree equationsasone. a )778(! F AonB = )]TJ 8.485 7.455 Td [()778(! F BonA standsfor F AonB, x = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F BonA, x F AonB, y = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F BonA, y F AonB, z = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F BonA, z b )778(! F total = )778(! F 1 + )778(! F 2 + ::: standsfor F total, x = F 1, x + F 2, x + ::: F total, y = F 1, y + F 2, y + ::: F total, z = F 1, z + F 2, z + ::: c )778(! a = )826(! v t standsfor a x = v x = t a y = v y = t a z = v z = t ExampleashowsbothwaysofwritingNewton'sthirdlaw.Which wouldyouratherwrite? Theideaisthateachofthealgebrasymbolswithanarrowwrit187

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tenontop,calledavector,isactuallyanabbreviationforthree dierentnumbers,the x y ,and z components.Thethreecomponentsarereferredtoasthecomponentsofthevector,e.g., F x isthe x componentofthevector )778(! F .Thenotationwithanarrowontop isgoodforhandwrittenequations,butisunattractiveinaprinted book,sobooksuseboldface, F ,torepresentvectors.Afterthis point,I'lluseboldfaceforvectorsthroughoutthisbook. Ingeneral,thevectornotationisusefulforanyquantitythat hasbothanamountandadirectioninspace.Evenwhenyouare notgoingtowriteanyactualvectornotation,theconceptitselfisa usefulone.Wesaythatforceandvelocity,forexample,arevectors. Aquantitythathasnodirectioninspace,suchasmassortime, iscalledascalar.Theamountofavectorquantityiscalledits magnitude.Thenotationforthemagnitudeofavector A is j A j liketheabsolutevaluesignusedwithscalars. Often,asinexampleb,wewishtousethevectornotationto representaddingupallthe x componentstogetatotal x component, etc.Theplussignisusedbetweentwovectorstoindicatethistype ofcomponent-by-componentaddition.Ofcourse,vectorsarereally tripletsofnumbers,notnumbers,sothisisnotthesameastheuse oftheplussignwithindividualnumbers.Butsincewedon'twantto havetoinventnewwordsandsymbolsforthisoperationonvectors, weusethesameoldplussign,andthesameoldaddition-related wordslikeadd,"sum,"andtotal."Combiningvectorsthisway iscalledvectoraddition. Similarly,theminussigninexampleawasusedtoindicate negatingeachofthevector'sthreecomponentsindividually.The equalssignisusedtomeanthatallthreecomponentsofthevector ontheleftsideofanequationarethesameasthecorresponding componentsontheright. Examplecshowshowweabusethedivisionsymbolinasimilar manner.Whenwewritethevector v dividedbythescalart, wemeanthenewvectorformedbydividingeachoneofthevelocity componentsby t It'snothardtoimagineavarietyofoperationsthatwouldcombinevectorswithvectorsorvectorswithscalars,butonlyfourof themarerequiredinordertoexpressNewton'slaws: operationdenition vector + vector Addcomponentbycomponentto makeanewsetofthreenumbers. vector )]TJ/F17 10.9091 Tf 10.909 0 Td [(vector Subtractcomponentbycomponent tomakeanewsetofthreenumbers. vector scalarMultiplyeachcomponentofthevectorbythescalar. vector = scalarDivideeachcomponentofthevector bythescalar. 188 Chapter7Vectors

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b / The x an y components ofavectorcanbethoughtofas theshadowsitcastsontothe x and y axes. c / Self-checkB. Asanexampleofanoperationthatisnotusefulforphysics,there justaren'tanyusefulphysicsapplicationsfordividingavectorby anothervectorcomponentbycomponent.Inoptionalsection7.5, wediscussinmoredetailthefundamentalreasonswhysomevector operationsareusefulandothersuseless. Wecandoalgebrawithvectors,orwithamixtureofvectors andscalarsinthesameequation.Basicallyallthenormalrulesof algebraapply,butifyou'renotsureifacertainstepisvalid,you shouldsimplytranslateitintothreecomponent-basedequationsand seeifitworks. Orderofadditionexample1 Ifweareaddingtwoforcevectors, F + G ,isitvalidtoassume asinordinaryalgebrathat F + G isthesameas G + F ? Totellifthisalgebrarulealsoappliestovectors,wesimply translatethevectornotationintoordinaryalgebranotation.In termsofordinarynumbers,thecomponentsofthevector F + G wouldbe F x + G x F y + G y ,and F z + G z ,whicharecertainlythe samethreenumbersas G x + F x G y + F y ,and G z + F z .Yes, F + G isthesameas G + F Itisusefultodeneasymbol r forthevectorwhosecomponents are x y ,and z ,andasymbol r madeoutof x y ,and z Althoughthismayallseemalittleformidable,keepinmindthat itamountstonothingmorethanawayofabbreviatingequations! Also,tokeepthingsfromgettingtooconfusingtheremainderofthis chapterfocusesmainlyonthe r vector,whichisrelativelyeasyto visualize. self-checkA Translatetheequations v x = x = t v y = y = t ,and v z = z = t for motionwithconstantvelocityintoasingleequationinvectornotation. Answer,p.268 Drawingvectorsasarrows Avectorintwodimensionscanbeeasilyvisualizedbydrawing anarrowwhoselengthrepresentsitsmagnitudeandwhosedirection representsitsdirection.The x componentofavectorcanthenbe visualizedasthelengthoftheshadowitwouldcastinabeamof lightprojectedontothe x axis,andsimilarlyforthe y component. Shadowswitharrowheadspointingbackagainstthedirectionofthe positiveaxiscorrespondtonegativecomponents. Inthistypeofdiagram,thenegativeofavectoristhevector withthesamemagnitudebutintheoppositedirection.Multiplying avectorbyascalarisrepresentedbylengtheningthearrowbythat factor,andsimilarlyfordivision. self-checkB Givenvector Q representedbyanarrowingurec,drawarrowsrepreSection7.1VectorNotation 189

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d / Example2. sentingthevectors1.5 Q and )]TJ/F64 9.9626 Tf 7.749 0 Td [(Q Answer,p. 268 DiscussionQuestions A Woulditmakesensetodeneazerovector?Discusswhatthe zerovector'scomponents,magnitude,anddirectionwouldbe;arethere anyissueshere?Ifyouwantedtodisqualifysuchathingfrombeinga vector,considerwhetherthesystemofvectorswouldbecomplete.For comparison,canyouthinkofasimplearithmeticproblemwithordinary numberswhereyouneedzeroastheresult?Doesthesamereasoning applytovectors,ornot? B Youdrivetoyourfriend'shouse.Howdoesthemagnitudeofyour r vectorcomparewiththedistanceyou'veaddedtothecar'sodometer? 7.2CalculationswithMagnitudeandDirection IfyouasksomeonewhereLasVegasiscomparedtoLosAngeles, theyareunlikelytosaythatthe x is290kmandthe y is230 km,inacoordinatesystemwherethepositive x axisiseastandthe y axispointsnorth.Theywillprobablysayinsteadthatit's370km tothenortheast.Iftheywerebeingprecise,theymightspecifythe directionas38 counterclockwisefromeast.Intwodimensions,we canalwaysspecifyavector'sdirectionlikethis,usingasingleangle. Amagnitudeplusananglesucetospecifyeverythingaboutthe vector.Thefollowingtwoexamplesshowhowweusetrigonometry andthePythagoreantheoremtogobackandforthbetweenthe x )]TJ/F20 10.9091 Tf 9.039 0 Td [(y andmagnitude-angledescriptionsofvectors. Findingmagnitudeandanglefromcomponentsexample2 Giventhatthe r vectorfromLAtoLasVegashas x =290km and y =230km,howwouldwendthemagnitudeanddirection of r? Wendthemagnitudeof r fromthePythagoreantheorem: j r j = q x 2 + y 2 =370km Weknowallthreesidesofthetriangle,sotheangle canbe foundusinganyoftheinversetrigfunctions.Forexample,we knowtheoppositeandadjacentsides,so =tan )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 y x =38 Findingcomponentsfrommagnitudeandangleexample3 Giventhatthestraight-linedistancefromLosAngelestoLas Vegasis370km,andthattheangle inthegureis38 ,how canthexand y componentsofthe r vectorbefound? 190 Chapter7Vectors

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e / Example4. Thesineandcosineof relatethegiveninformationtothe informationwewishtond: cos = x j r j sin = y j r j Solvingfortheunknownsgives x = j r j cos =290km and y = j r j sin =230km. Thefollowingexampleshowsthecorrecthandlingoftheplus andminussigns,whichisusuallythemaincauseofmistakes. Negativecomponentsexample4 SanDiegois120kmeastand150kmsouthofLosAngeles.An airplanepilotissettingcoursefromSanDiegotoLosAngeles.At whatangleshouldshesethercourse,measuredcounterclockwisefromeast,asshowninthegure? Ifwemakethetraditionalchoiceofcoordinateaxes,with x pointingtotherightand y pointinguponthemap,thenher x is negative,becausehernal x valueislessthanherinitial x value. Her y ispositive,sowehave x = )]TJ/F39 10.9091 Tf 8.485 0 Td [(120km y =150km. Ifweworkbyanalogywiththepreviousexample,weget =tan )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 y x =tan )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 )]TJ/F39 10.9091 Tf 8.485 0 Td [(1.25 = )]TJ/F39 10.9091 Tf 8.485 0 Td [(51 Accordingtotheusualwayofdeninganglesintrigonometry, anegativeresultmeansananglethatliesclockwisefromthex axis,whichwouldhaveherheadingfortheBajaCalifornia.What wentwrong?Theansweristhatwhenyouaskyourcalculatorto takethearctangentofanumber,therearealwaystwovalidpossibilitiesdifferingby180 .Thatis,therearetwopossibleangles whosetangentsequal-1.25: tan129 = )]TJ/F39 10.9091 Tf 8.484 0 Td [(1.25 tan )]TJ/F39 10.9091 Tf 8.485 0 Td [(51 = )]TJ/F39 10.9091 Tf 8.484 0 Td [(1.25 Youcalculatordoesn'tknowwhichisthecorrectone,soitjust picksone.Inthiscase,theoneitpickedwasthewrongone,and itwasuptoyoutoadd180 toittondtherightanswer. Section7.2CalculationswithMagnitudeandDirection 191

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f / Example5. g / Vectorscanbeaddedgraphicallybyplacingthemtiptotail, andthendrawingavectorfrom thetailoftherstvectortothetip ofthesecondvector. DiscussionQuestion A Intheexampleabove,wedealtwith components thatwerenegative. Doesitmakesensetotalkaboutpositiveandnegative vectors ? 7.3TechniquesforAddingVectors Additionofvectorsgiventheircomponents Theeasiesttypeofvectoradditioniswhenyouareinpossession ofthecomponents,andwanttondthecomponentsoftheirsum. Addingcomponentsexample5 Giventhe x and y valuesfromthepreviousexamples,nd the x and y fromSanDiegotoLasVegas. x total = x 1 + x 2 = )]TJ/F39 10.9091 Tf 8.485 0 Td [(120km+290km =170km y total = y 1 + y 2 =150km+230km =380 Notehowthesignsofthe x componentstakecareofthewestwardandeastwardmotions,whichpartiallycancel. Additionofvectorsgiventheirmagnitudesanddirections Inthiscase,youmustrsttranslatethemagnitudesanddirectionsintocomponents,andtheaddthecomponents. Graphicaladditionofvectors Oftentheeasiestwaytoaddvectorsisbymakingascaledrawing onapieceofpaper.Thisisknownasgraphicaladdition,asopposed totheanalytictechniquesdiscussedpreviously. LAtoVegas,graphicallyexample6 Giventhemagnitudesandanglesofthe r vectorsfromSan DiegotoLosAngelesandfromLosAngelestoLasVegas,nd themagnitudeandangleofthe r vectorfromSanDiegotoLas Vegas. Usingaprotractorandaruler,wemakeacarefulscaledrawing, asshowningureh.Ascaleof1mm 2kmwaschosenforthis solution.Witharuler,wemeasurethedistancefromSanDiego toLasVegastobe206mm,whichcorrespondsto412km.With aprotractor,wemeasuretheangle tobe65 Evenwhenwedon'tintendtodoanactualgraphicalcalculation witharulerandprotractor,itcanbeconvenienttodiagramthe additionofvectorsinthisway.With r vectors,itintuitivelymakes sensetolaythevectorstip-to-tailanddrawthesumvectorfromthe 192 Chapter7Vectors

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tailoftherstvectortothetipofthesecondvector.Wecando thesamewhenaddingothervectorssuchasforcevectors. h / Example6. self-checkC Howwouldyousubtractvectorsgraphically? Answer,p.268 Section7.3TechniquesforAddingVectors 193

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DiscussionQuestions A Ifyou'redoing graphical additionofvectors,doesitmatterwhich vectoryoustartwithandwhichvectoryoustartfromtheothervector's tip? B Ifyouaddavectorwithmagnitude1toavectorofmagnitude2, whatmagnitudesarepossibleforthevectorsum? C Whichoftheseexamplesofvectoradditionarecorrect,andwhich areincorrect? 7.4 ? UnitVectorNotation Whenwewanttospecifyavectorbyitscomponents,itcanbecumbersometohavetowritethealgebrasymbolforeachcomponent: x =290km, y =230km Amorecompactnotationistowrite r =km ^ x +km ^ y wherethevectors ^ x ^ y ,and ^ z ,calledtheunitvectors,aredened asthevectorsthathavemagnitudeequalto1anddirectionslying alongthe x y ,and z axes.Inspeech,theyarereferredtoasx-hat" andsoon. Aslightlydierent,andhardertoremember,versionofthis notationisunfortunatelymoreprevalent.Inthisversion,theunit vectorsarecalled ^ i ^ j ,and ^ k : r =km ^ i +km ^ j 7.5 ? RotationalInvariance Let'stakeacloserlookatwhycertainvectoroperationsareusefulandothersarenot.Considertheoperationofmultiplyingtwo vectorscomponentbycomponenttoproduceathirdvector: R x = P x Q x R y = P y Q y R z = P z Q z Asasimpleexample,wechoosevectors P and Q tohavelength 1,andmakethemperpendiculartoeachother,asshowningure 194 Chapter7Vectors

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i / Component-by-component multiplicationofthevectorsin1 wouldproducedifferentvectors incoordinatesystems2and3. i/1.Ifwecomputetheresultofournewvectoroperationusingthe coordinatesystemini/2,wend: R x =0 R y =0 R z =0 The x componentiszerobecause P x =0,the y componentiszero because Q y =0,andthe z componentisofcoursezerobecauseboth vectorsareinthe x )]TJ/F20 10.9091 Tf 11.266 0 Td [(y plane.However,ifwecarryoutthesame operationsincoordinatesystemi/3,rotated45degreeswithrespect tothepreviousone,wend R x =1 = 2 R y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2 R z =0 Theoperation'sresultdependsonwhatcoordinatesystemweuse, andsincethetwoversionsof R havedierentlengthsonebeingzero andtheothernonzero,theydon'tjustrepresentthesameanswer expressedintwodierentcoordinatesystems.Suchanoperation willneverbeusefulinphysics,becauseexperimentsshowphysics worksthesameregardlessofwhichwayweorientthelaboratory building!The useful vectoroperations,suchasadditionandscalar multiplication,arerotationallyinvariant,i.e.,comeoutthesame regardlessoftheorientationofthecoordinatesystem. Section7.5 ? RotationalInvariance 195

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Summary SelectedVocabulary vector.......aquantitythathasbothanamountmagnitudeandadirectioninspace magnitude....theamount"associatedwithavector scalar.......aquantitythathasnodirectioninspace,only anamount Notation A .........avectorwithcomponents A x A y ,and A z )778(! A .........handwrittennotationforavector j A j ........themagnitudeofvector A r ..........thevectorwhosecomponentsare x y ,and z r .........thevectorwhosecomponentsare x y ,and z ^ x ^ y ^ z ......optionaltopicunitvectors;thevectorswith magnitude1lyingalongthe x y ,and z axes ^ i ^ j ^ k .......ahardertoremembernotationfortheunit vectors OtherTerminologyandNotation displacementvector......... anameforthesymbol r speed.......themagnitudeofthevelocityvector,i.e.,the velocitystrippedofanyinformationaboutits direction Summary Avectorisaquantitythathasbothamagnitudeamountand adirectioninspace,asopposedtoascalar,whichhasnodirection. Thevectornotationamountssimplytoanabbreviationforwriting thevector'sthreecomponents. Intwodimensions,avectorcanberepresentedeitherbyitstwo componentsorbyitsmagnitudeanddirection.Thetwowaysof describingavectorcanberelatedbytrigonometry. Thetwomainoperationsonvectorsareadditionofavectorto avector,andmultiplicationofavectorbyascalar. Vectoradditionmeansaddingthecomponentsoftwovectors toformthecomponentsofanewvector.Ingraphicalterms,this correspondstodrawingthevectorsastwoarrowslaidtip-to-tailand drawingthesumvectorfromthetailoftherstvectortothetip ofthesecondone.Vectorsubtractionisperformedbynegatingthe vectortobesubtractedandthenadding. Multiplyingavectorbyascalarmeansmultiplyingeachofits componentsbythescalartocreateanewvector.Divisionbya scalarisdenedsimilarly. 196 Chapter7Vectors

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Problem4. Problem1. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Thegureshowsvectors A and B .Graphicallycalculatethe following: A + B A )]TJ/F17 10.9091 Tf 10.909 0 Td [(B B )]TJ/F17 10.9091 Tf 10.91 0 Td [(A )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 B A )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 B Nonumbersareinvolved. 2 PhnomPenhis470kmeastand250kmsouthofBangkok. Hanoiis60kmeastand1030kmnorthofPhnomPenh. aChooseacoordinatesystem,andtranslatethesedatainto x and y valueswiththeproperplusandminussigns. bFindthecomponentsofthe r vectorpointingfromBangkok toHanoi. p 3 Ifyouwalk35kmatanangle25 counterclockwisefromeast, andthen22kmat230 counterclockwisefromeast,ndthedistance anddirectionfromyourstartingpointtoyourdestination. p 4 Amachinistisdrillingholesinapieceofaluminumaccording totheplanshowninthegure.Shestartswiththetophole,then movestotheoneontheleft,andthentotheoneontheright.Since thisisahigh-precisionjob,shenishesbymovinginthedirection andattheanglethatshouldtakeherbacktothetophole,and checksthatsheendsupinthesameplace.Whatarethedistance anddirectionfromtheright-handholetothetopone? p 5 Supposesomeoneproposesanewoperationinwhichavector A andascalar B areaddedtogethertomakeanewvector C like this: C x = A x + B C y = A y + B C y = A y + B Provethatthisoperationwon'tbeusefulinphysics,becauseit's notrotationallyinvariant. Problems 197

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Chapter8 VectorsandMotion In1872,capitalistandformerCaliforniagovernorLelandStanford askedphotographerEadweardMuybridgeifhewouldworkforhim onaprojecttosettlea$25,000betaprincelysumatthattime. Stanford'sfriendswereconvincedthatagallopinghorsealwayshad atleastonefootontheground,butStanfordclaimedthattherewas amomentduringeachcycleofthemotionwhenallfourfeetwere intheair.Thehumaneyewassimplynotfastenoughtosettlethe question.In1878,Muybridgenallysucceededinproducingwhat amountedtoamotionpictureofthehorse,showingconclusively thatallfourfeetdidleavethegroundatonepoint.Muybridgewas acolorfulgureinSanFranciscohistory,andhisacquittalforthe murderofhiswife'sloverwasconsideredthetrialofthecenturyin California. ThelosersofthebethadprobablybeeninuencedbyAristotelianreasoning,forinstancetheexpectationthataleapinghorse wouldlosehorizontalvelocitywhileintheairwithnoforcetopush itforward,sothatitwouldbemoreecientforthehorsetorun withoutleaping.Butevenforstudentswhohaveconvertedwhole199

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a / Example1. heartedlytoNewtonianism,therelationshipbetweenforceandaccelerationleadstosomeconceptualdiculties,themainonebeing aproblemwiththetruebutseeminglyabsurdstatementthatan objectcanhaveanaccelerationvectorwhosedirectionisnotthe sameasthedirectionofmotion.Thehorse,forinstance,hasnearly constanthorizontalvelocity,soits a x iszero.Butasanyonecantell youwhohasriddenagallopinghorse,thehorseacceleratesupand down.Thehorse'saccelerationvectorthereforechangesbackand forthbetweentheupanddowndirections,butisneverinthesame directionasthehorse'smotion.Inthischapter,wewillexamine morecarefullythepropertiesofthevelocity,acceleration,andforce vectors.Nonewprinciplesareintroduced,butanattemptismade totiethingstogetherandshowexamplesofthepowerofthevector formulationofNewton'slaws. 8.1TheVelocityVector Formotionwithconstantvelocity,thevelocityvectoris v = r = t .[onlyforconstantvelocity] The r vectorpointsinthedirectionofthemotion,anddividing itbythescalar t onlychangesitslength,notitsdirection,sothe velocityvectorpointsinthesamedirectionasthemotion.When thevelocityisnotconstant,i.e.,whenthe x )]TJ/F20 10.9091 Tf 9.002 0 Td [(t y )]TJ/F20 10.9091 Tf 9.002 0 Td [(t ,and z )]TJ/F20 10.9091 Tf 9.001 0 Td [(t graphs arenotalllinear,weusetheslope-of-the-tangent-lineapproachto denethecomponents v x v y ,and v z ,fromwhichweassemblethe velocityvector.Evenwhenthevelocityvectorisnotconstant,it stillpointsalongthedirectionofmotion. Vectoradditionisthecorrectwaytogeneralizetheone-dimensional conceptofaddingvelocitiesinrelativemotion,asshowninthefollowingexample: Velocityvectorsinrelativemotionexample1 Youwishtocrossariverandarriveatadockthatisdirectly acrossfromyou,buttheriver'scurrentwilltendtocarryyou downstream.Tocompensate,youmuststeertheboatatanangle.Findtheangle ,giventhemagnitude, j v WL j ,ofthewater's velocityrelativetotheland,andthemaximumspeed, j v BW j ,of whichtheboatiscapablerelativetothewater. Theboat'svelocityrelativetothelandequalsthevectorsumof itsvelocitywithrespecttothewaterandthewater'svelocitywith respecttotheland, v BL = v BW + v W L Iftheboatistotravelstraightacrosstheriver,i.e.,alongthe y axis,thenweneedtohave v BL x =0.This x componentequals thesumofthe x componentsoftheothertwovectors, v B L x = v B W x + v W L x 200 Chapter8VectorsandMotion

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or 0= j v BW j sin + j v WL j Solvingfor ,wend sin = j v WL j = j v BW j so =sin )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 j v WL j v BW Solvedproblem:AnnieOakleypage212,problem8 DiscussionQuestions A Isitpossibleforanairplanetomaintainaconstantvelocityvector butnotaconstant j v j ?Howabouttheoppositeaconstant j v j butnota constantvelocityvector?Explain. B NewYorkandRomeareataboutthesamelatitude,sotheearth's rotationcarriesthembotharoundnearlythesamecircle.Dothetwocities havethesamevelocityvectorrelativetothecenteroftheearth?Ifnot, isthereanywayfortwocitiestohavethesamevelocityvector? Section8.1TheVelocityVector 201

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b / Achangeinthemagnitudeofthevelocityvectorimplies anacceleration. c / Achangeinthedirection ofthevelocityvectoralsoproducesanonzero v vector,and thusanonzeroacceleration vector, v = t 8.2TheAccelerationVector Whenallthreeaccelerationcomponentsareconstant,i.e.,when the v x )]TJ/F20 10.9091 Tf 10.564 0 Td [(t v y )]TJ/F20 10.9091 Tf 10.564 0 Td [(t ,and v z )]TJ/F20 10.9091 Tf 10.564 0 Td [(t graphsarealllinear,wecandenethe accelerationvectoras a = v = t ,[onlyforconstantacceleration] whichcanbewrittenintermsofinitialandnalvelocitiesas a = v f )]TJ/F17 10.9091 Tf 10.91 0 Td [(v i = t .[onlyforconstantacceleration] Iftheaccelerationisnotconstant,wedeneitasthevectormade outofthe a x a y ,and a z componentsfoundbyapplyingtheslopeof-the-tangent-linetechniquetothe v x )]TJ/F20 10.9091 Tf 10.209 0 Td [(t v y )]TJ/F20 10.9091 Tf 10.209 0 Td [(t ,and v z )]TJ/F20 10.9091 Tf 10.209 0 Td [(t graphs. Nowtherearetwowaysinwhichwecouldhaveanonzeroacceleration.Eitherthemagnitudeorthedirectionofthevelocityvector couldchange.Thiscanbevisualizedwitharrowdiagramsasshown inguresbandc.Boththemagnitudeanddirectioncanchange simultaneously,aswhenacaraccelerateswhileturning.Onlywhen themagnitudeofthevelocitychangeswhileitsdirectionstaysconstantdowehavea v vectorandanaccelerationvectoralongthe samelineasthemotion. self-checkA Ingureb,istheobjectspeedingup,orslowingdown?What wouldthediagramlooklikeif v i wasthesameas v f ?Describehow the v vectorisdifferentdependingonwhetheranobjectisspeeding uporslowingdown. Answer,p.268 Ifthisallseemsalittlestrangeandabstracttoyou,you'renot alone.Itdoesn'tmeanmuchtomostphysicsstudentstherst timesomeonetellsthemthataccelerationisavector,andthatthe accelerationvectordoesnothavetobeinthesamedirectionasthe velocityvector.Onewaytounderstandthosestatementsbetteris toimagineanobjectsuchasanairfreshenerorapairoffuzzydice hangingfromtherear-viewmirrorofacar.Suchahangingobject, calledabob,constitutesanaccelerometer.Ifyouwatchthebob asyouacceleratefromastoplight,you'llseeitswingbackward. Thehorizontaldirectioninwhichthebobtiltsisoppositetothe directionoftheacceleration.Ifyouapplythebrakesandthecar's accelerationvectorpointsbackward,thebobtiltsforward. Afteracceleratingandslowingdownafewtimes,youthink you'veputyouraccelerometerthroughitspaces,butthenyoumake arightturn.Surprise!Accelerationisavector,andneedn'tpoint inthesamedirectionasthevelocityvector.Asyoumakearight turn,thebobswingsoutward,toyourleft.Thatmeansthecar's accelerationvectoristoyourright,perpendiculartoyourvelocity vector.Ausefuldenitionofanaccelerationvectorshouldrelate inasystematicwaytotheactualphysicaleectsproducedbythe 202 Chapter8VectorsandMotion

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acceleration,soaphysicallyreasonabledenitionoftheacceleration vectormustallowforcaseswhereitisnotinthesamedirectionas themotion. self-checkB Inprojectilemotion,whatdirectiondoestheaccelerationvectorhave? Answer,p.268 d / Example2. Rappellingexample2 Ingured,therappeller'svelocityhaslongperiodsofgradual changeinterspersedwithshortperiodsofrapidchange.These correspondtoperiodsofsmallaccelerationandforce,andperiodsoflargeaccelerationandforce. Thegallopinghorseexample3 Figureeonpage204showsoutlinestracedfromtherst,third, fth,seventh,andninthframesinMuybridge'sseriesofphotographsofthegallopinghorse.Theestimatedlocationofthe horse'scenterofmassisshownwithacircle,whichbobsabove andbelowthehorizontaldashedline. Ifwedon'tcareaboutcalculatingvelocitiesandaccelerationsin anyparticularsystemofunits,thenwecanpretendthatthetime betweenframesisoneunit.Thehorse'svelocityvectorasit movesfromonepointtothenextcanthenbefoundsimplyby drawinganarrowtoconnectonepositionofthecenterofmassto thenext.ThisproducesaseriesofvelocityvectorswhichalterSection8.2TheAccelerationVector 203

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e / Example3. natebetweenpointingaboveandbelowhorizontal. The v vectoristhevectorwhichwewouldhavetoaddontoone velocityvectorinordertogetthenextvelocityvectorintheseries. The v vectoralternatesbetweenpointingdownaroundthetime whenthehorseisintheair,Banduparoundthetimewhenthe horsehastwofeetontheground,D. DiscussionQuestions A Whenacaraccelerates,whydoesabobhangingfromtherearview mirrorswingtowardthebackofthecar?Isitbecauseaforcethrowsit backward?Ifso,whatforce?Similarly,describewhathappensinthe othercasesdescribedabove. B Supermanisguidingacrippledspaceshipintoport.Theship's enginesarenotworking.IfSupermansuddenlychangesthedirectionof hisforceontheship,doestheship'svelocityvectorchangesuddenly?Its accelerationvector?Itsdirectionofmotion? 204 Chapter8VectorsandMotion

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f / Example4. g / Theappliedforce F A pushes theblockupthefrictionlessramp. h / Threeforcesactonthe block.Theirvectorsumiszero. i / Iftheblockistomoveat constantvelocity,Newton'srst lawsaysthatthethreeforce vectorsactingonitmustadd uptozero.Toperformvector addition,weputthevectorstip totail,andinthiscaseweare addingthreevectors,soeach one'stailgoesagainstthetipof thepreviousone.Sincetheyare supposedtoadduptozero,the thirdvector'stipmustcomeback totouchthetailoftherstvector. Theyformatriangle,andsince theappliedforceisperpendicular tothenormalforce,itisaright triangle. 8.3TheForceVectorandSimpleMachines Forceisrelativelyeasytointuitasavector.Theforcevectorpoints inthedirectioninwhichitistryingtoacceleratetheobjectitis actingon. Sinceforcevectorsaresomucheasiertovisualizethanaccelerationvectors,itisoftenhelpfultorstndthedirectionofthe totalforcevectoractingonanobject,andthenusethatinformationtodeterminethedirectionoftheaccelerationvector.Newton's secondlaw, F total = m a ,tellsusthatthetwomustbeinthesame direction. Acomponentofaforcevectorexample4 Figuref,redrawnfromaclassic1920textbook,showsaboy pullinganotherchildonasled.Hisforcehasbothahorizontal componentandaverticalone,butonlythehorizontaloneacceleratesthesled.Theverticalcomponentjustpartiallycancelsthe forceofgravity,causingadecreaseinthenormalforcebetween therunnersandthesnow.Therearetwotrianglesinthegure. Onetriangle'shypotenuseistherope,andtheother'sisthemagnitudeoftheforce.Thesetrianglesaresimilar,sotheirinternal anglesareallthesame,buttheyarenotthesametriangle.One isadistancetriangle,withsidesmeasuredinmeters,theother aforcetriangle,withsidesinnewtons.Inbothcases,thehorizontallegis93%aslongasthehypotenuse.Itdoesnotmake sense,however,tocomparethesizesofthetrianglestheforce triangleisnotsmallerinanymeaningfulsense. Pushingablockuparampexample5 Figureashowsablockbeingpushedupafrictionlessramp atconstantspeedbyanappliedforce F A .Howmuchforceis required,intermsoftheblock'smass, m ,andtheangleofthe ramp, ? Figurebshowstheothertwoforcesactingontheblock:a normalforce, F N ,createdbytheramp,andtheweightforce, F W createdbytheearth'sgravity.Becausetheblockisbeingpushed upatconstantspeed,ithaszeroacceleration,andthetotalforce onitmustbezero.Fromgurec,wend j F A j = j F W j sin = mg sin Sincethesineisalwayslessthanone,theappliedforceisalways lessthan mg ,i.e.,pushingtheblockuptherampiseasierthan liftingitstraightup.Thisispresumablytheprincipleonwhichthe pyramidswereconstructed:theancientEgyptianswouldhave hadahardtimeapplyingtheforcesofenoughslavestoequalthe fullweightofthehugeblocksofstone. EssentiallythesameanalysisappliestoseveralothersimplemaSection8.3TheForceVectorandSimpleMachines 205

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DiscussionquestionA. j / DiscussionquestionB. chines,suchasthewedgeandthescrew. Solvedproblem:Acargoplanepage212,problem9 Solvedproblem:Theangleofreposepage213,problem11 Solvedproblem:Awagonpage212,problem10 DiscussionQuestions A Thegureshowsablockbeingpresseddiagonallyupwardagainsta wall,causingittoslideupthewall.Analyzetheforcesinvolved,including theirdirections. B Thegureshowsarollercoastercarrollingdownandthenupunder theinuenceofgravity.Sketchthecar'svelocityvectorsandacceleration vectors.Pickaninterestingpointinthemotionandsketchasetofforce vectorsactingonthecarwhosevectorsumcouldhaveresultedinthe rightaccelerationvector. 8.4 R CalculusWithVectors Usingtheunitvectornotationintroducedinsection7.4,thedenitionsofthevelocityandaccelerationcomponentsgiveninchapter 6canbetranslatedintocalculusnotationas v = d x d t ^ x + d y d t ^ y + d z d t ^ z and a = d v x d t ^ x + d v y d t ^ y + d v z d t ^ z Tomakethenotationlesscumbersome,wegeneralizetheconcept ofthederivativetoincludederivativesofvectors,sothatwecan abbreviatetheaboveequationsas v = d r d t and a = d v d t Inwords,totakethederivativeofavector,youtakethederivatives ofitscomponentsandmakeanewvectoroutofthose.Thisdenitionmeansthatthederivativeofavectorfunctionhasthefamiliar properties d c f d t = c d f d t [ c isaconstant] and d f + g d t = d f d t + d g d t Theintegralofavectorislikewisedenedasintegratingcomponent bycomponent. 206 Chapter8VectorsandMotion

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Thesecondderivativeofavectorexample6 Twoobjectshavepositionsasfunctionsoftimegivenbythe equations r 1 =3 t 2 x + t y and r 2 =3 t 4 x + t y Findbothobjects'accelerationsusingcalculus.Couldeitheranswerhavebeenfoundwithoutcalculus? Takingtherstderivativeofeachcomponent,wend v 1 =6 t x + y v 2 =12 t 3 x + y andtakingthederivativesagaingivesacceleration, a 1 =6 x a 2 =36 t 2 x Therstobject'saccelerationcouldhavebeenfoundwithoutcalculus,simplybycomparingthe x and y coordinateswiththe constant-accelerationequation x = v o t + 1 2 a t 2 .Thesecond equation,however,isn'tjustasecond-orderpolynomialin t ,so theaccelerationisn'tconstant,andwereallydidneedcalculusto ndthecorrespondingacceleration. Theintegralofavectorexample7 Startingfromrest,ayingsaucerofmass m isobservedto varyitspropulsionwithmathematicalprecisionaccordingtothe equation F = bt 42 x + ct 137 y Thealiensinformusthatthenumbers42and137haveaspecial religioussignicanceforthem.Findthesaucer'svelocityasa functionoftime. Fromthegivenforce,wecaneasilyndtheacceleration a = F m = b m t 42 x + c m t 137 y Thevelocityvector v istheintegralwithrespecttotimeofthe acceleration, v = Z a d t = Z b m t 42 x + c m t 137 y d t Section8.4 R CalculusWithVectors 207

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andintegratingcomponentbycomponentgives = Z b m t 42 d t x + Z c m t 137 d t y = b 43 m t 43 x + c 138 m t 138 y wherewehaveomittedtheconstantsofintegration,sincethe saucerwasstartingfromrest. Are-extinguisherstuntoniceexample8 Prof.Puerilesmugglesareextinguisherintoaskatingrink. Climbingoutontotheicewithoutanyskateson,hesitsdownand pushesofffromthewallwithhisfeet,acquiringaninitialvelocity v o y .At t =0,hethendischargesthereextinguisherata45degreeanglesothatitappliesaforcetohimthatisbackward andtotheleft,i.e.,alongthenegative y axisandthepositive x axis.Thereextinguisher'sforceisstrongatrst,butthendies downaccordingtotheequation j F j = b )]TJ/F78 10.9091 Tf 11.275 0 Td [(ct ,where b and c are constants.Findtheprofessor'svelocityasafunctionoftime. Measuredcounterclockwisefromthe x axis,theangleofthe forcevectorbecomes315 .Breakingtheforcedowninto x and y components,wehave F x = j F j cos315 = b )]TJ/F78 10.9091 Tf 10.909 0 Td [(ct F y = j F j sin315 = )]TJ/F78 10.9091 Tf 8.485 0 Td [(b + ct Inunitvectornotation,thisis F = b )]TJ/F78 10.9091 Tf 10.909 0 Td [(ct x + )]TJ/F78 10.9091 Tf 8.485 0 Td [(b + ct y Newton'ssecondlawgives a = F = m = b )]TJ/F78 10.9091 Tf 10.909 0 Td [(ct p 2 m x + )]TJ/F78 10.9091 Tf 8.485 0 Td [(b + ct p 2 m y Tondthevelocityvectorasafunctionoftime,weneedtointegratetheaccelerationvectorwithrespecttotime, v = Z a d t = Z b )]TJ/F78 10.9091 Tf 10.909 0 Td [(ct p 2 m x + )]TJ/F78 10.9091 Tf 8.485 0 Td [(b + ct p 2 m y d t = 1 p 2 m Z b )]TJ/F78 10.9091 Tf 10.91 0 Td [(ct x + )]TJ/F78 10.9091 Tf 8.485 0 Td [(b + ct y d t 208 Chapter8VectorsandMotion

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Avectorfunctioncanbeintegratedcomponentbycomponent,so thiscanbebrokendownintotwointegrals, v = x p 2 m Z b )]TJ/F78 10.9091 Tf 10.909 0 Td [(ct d t + y p 2 m Z )]TJ/F78 10.9091 Tf 8.485 0 Td [(b + ct d t = bt )]TJ/F39 7.9701 Tf 12.105 4.295 Td [(1 2 ct 2 p 2 m +constant#1 x + )]TJ/F78 10.9091 Tf 8.485 0 Td [(bt + 1 2 ct 2 p 2 m +constant#2 y Herethephysicalsignicanceofthetwoconstantsofintegration isthattheygivetheinitialvelocity.Constant#1isthereforezero, andconstant#2mustequal v o .Thenalresultis v = bt )]TJ/F39 7.9701 Tf 12.105 4.296 Td [(1 2 ct 2 p 2 m x + )]TJ/F78 10.9091 Tf 8.485 0 Td [(bt + 1 2 ct 2 p 2 m + v o y Section8.4 R CalculusWithVectors 209

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Summary Thevelocityvectorpointsinthedirectionoftheobject'smotion. Relativemotioncanbedescribedbyvectoradditionofvelocities. Theaccelerationvectorneednotpointinthesamedirectionas theobject'smotion.Weusethewordacceleration"todescribeany changeinanobject'svelocityvector,whichcanbeeitherachange initsmagnitudeorachangeinitsdirection. Animportantapplicationofthevectoradditionofforcesisthe useofNewton'srstlawtoanalyzemechanicalsystems. 210 Chapter8VectorsandMotion

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. Problem1. 1 Adinosaurfossilisslowlymovingdowntheslopeofaglacier undertheinuenceofwind,rainandgravity.Atthesametime, theglacierismovingrelativetothecontinentunderneath.The dashedlinesrepresentthedirectionsbutnotthemagnitudesofthe velocities.Pickascale,andusegraphicaladditionofvectorstond themagnitudeandthedirectionofthefossil'svelocityrelativeto thecontinent.Youwillneedarulerandprotractor. p 2 Isitpossibleforahelicoptertohaveanaccelerationdueeast andavelocityduewest?Ifso,whatwouldbegoingon?Ifnot,why not? 3 Abirdisinitiallyyinghorizontallyeastat21.1m/s,butone secondlaterithaschangeddirectionsothatitisyinghorizontally and7 northofeast,atthesamespeed.Whatarethemagnitude anddirectionofitsaccelerationvectorduringthatonesecondtime interval?Assumeitsaccelerationwasroughlyconstant. p Problem4. 4 Apersonofmass M standsinthemiddleofatightrope, whichisxedattheendstotwobuildingsseparatedbyahorizontal distance L .Theropesagsinthemiddle,stretchingandlengthening theropeslightly. Problems 211

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Problem9. Problem10. Problem5. aIfthetightropewalkerwantstheropetosagverticallybyno morethanaheight h ,ndtheminimumtension, T ,thattherope mustbeabletowithstandwithoutbreaking,intermsof h g M and L p bBasedonyourequation,explainwhyitisnotpossibletoget h =0,andgiveaphysicalinterpretation. 5 Yourhandpressesablockofmass m againstawallwitha force F H actingatanangle .Findtheminimumandmaximum possiblevaluesof j F H j thatcankeeptheblockstationary,interms of m g ,and s ,thecoecientofstaticfrictionbetweentheblock andthewall. p ? 6 Askierofmass m iscoastingdownaslopeinclinedatanangle comparedtohorizontal.Assumeforsimplicitythatthetreatment ofkineticfrictiongiveninchapter5isappropriatehere,althougha softandwetsurfaceactuallybehavesalittledierently.Thecoecientofkineticfrictionactingbetweentheskisandthesnowis k andinadditiontheskierexperiencesanairfrictionforceofmagnitude bv 2 ,where b isaconstant. aFindthemaximumspeedthattheskierwillattain,intermsof thevariables m g k ,and b p bForanglesbelowacertainminimumangle min ,theequation givesaresultthatisnotmathematicallymeaningful.Findanequationfor min ,andgiveaphysicalexplanationofwhatishappening for < min 7 Agunisaimedhorizontallytothewest,andredat t =0.The bullet'spositionvectorasafunctionoftimeis r = b ^ x + ct ^ y + dt 2 ^ z where b c ,and d arepositiveconstants. aWhatunitswould b c ,and d needtohavefortheequationto makesense? bFindthebullet'svelocityandaccelerationasfunctionsoftime. cGivephysicalinterpretationsof b c d ^ x ^ y ,and ^ z R 8 AnnieOakley,ridingnorthonhorsebackat30mi/hr,shoots herrie,aiminghorizontallyandtothenortheast.Themuzzlespeed oftherieis140mi/hr.Whenthebullethitsadefenselessfuzzy animal,whatisitsspeedofimpact?Neglectairresistance,and ignoretheverticalmotionofthebullet. Solution,p.275 9 AcargoplanehastakenofromatinyairstripintheAndes, andisclimbingatconstantspeed,atanangleof =17 withrespect tohorizontal.Itsenginessupplyathrustof F thrust =200kN,and theliftfromitswingsis F lift =654kN.Assumethatairresistance dragisnegligible,sotheonlyforcesactingarethrust,lift,and weight.Whatisitsmass,inkg? Solution,p.276 10 Awagonisbeingpulledatconstantspeedupaslope bya ropethatmakesanangle withthevertical. aAssumingnegligiblefriction,showthatthetensionintherope 212 Chapter8VectorsandMotion

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Problem13MillikanandGale, 1920. Problem12. isgivenbytheequation F T = sin sin + F W where F W istheweightforceactingonthewagon. bInterpretthisequationinthespecialcasesof =0and = 180 )]TJ/F20 10.9091 Tf 10.909 0 Td [( Solution,p.276 11 Theangleofreposeisthemaximumslopeonwhichanobject willnotslide.Onairless,geologicallyinertbodieslikethemoonor anasteroid,theonlythingthatdetermineswhetherdustorrubble willstayonaslopeiswhethertheslopeislesssteepthantheangle ofrepose. aFindanequationfortheangleofrepose,decidingforyourself whataretherelevantvariables. bOnanasteroid,where g canbethousandsoftimeslowerthan onEarth,wouldrubblebeabletolieatasteeperangleofrepose? Solution,p.277 12 Thegureshowsanexperimentinwhichacartisreleased fromrestatA,andacceleratesdowntheslopethroughadistance x untilitpassesthroughasensor'slightbeam.Thepointofthe experimentistodeterminethecart'sacceleration.AtB,acardboardvanemountedonthecartentersthelightbeam,blockingthe lightbeam,andstartsanelectronictimerrunning.AtC,thevane emergesfromthebeam,andthetimerstops. aFindthenalvelocityofthecartintermsofthewidth w of thevaneandthetime t b forwhichthesensor'slightbeamwas blocked. p bFindthemagnitudeofthecart'saccelerationintermsofthe measurablequantities x t b ,and w p cAnalyzetheforcesinwhichthecartparticipates,usingatablein theformatintroducedinsection5.3.Assumefrictionisnegligible. dFindatheoreticalvaluefortheaccelerationofthecart,which couldbecomparedwiththeexperimentallyobservedvalueextracted inpart b .Expressthetheoreticalvalueintermsoftheangle of theslope,andthestrength g ofthegravitationaleld. p 13 Thegureshowsaboyhanginginthreepositions:with hisarmsstraightup,withhisarmsat45degrees,andwith hisarmsat60degreeswithrespecttothevertical.Comparethe tensioninhisarmsinthethreecases. Problems 213

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Problem15. 14 Drivingdownahillinclinedatanangle withrespectto horizontal,youslamonthebrakestokeepfromhittingadeer. aAnalyzetheforces.Ignorerollingresistanceandairfriction. bFindthecar'smaximumpossibledeceleration, a expressedas apositivenumber,intermsof g ,andtherelevantcoecientof friction. p cExplainphysicallywhythecar'smasshasnoeectonyour answer. dDiscussthemathematicalbehaviorandphysicalinterpretation ofyourresultfornegativevaluesof eDothesameforverylargepositivevaluesof 15 ThegureshowsthepathfollowedbyHurricaneIrenein 2005asitmovednorth.Thedotsshowthelocationofthecenter ofthestormatsix-hourintervals,withlighterdotsatthetime whenthestormreacheditsgreatestintensity.Findthetimewhen thestorm'scenterhadavelocityvectortothenortheastandan accelerationvectortothesoutheast. 214 Chapter8VectorsandMotion

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Chapter9 CircularMotion 9.1ConceptualFrameworkforCircularMotion InowlivefteenminutesfromDisneyland,somyfriendsandfamily inmynativeNorthernCaliforniathinkit'salittlestrangethatI've nevervisitedtheMagicKingdomagainsinceachildhoodtriptothe south.Thetruthisthatformeasapreschooler,Disneylandwas nottheHappiestPlaceonEarth.Mymothertookmeonaridein whichlittlecarsshapedlikerocketshipscircledrapidlyarounda centralpillar.IknewIwasgoingtodie.Therewasaforcetryingto throwmeoutward,andthesafetyfeaturesoftheridewouldsurely havebeeninadequateifIhadn'tscreamedthewholetimetomake sureMomwouldholdontome.Afterward,sheseemedsurprisingly indierenttotheextremedangerwehadexperienced. Circularmotiondoesnotproduceanoutwardforce Myyoungerself'sunderstandingofcircularmotionwaspartly rightandpartlywrong.Iwaswronginbelievingthattherewasa forcepullingmeoutward,awayfromthecenterofthecircle.The easiestwaytounderstandthisistobringbacktheparableofthe bowlingballinthepickuptruckfromchapter4.Asthetruckmakes aleftturn,thedriverlooksintherearviewmirrorandthinksthat somemysteriousforceispullingtheballoutward,butthetruck isaccelerating,sothedriver'sframeofreferenceisnotaninertial frame.Newton'slawsareviolatedinanoninertialframe,sotheball appearstoacceleratewithoutanyactualforceactingonit.Because weareusedtoinertialframes,inwhichaccelerationsarecausedby 215

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b / Thiscraney'shalteres helpittomaintainitsorientation inight. forces,theball'saccelerationcreatesavividillusionthattheremust beanoutwardforce. a / 1.Intheturningtruck'sframe ofreference,theballappearsto violateNewton'slaws,displayingasidewaysaccelerationthat isnotthere-sultofaforceinteractionwithanyotherobject. 2.Inaninertialframeofreference,suchastheframexedto theearth'ssurface,theballobeys Newton'srstlaw.Noforcesare actingonit,anditcontinuesmovinginastraightline.Itisthetruck thatisparticipatinginaninteractionwiththeasphalt,thetruckthat acceleratesasitshouldaccording toNewton'ssecondlaw. Inaninertialframeeverythingmakesmoresense.Theballhas noforceonit,andgoesstraightasrequiredbyNewton'srstlaw. Thetruckhasaforceonitfromtheasphalt,andrespondstoit byacceleratingchangingthedirectionofitsvelocityvectoras Newton'ssecondlawsaysitshould. Thehalteresexample1 Anotherinterestingexampleisaninsectorgancalledthehalteres,apairofsmallknobbedlimbsbehindthewings,whichvibrateupanddownandhelptheinsecttomaintainitsorientation inight.Thehalteresevolvedfromasecondpairofwingspossessedbyearlierinsects.Suppose,forexample,thatthehalteres areontheirupwardstroke,andatthatmomentanaircurrent causestheytopitchitsnosedown.ThehalteresfollowNewton'srstlaw,continuingtorisevertically,butinthey'srotating frameofreference,itseemsasthoughtheyhavebeensubjected toabackwardforce.Theyhasspecialsensoryorgansthatperceivethistwist,andhelpittocorrectitselfbyraisingitsnose. Circularmotiondoesnotpersistwithoutaforce Iwascorrect,however,onadierentpointabouttheDisneyland ride.Tomakemecurvearoundwiththecar,Ireallydidneedsome forcesuchasaforcefrommymother,frictionfromtheseat,ora normalforcefromthesideofthecar.Infact,allthreeforceswere probablyaddingtogether.OneofthereasonswhyGalileofailedto 216 Chapter9CircularMotion

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c / 1.Anoverheadviewofapersonswingingarockonarope. Aforcefromthestringisrequiredtomaketherock'svelocity vectorkeepchangingdirec-tion. 2.Ifthestringbreaks,therock willfollowNewton'srstlawand gostraightinsteadofcontinuing aroundthecircle. renetheprincipleofinertiaintoaquantitativestatementlikeNewton'srstlawisthathewasnotsurewhethermotionwithoutaforce wouldnaturallybecircularorlinear.Infact,themostimpressive examplesheknewofthepersistenceofmotionweremostlycircular: thespinningofatoportherotationoftheearth,forexample.Newtonrealizedthatinexamplessuchasthese,therereallywereforces atwork.Atomsonthesurfaceofthetoparepreventedfromying ostraightbytheordinaryforcethatkeepsatomsstucktogetherin solidmatter.Theearthisnearlyallliquid,butgravitationalforces pullallitspartsinward. Uniformandnonuniformcircularmotion Circularmotionalwaysinvolvesachangeinthedirectionofthe velocityvector,butitisalsopossibleforthemagnitudeofthevelocitytochangeatthesametime.Circularmotionisreferredtoas uniform if j v j isconstant,and nonuniform ifitischanging. Yourspeedometertellsyouthemagnitudeofyourcar'svelocity vector,sowhenyougoaroundacurvewhilekeepingyourspeedometerneedlesteady,youareexecutinguniformcircularmotion.Ifyour speedometerreadingischangingasyouturn,yourcircularmotion isnonuniform.Uniformcircularmotionissimplertoanalyzemathematically,sowewillattackitrstandthenpasstothenonuniform case. self-checkA Whichoftheseareexamplesofuniformcircularmotionandwhichare nonuniform? theclothesinaclothesdryerassumingtheyremainagainstthe insideofthedrum,evenatthetop arockontheendofastringbeingwhirledinaverticalcircle Answer,p.268 Section9.1ConceptualFrameworkforCircularMotion 217

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f / Whenacarisgoingstraight atconstantspeed,theforward andbackwardforcesonitare cancelingout,producingatotal forceofzero.Whenitmoves inacircleatconstantspeed, therearethreeforcesonit,but theforwardandbackwardforces cancelout,sothevectorsumis aninwardforce. d / Tomakethebrickgoina circle,Ihadtoexertaninward forceontherope. Onlyaninwardforceisrequiredforuniformcircularmotion. Figurecshowedthestringpullinginstraightalongaradiusof thecircle,butmanypeoplebelievethatwhentheyaredoingthis theymustbeleading"therockalittletokeepitmovingalong. Thatis,theybelievethattheforcerequiredtoproduceuniform circularmotionisnotdirectlyinwardbutataslightangletothe radiusofthecircle.Thisintuitionisincorrect,whichyoucaneasily verifyforyourselfnowifyouhavesomestringhandy.Itisonly whileyouaregettingtheobjectgoingthatyourforceneedstobeat anangletotheradius.Duringthisinitialperiodofspeedingup,the motionisnotuniform.Onceyousettledownintouniformcircular motion,youonlyapplyaninwardforce. Ifyouhavenotdonetheexperimentforyourself,hereisatheoreticalargumenttoconvinceyouofthisfact.Wehavediscussedin chapter6theprinciplethatforceshavenoperpendiculareects.To keeptherockfromspeedinguporslowingdown,weonlyneedto makesurethatourforceisperpendiculartoitsdirectionofmotion. Wearethenguaranteedthatitsforwardmotionwillremainunaffected:ourforcecanhavenoperpendiculareect,andthereisno otherforceactingontherockwhichcouldslowitdown.Therock requiresnoforwardforcetomaintainitsforwardmotion,anymore thanaprojectileneedsahorizontalforcetohelpitoverthetop" ofitsarc. e / Aseriesofthreehammertapsmakestherollingballtraceatriangle,sevenhammersaheptagon.Ifthenumberofhammerswaslarge enough,theballwouldessentiallybeexperiencingasteadyinwardforce, anditwouldgoinacircle.Innocaseisanyforwardforcenecessary. 218 Chapter9CircularMotion

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g / Example2. Why,then,doesacardrivingincirclesinaparkinglotstop executinguniformcircularmotionifyoutakeyourfootothegas? ThesourceofconfusionhereisthatNewton'slawspredictanobject'smotionbasedonthe total forceactingonit.Acardrivingin circleshasthreeforcesonit aninwardforcefromtheasphalt,controlledwiththesteering wheel; aforwardforcefromtheasphalt,controlledwiththegas pedal;and backwardforcesfromairresistanceandrollingresistance. Youneedtomakesurethereisaforwardforceonthecarsothat thebackwardforceswillbeexactlycanceledout,creatingavector sumthatpointsdirectlyinward. Amotorcyclemakingaturnexample2 Themotorcyclistinguregismovingalonganarcofacircle.It lookslikehe'schosentoridetheslantedsurfaceofthedirtata placewhereitmakesjusttheanglehewants,allowinghimtoget theforceheneedsonthetiresasanormalforce,withoutneeding anyfrictionalforce.Thedirt'snormalforceonthetirespointsup andtoourleft.Theverticalcomponentofthatforceiscanceled bygravity,whileitshorizontalcomponentcauseshimtocurve. Inuniformcircularmotion,theaccelerationvectorisinward Sinceexperimentsshowthattheforcevectorpointsdirectly inward,Newton'ssecondlawimpliesthattheaccelerationvector pointsinwardaswell.Thisfactcanalsobeprovenonpurelykinematicalgrounds,andwewilldosointhenextsection. Section9.1ConceptualFrameworkforCircularMotion 219

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DiscussionquestionsA-D DiscussionquestionE. DiscussionQuestions A Inthegameofcrackthewhip,alineofpeoplestandholdinghands, andthentheystartsweepingoutacircle.Onepersonisatthecenter,and rotateswithoutchanginglocation.Attheoppositeendisthepersonwho isrunningthefastest,inawidecircle.Inthisgame,someonealwaysends uplosingtheirgripandyingoff.Supposethepersonontheendloses hergrip.Whatpathdoesshefollowasshegoesyingoff?Assumeshe isgoingsofastthatsheisreallyjusttryingtoputonefootinfrontofthe otherfastenoughtokeepfromfalling;sheisnotabletogetanysignicant horizontalforcebetweenherfeetandtheground. B Supposethepersonontheoutsideisstillholdingon,butfeelsthat shemayloosehergripatanymoment.Whatforceorforcesareacting onher,andinwhatdirectionsarethey?Wearenotinterestedinthe verticalforces,whicharetheearth'sgravitationalforcepullingdown,and theground'snormalforcepushingup. C Supposethepersonontheoutsideisstillholdingon,butfeelsthat shemayloosehergripatanymoment.Whatiswrongwiththefollowing analysisofthesituation?Thepersonwhosehandshe'sholdingexerts aninwardforceonher,andbecauseofNewton'sthirdlaw,there'san equalandoppositeforceactingoutward.Thatoutwardforceistheone shefeelsthrowingheroutward,andtheoutwardforceiswhatmightmake hergoyingoff,ifit'sstrongenough. D Iftheonlyforcefeltbythepersonontheoutsideisaninwardforce, whydoesn'tshegostraightin? E Intheamusementparkrideshowninthegure,thecylinderspins fasterandfasteruntilthecustomercanpickherfeetupofftheoorwithoutfalling.IntheoldConeyIslandversionoftheride,theooractually droppedoutlikeatrapdoor,showingtheoceanbelow.Thereisalsoa versioninwhichthewholethingtiltsupdiagonally,butwe'rediscussing theversionthatstaysat.Ifthereisnooutwardforceactingonher,why doesshesticktothewall?Analyzealltheforcesonher. F Whatisanexampleofcircularmotionwheretheinwardforceisa normalforce?Whatisanexampleofcircularmotionwheretheinward forceisfriction?Whatisanexampleofcircularmotionwheretheinward forceisthesumofmorethanoneforce? G Doestheaccelerationvectoralwayschangecontinuouslyincircular motion?Thevelocityvector? 220 Chapter9CircularMotion

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h / Thelawofsines. i / Deriving j a j = j v j 2 = r for uniformcircularmotion. 9.2UniformCircularMotion InthissectionIderiveasimpleandveryusefulequationfor themagnitudeoftheaccelerationofanobjectundergoingconstant acceleration.Thelawofsinesisinvolved,soI'verecappeditin gureh. Thederivationisbrief,butthemethodrequiressomeexplanationandjustication.Theideaistocalculatea v vectordescribing thechangeinthevelocityvectorastheobjectpassesthroughan angle .Wethencalculatetheacceleration, a = v = t .Theastutereaderwillrecall,however,thatthisequationisonlyvalidfor motionwithconstantacceleration.Althoughthemagnitudeofthe accelerationisconstantforuniformcircularmotion,theacceleration vectorchangesitsdirection,soitisnotaconstantvector,andthe equation a = v = t doesnotapply.Thejusticationforusingit isthatwewillthenexamineitsbehaviorwhenwemakethetime intervalveryshort,whichmeansmakingtheangle verysmall.For smallerandsmallertimeintervals,the v = t expressionbecomes abetterandbetterapproximation,sothatthenalresultofthe derivationisexact. Ingurei/1,theobjectsweepsoutanangle .Itsdirectionof motionalsotwistsaroundbyanangle ,fromtheverticaldashed linetothetiltedone.Figurei/2showstheinitialandnalvelocity vectors,whichhaveequalmagnitude,butdirectionsdieringby Ini/3,I'vereassembledthevectorsintheproperpositionsforvector subtraction.Theyformanisoscelestrianglewithinteriorangles ,and .Eta, ,ismyfavoriteGreekletter.Thelawofsines gives j v j sin = j v j sin Thistellsusthemagnitudeof v ,whichisoneofthetwoingredients weneedforcalculatingthemagnitudeof a = v = t .Theother ingredientis t .Thetimerequiredfortheobjecttomovethrough theangle is t = lengthofarc j v j Nowifwemeasureouranglesinradianswecanusethedenitionof radianmeasure,whichisangle=lengthofarc = radius,giving t = r= j v j .Combiningthiswiththerstexpressioninvolving j v j gives j a j = j v j = t = j v j 2 r sin 1 sin When becomesverysmall,thesmall-angleapproximationsin applies,andalso becomescloseto90 ,sosin 1,andwehave Section9.2UniformCircularMotion 221

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j / Example6. anequationfor j a j : j a j = j v j 2 r .[uniformcircularmotion] Forcerequiredtoturnonabikeexample3 Abicyclistismakingaturnalonganarcofacirclewithradius 20m,ataspeedof5m/s.Ifthecombinedmassofthecyclist plusthebikeis60kg,howgreatastaticfrictionforcemustthe roadbeabletoexertonthetires? TakingthemagnitudesofbothsidesofNewton'ssecondlaw gives j F j = j m a j = m j a j Substituting j a j = j v j 2 = r gives j F j = m j v j 2 = r 80N roundedofftoonesigg. Don'thugthecenterlineonacurve!example4 You'redrivingonamountainroadwithasteepdroponyour right.Whenmakingaleftturn,isitsafertohugthecenterlineor tostayclosertotheoutsideoftheroad? Youwantwhicheverchoiceinvolvestheleastacceleration,becausethatwillrequiretheleastforceandentailtheleastriskof exceedingthemaximumforceofstaticfriction.Assumingthe curveisanarcofacircleandyourspeedisconstant,yourcar isperforminguniformcircularmotion,with j a j = j v j 2 = r .Thedependenceonthesquareofthespeedshowsthatdrivingslowly isthemainsafetymeasureyoucantake,butforanygivenspeed youalsowanttohavethelargestpossiblevalueof r .Eventhough yourinstinctistokeepawayfromthatscaryprecipice,youareactuallylesslikelytoskidifyoukeeptowardtheoutside,because thenyouaredescribingalargercircle. Accelerationrelatedtoradiusandperiodofrotationexample5 Howcantheequationfortheaccelerationinuniformcircular motionberewrittenintermsoftheradiusofthecircleandthe period, T ,ofthemotion,i.e.,thetimerequiredtogoaroundonce? Theperiodcanberelatedtothespeedasfollows: j v j = circumference T =2 r = T Substitutingintotheequation j a j = j v j 2 = r gives j a j = 4 2 r T 2 222 Chapter9CircularMotion

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Aclothesdryerexample6 Myclothesdryerhasadrumwithaninsideradiusof35cm,and itspinsat48revolutionsperminute.Whatistheaccelerationof theclothesinside? Wecansolvethisbyndingtheperiodandpluggingintothe resultofthepreviousexample.Ifitmakes48revolutionsinone minute,thentheperiodis1/48ofaminute,or1.25s.Togetan accelerationinmksunits,wemustconverttheradiusto0.35m. Pluggingin,theresultis8.8m = s 2 Moreaboutclothesdryers!example7 Inadiscussionquestionintheprevioussection,wemadethe assumptionthattheclothesremainagainsttheinsideofthedrum astheygooverthetop.Inlightofthepreviousexample,isthisa correctassumption? No.Weknowthattheremustbesomeminimumspeedatwhich themotorcanrunthatwillresultintheclothesjustbarelystayingagainsttheinsideofthedrumastheygooverthetop.Ifthe clothesdryerranatjustthisminimumspeed,thentherewouldbe nonormalforceontheclothesatthetop:theywouldbeonthe vergeoflosingcontact.Theonlyforceactingonthematthetop wouldbetheforceofgravity,whichwouldgivethemanaccelerationof g =9.8m = s 2 .Theactualdryermustberunningslower thanthisminimumspeed,becauseitproducesanaccelerationof only8.8m = s 2 .Mytheoryisthatthisisdoneintentionally,tomake theclothesmixandtumble. Solvedproblem:Thetilt-a-whirlpage227,problem6 Solvedproblem:Anoff-ramppage227,problem7 DiscussionQuestions A Acertainamountofforceisneededtoprovidetheaccelerationof circularmotion.Whatifwereareexertingaforceperpendiculartothe directionofmotioninanattempttomakeanobjecttraceacircleofradius r ,buttheforceisn'tasbigas m j v j 2 = r ? B Supposearotatingspacestation,asingurekonpage224,isbuilt. Itgivesitsoccupantstheillusionofordinarygravity.Whathappenswhen apersoninthestationletsgoofaball?Whathappenswhenshethrows aballstraightupintheairi.e.,towardsthecenter? Section9.2UniformCircularMotion 223

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l / 1.Movinginacirclewhile speedingup.2.Uniformcircular motion.3.Slowingdown. k / DiscussionquestionB.Anartist'sconceptionofarotatingspace colonyintheformofagiantwheel.Apersonlivinginthisnoninertial frameofreferencehasanillusionofaforcepullingheroutward,toward thedeck,forthesamereasonthatapersoninthepickuptruckhasthe illusionofaforcepullingthebowlingball.Byadjustingthespeedofrotation,thedesignerscanmakeanacceleration j v j 2 = r equaltotheusual accelerationofgravityonearth.Onearth,youraccelerationstandingon thegroundiszero,andafallingrockheadsforyourfeetwithanaccelerationof9.8m = s 2 .Apersonstandingonthedeckofthespacecolonyhas an upward accelerationof9.8m = s 2 ,andwhensheletsgoofarock,her feethead up atthenonacceleratingrock.Toher,itseemsthesameas truegravity. 9.3NonuniformCircularMotion Whataboutnonuniformcircularmotion?Althoughsofarwe havebeendiscussingcomponentsofvectorsalongxed x and y axes,itnowbecomesconvenienttodiscusscomponentsoftheaccelerationvectoralongtheradiallinein-outandthetangentialline alongthedirectionofmotion.Fornonuniformcircularmotion, theradialcomponentoftheaccelerationobeysthesameequation asforuniformcircularmotion, a r = j v j 2 =r buttheaccelerationvectoralsohasatangentialcomponent, a t =slopeofthegraphof j v j versus t Thelatterquantityhasasimpleinterpretation.Ifyouaregoing aroundacurveinyourcar,andthespeedometerneedleismoving,thetangentialcomponentoftheaccelerationvectorissimply whatyouwouldhavethoughttheaccelerationwasifyousawthe speedometeranddidn'tknowyouweregoingaroundacurve. Slowdownbeforeaturn,notduringit.example8 Whenyou'remakingaturninyourcarandyou'reafraidyou mayskid,isn'titagoodideatoslowdown? Iftheturnisanarcofacircle,andyou'vealreadycompleted partoftheturnatconstantspeedwithoutskidding,thentheroad andtiresareapparentlycapableofenoughstaticfrictiontosupplyanaccelerationof j v j 2 = r .Thereisnoreasonwhyyouwould skidoutnowifyouhaven'talready.Ifyougetnervousandbrake, however,thenyouneedtohaveatangentialaccelerationcomponentinadditiontotheradialcomponentyouwerealreadyable toproducesuccessfully.Thiswouldrequireanaccelerationvectorwithagreatermagnitude,whichinturnwouldrequirealarger force.Staticfrictionmightnotbeabletosupplythatmuchforce, andyoumightskidout.Asinthepreviousexampleonasimilar topic,thesafethingtodoistoapproachtheturnatacomfortably lowspeed. Solvedproblem:Abikeracepage226,problem5 224 Chapter9CircularMotion

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Summary SelectedVocabulary uniformcircular motion...... circularmotioninwhichthemagnitudeofthe velocityvectorremainsconstant nonuniformcircularmotion.... circularmotioninwhichthemagnitudeofthe velocityvectorchanges radial.......paralleltotheradiusofacircle;thein-out direction tangential....tangenttothecircle,perpendiculartotheradialdirection Notation a r .........radialacceleration;thecomponentoftheaccelerationvectoralongthein-outdirection a t .........tangentialacceleration;thecomponentofthe accelerationvectortangenttothecircle Summary Ifanobjectistohavecircularmotion,aforcemustbeexertedon ittowardthecenterofthecircle.Thereisnooutwardforceonthe object;theillusionofanoutwardforcecomesfromourexperiences inwhichourpointofviewwasrotating,sothatwewereviewing thingsinanoninertialframe. Anobjectundergoinguniformcircularmotionhasaninward accelerationvectorofmagnitude j a j = j v j 2 =r Innonuniformcircularmotion,theradialandtangentialcomponentsoftheaccelerationvectorare a r = j v j 2 =r a t =slopeofthegraphof j v j versus t Summary 225

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Problem5. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Whenyou'redoneusinganelectricmixer,youcangetmost ofthebatteroofthebeatersbyliftingthemoutofthebatterwith themotorrunningatahighenoughspeed.Let'simagine,tomake thingseasiertovisualize,thatweinsteadhaveapieceoftapestuck tooneofthebeaters. aExplainwhystaticfrictionhasnoeectonwhetherornotthe tapeieso. bSupposeyoundthatthetapedoesn'tyowhenthemotor isonalowspeed,butatagreaterspeed,thetapewon'tstayon. Whywouldthegreaterspeedchangethings? 2 Showthattheexpression j v j 2 =r hastheunitsofacceleration. 3 Aplaneisowninaloop-the-loopofradius1.00km.The planestartsoutyingupside-down,straightandlevel,thenbegins curvingupalongthecircularloop,andisright-sideupwhenit reachesthetop.Theplanemayslowdownsomewhatontheway up.Howfastmusttheplanebegoingatthetopifthepilotisto experiencenoforcefromtheseatortheseatbeltwhileatthetopof theloop? p 4 Inthisproblem,you'llderivetheequation j a j = j v j 2 =r usingcalculus.Insteadofcomparingvelocitiesattwopointsinthe particle'smotionandthentakingalimitwherethepointsareclose together,you'lljusttakederivatives.Theparticle'spositionvector is r = r cos ^ x + r sin ^ y ,whereandaretheunitvectorsalong the x and y axes.Bythedenitionofradians,thedistancetraveled since t =0is r ,soiftheparticleistravelingatconstantspeed v = j v j ,wehave v = r /t. aEliminate togettheparticle'spositionvectorasafunctionof time. bFindtheparticle'saccelerationvector. cShowthatthemagnitudeoftheaccelerationvectorequals v 2 =r R 5 Threecyclistsinaraceareroundingasemicircularcurve. Atthemomentdepicted,cyclistAisusingherbrakestoapplya forceof375Ntoherbike.CyclistBiscoasting.CyclistCis pedaling,resultinginaforceof375NonherbikeEachcyclist, withherbike,hasamassof75kg.Attheinstantshown,the instantaneousspeedofallthreecyclistsis10m/s.Onthediagram, draweachcyclist'saccelerationvectorwithitstailontopofher presentposition,indicatingthedirectionsandlengthsreasonably accurately.Indicateapproximatelytheconsistentscaleyouareusing 226 Chapter9CircularMotion

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Problem9. Problem7. Problem6. forallthreeaccelerationvectors.Extremeprecisionisnotnecessary aslongasthedirectionsareapproximatelyright,andlengthsof vectorsthatshouldbeequalappearroughlyequal,etc.Assumeall threecyclistsaretravelingalongtheroadallthetime,notwandering acrosstheirlaneorwipingoutandgoingotheroad. Solution,p.277 6 Theamusementparkrideshowninthegureconsistsofa cylindricalroomthatrotatesaboutitsverticalaxis.Whentherotationisfastenough,apersonagainstthewallcanpickhisorher feetupotheoorandremainstuck"tothewallwithoutfalling. aSupposetherotationresultsinthepersonhavingaspeed v .The radiusofthecylinderis r ,theperson'smassis m ,thedownward accelerationofgravityis g ,andthecoecientofstaticfrictionbetweenthepersonandthewallis s .Findanequationforthespeed, v ,required,intermsoftheothervariables.Youwillndthatone ofthevariablescancelsout. bNowsupposetwopeopleareridingtheride.Huyiswearing denim,andGinaiswearingpolyester,soHuy'scoecientofstatic frictionisthreetimesgreater.Theridestartsfromrest,andasit beginsrotatingfasterandfaster,Ginamustwaitlongerbeforebeing abletoliftherfeetwithoutslidingtotheoor.Basedonyourequationfromparta,howmanytimesgreatermustthespeedbebefore Ginacanliftherfeetwithoutslidingdown? Solution,p.277 ? 7 Anengineerisdesigningacurvedo-rampforafreeway. Sincetheo-rampiscurved,shewantstobankittomakeitless likelythatmotoristsgoingtoofastwillwipeout.Iftheradiusof thecurveis r ,howgreatshouldthebankingangle, ,besothat foracargoingataspeed v ,nostaticfrictionforcewhatsoeveris requiredtoallowthecartomakethecurve?Stateyouranswerin termsof v r ,and g ,andshowthatthemassofthecarisirrelevant. Solution,p.277 8 Lionelbrandtoytrainscomewithsectionsoftrackinstandard lengthsandshapes.Forcirculararcs,themostcommonlyused sectionshavediametersof662and1067mmattheinsideoftheouter rail.Themaximumspeedatwhichatraincantakethebroader curvewithoutyingothetracksis0.95m/s.Atwhatspeedmust thetrainbeoperatedtoavoidderailingonthetightercurve? p 9 Thegureshowsaballontheendofastringoflength L attachedtoaverticalrodwhichisspunaboutitsverticalaxisbya motor.Theperiodtimeforonerotationis P aAnalyzetheforcesinwhichtheballparticipates. bFindhowtheangle dependson P g ,and L .[Hints: WritedownNewton'ssecondlawfortheverticalandhorizontal componentsofforceandacceleration.Thisgivestwoequations, whichcanbesolvedforthetwounknowns, andthetensionin thestring.Ifyouintroducevariableslike v and r ,relatethem tothevariablesyoursolutionissupposedtocontain,andeliminate Problems 227

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Problem12. Problem11. Problem10. them.] p cWhathappensmathematicallytoyoursolutionifthemotoris runveryslowlyverylargevaluesof P ?Physically,whatdoyou thinkwouldactuallyhappeninthiscase? 10 PsychologyprofessorR.O.Dentrequestsfundingforanexperimentoncompulsivethrill-seekingbehaviorinhamsters,inwhich thesubjectistobeattachedtotheendofaspringandwhirled aroundinahorizontalcircle.Thespringhasequilibriumlength b andobeysHooke'slawwithspringconstant k .Itisstienoughto keepfrombendingsignicantlyunderthehamster'sweight. aCalculatethelengthofthespringwhenitisundergoingsteady circularmotioninwhichonerotationtakesatime T .Expressyour resultintermsof k m b T ,andthehamster'smass m p bTheethicscommitteesomehowfailstovetotheexperiment,but thesafetycommitteeexpressesconcern.Why?Doesyourequationdoanythingunusual,orevenspectacular,foranyparticular valueof T ?Whatdoyouthinkisthephysicalsignicanceofthis mathematicalbehavior? 11 Thegureshowsanold-fashioneddevicecalledayball governor,usedforkeepinganenginerunningatthecorrectspeed. Thewholethingrotatesabouttheverticalshaft,andthemass M isfreetoslideupanddown.Thismasswouldhaveaconnection notshowntoavalvethatcontrolledtheengine.If,forinstance, theenginerantoofast,themasswouldrise,causingtheengineto slowbackdown. aShowthatinthespecialcaseof a =0,theangle isgivenby =cos )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 g m + M P 2 4 2 mL where P istheperiodofrotationtimerequiredforonecomplete rotation. bThereisnoclosed-formsolutionfor inthegeneralcasewhere a isnotzero.However,explainhowtheundesirablelow-speedbehaviorofthe a =0devicewouldbeimprovedbymaking a nonzero. ? 12 Thegureshowstwoblocksofmasses m 1 and m 2 sliding incirclesonafrictionlesstable.Findthetensioninthestringsif theperiodofrotationtimerequiredforonecompleterotationis P p 13 Theaccelerationofanobjectinuniformcircularmotioncan begiveneitherby j a j = j v j 2 =r or,equivalently,by j a j =4 2 r=T 2 where T isthetimerequiredforonecycleexample5onpage222. PersonAsaysbasedontherstequationthattheaccelerationin circularmotionisgreaterwhenthecircleissmaller.PersonB,arguingfromthesecondequation,saysthattheaccelerationissmaller whenthecircleissmaller.Rewritethetwostatementssothatthey arelessmisleading,eliminatingthesupposedparadox.[Basedona problembyArnoldArons.] 228 Chapter9CircularMotion

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a / JohannesKeplerfounda mathematicaldescriptionofthe motionoftheplanets,whichled toNewton'stheoryofgravity. Gravityistheonlyreallyimportantforceonthecosmicscale.Thisfalsecolorrepresentationofsaturn'sringswasmadefromanimagesentback bytheVoyager2spaceprobe.Theringsarecomposedofinnumerable tinyiceparticlesorbitingincirclesundertheinuenceofsaturn'sgravity. Chapter10 Gravity Cruiseyourradiodialtodayandtrytondanypopularsongthat wouldhavebeenimaginablewithoutLouisArmstrong.Byintroducingsoloimprovisationintojazz,Armstrongtookapartthejigsaw puzzleofpopularmusicandtthepiecesbacktogetherinadifferentway.Inthesameway,Newtonreassembledourviewofthe universe.Considerthetitlesofsomerecentphysicsbookswritten forthegeneralreader:TheGodParticle,DreamsofaFinalTheory.Whenthesubatomicparticlecalledtheneutrinowasrecently provenforthersttimetohavemass,specialistsincosmologybegandiscussingseriouslywhateectthiswouldhaveoncalculations oftheultimatefateoftheuniverse:wouldtheneutrinos'masscause enoughextragravitationalattractiontomaketheuniverseeventuallystopexpandingandfallbacktogether?WithoutNewton,such attemptsatuniversalunderstandingwouldnotmerelyhaveseemed alittlepretentious,theysimplywouldnothaveoccurredtoanyone. ThischapterisaboutNewton'stheoryofgravity,whichheused toexplainthemotionoftheplanetsastheyorbitedthesun.Whereas 229

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b / TychoBrahemadehisname asanastronomerbyshowingthat thebrightnewstar,todaycalled asupernova,thatappearedin theskiesin1572wasfarbeyond theEarth'satmosphere.This, alongwithGalileo'sdiscoveryof sunspots,showedthatcontrary toAristotle,theheavenswere notperfectandunchanging. Brahe'sfameasanastronomer broughthimpatronagefromKing FrederickII,allowinghimtocarry outhishistorichigh-precision measurementsoftheplanets' motions.Acontradictorycharacter,Braheenjoyedlecturingother noblesabouttheevilsofdueling, buthadlosthisownnoseina youthfulduelandhaditreplaced withaprosthesismadeofan alloyofgoldandsilver.Willingto endurescandalinordertomarry apeasant,heneverthelessused thefeudalpowersgiventohimby thekingtoimposeharshforced laborontheinhabitantsofhis parishes.Theresultoftheirwork, anItalian-stylepalacewithan observatoryontop,surelyranks asoneofthemostluxurious sciencelabseverbuilt.Hedied ofarupturedbladderafterfalling fromawagononthewayhome fromapartyinthosedays,it wasconsideredrudetoleavethe dinnertabletorelieveoneself. thisbookhasconcentratedonNewton'slawsofmotion,leaving gravityasadessert,Newtontossesothelawsofmotioninthe rst20pagesofthePrincipiaMathematicaandthenspendsthe next130discussingthemotionoftheplanets.Clearlyhesawthis asthecrucialscienticfocusofhiswork.Why?Becauseinithe showedthatthesamelawsofmotionappliedtotheheavensasto theearth,andthatthegravitationalforcethatmadeanapplefall wasthesameastheforcethatkepttheearth'smotionfromcarrying itawayfromthesun.WhatwasradicalaboutNewtonwasnothis lawsofmotionbuthisconceptofauniversalscienceofphysics. 10.1Kepler'sLaws Newtonwouldn'thavebeenabletogureout why theplanets movethewaytheydoifithadn'tbeenfortheastronomerTycho Brahe-1601andhisprotegeJohannesKepler-1630, whotogethercameupwiththerstsimpleandaccuratedescription of how theplanetsactuallydomove.Thedicultyoftheirtaskis suggestedbygurec,whichshowshowtherelativelysimpleorbital motionsoftheearthandMarscombinesothatasseenfromearth Marsappearstobestaggeringinloopslikeadrunkensailor. c / AstheEarthandMarsrevolvearoundthesunatdifferentrates, thecombinedeffectoftheirmotionsmakesMarsappeartotracea strange,loopedpathacrossthebackgroundofthedistantstars. Brahe,thelastofthegreatnaked-eyeastronomers,collectedex230 Chapter10Gravity

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tensivedataonthemotionsoftheplanetsoveraperiodofmany years,takingthegiantstepfromthepreviousobservations'accuracy ofabout10minutesofarc/60ofadegreetoanunprecedented 1minute.Thequalityofhisworkisallthemoreremarkableconsideringthathisobservatoryconsistedoffourgiantbrassprotractors mounteduprightinhiscastleinDenmark.Fourdierentobservers wouldsimultaneouslymeasurethepositionofaplanetinorderto checkformistakesandreducerandomerrors. WithBrahe'sdeath,itfelltohisformerassistantKeplertotry tomakesomesenseoutofthevolumesofdata.Kepler,incontradictiontohislateboss,hadformedaprejudice,acorrectone asitturnedout,infavorofthetheorythattheearthandplanets revolvedaroundthesun,ratherthantheearthstayingxedand everythingrotatingaboutit.Althoughmotionisrelative,itisnot justamatterofopinionwhatcircleswhat.Theearth'srotation andrevolutionaboutthesunmakeitanoninertialreferenceframe, whichcausesdetectableviolationsofNewton'slawswhenoneattemptstodescribesucientlypreciseexperimentsintheearth-xed frame.Althoughsuchdirectexperimentswerenotcarriedoutuntil the19thcentury,whatconvincedeveryoneofthesun-centeredsysteminthe17thcenturywasthatKeplerwasabletocomeupwith asurprisinglysimplesetofmathematicalandgeometricalrulesfor describingtheplanets'motionusingthesun-centeredassumption. After900pagesofcalculationsandmanyfalsestartsanddead-end ideas,Keplernallysynthesizedthedataintothefollowingthree laws: Kepler'sellipticalorbitlaw Theplanetsorbitthesuninellipticalorbitswiththesunat onefocus. Kepler'sequal-arealaw Thelineconnectingaplanettothesunsweepsoutequalareas inequalamountsoftime. Kepler'slawofperiods Thetimerequiredforaplanettoorbitthesun,calledits period,isproportionaltothelongaxisoftheellipseraisedto the3/2power.Theconstantofproportionalityisthesame foralltheplanets. Althoughtheplanets'orbitsareellipsesratherthancircles,most areveryclosetobeingcircular.Theearth'sorbit,forinstance,is onlyattenedby1.7%relativetoacircle.Inthespecialcaseofa planetinacircularorbit,thetwofocipluraloffocus"coincide atthecenterofthecircle,andKepler'sellipticalorbitlawthussays thatthecircleiscenteredonthesun.Theequal-arealawimplies thataplanetinacircularorbitmovesaroundthesunwithconstant speed.Foracircularorbit,thelawofperiodsthenamountstoa statementthatthetimeforoneorbitisproportionalto r 3 = 2 ,where Section10.1Kepler'sLaws 231

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d / Anellipseisacirclethat hasbeendistortedbyshrinking andstretchingalongperpendicularaxes. e / Anellipsecanbeconstructedbytyingastringtotwo pinsanddrawinglikethiswiththe pencilstretchingthestringtaut. Eachpinconstitutesonefocusof theellipse. f / Ifthetimeintervaltaken bytheplanettomovefromPtoQ isequaltothetimeintervalfrom RtoS,thenaccordingtoKepler's equal-arealaw,thetwoshaded areasareequal.Theplanet ismovingfasterduringinterval RSthanitdidduringPQ,which Newtonlaterdeterminedwasdue tothesun'sgravitationalforce acceleratingit.Theequal-area lawpredictsexactlyhowmuchit willspeedup. r istheradius.Ifalltheplanetsweremovingintheirorbitsatthe samespeed,thenthetimeforoneorbitwouldsimplydependon thecircumferenceofthecircle,soitwouldonlybeproportionalto r totherstpower.Themoredrasticdependenceon r 3 = 2 means thattheouterplanetsmustbemovingmoreslowlythantheinner planets. 10.2Newton'sLawofGravity Thesun'sforceontheplanetsobeysaninversesquarelaw. Kepler'slawswereabeautifullysimpleexplanationofwhatthe planetsdid,buttheydidn'taddresswhytheymovedastheydid. Didthesunexertaforcethatpulledaplanettowardthecenterof itsorbit,or,assuggestedbyDescartes,weretheplanetscirculating inawhirlpoolofsomeunknownliquid?Kepler,workinginthe Aristoteliantradition,hypothesizednotjustaninwardforceexerted bythesunontheplanet,butalsoasecondforceinthedirection ofmotiontokeeptheplanetfromslowingdown.Somespeculated thatthesunattractedtheplanetsmagnetically. OnceNewtonhadformulatedhislawsofmotionandtaught themtosomeofhisfriends,theybegantryingtoconnectthem toKepler'slaws.Itwasclearnowthataninwardforcewouldbe neededtobendtheplanets'paths.Thisforcewaspresumablyan attractionbetweenthesunandeachplanet.Althoughthesundoes accelerateinresponsetotheattractionsoftheplanets,itsmassisso greatthattheeecthadneverbeendetectedbytheprenewtonian astronomers.Sincetheouterplanetsweremovingslowlyalong moregentlycurvingpathsthantheinnerplanets,theiraccelerations wereapparentlyless.Thiscouldbeexplainedifthesun'sforcewas determinedbydistance,becomingweakerforthefartherplanets. Physicistswerealsofamiliarwiththenoncontactforcesofelectricity andmagnetism,andknewthattheyfellorapidlywithdistance, sothismadesense. Intheapproximationofacircularorbit,themagnitudeofthe sun'sforceontheplanetwouldhavetobe [1] F = ma = mv 2 =r Nowalthoughthisequationhasthemagnitude, v ,ofthevelocity vectorinit,whatNewtonexpectedwasthattherewouldbeamore fundamentalunderlyingequationfortheforceofthesunonaplanet, andthatthatequationwouldinvolvethedistance, r ,fromthesun totheobject,butnottheobject'sspeed, v |motiondoesn'tmake objectslighterorheavier. self-checkA Ifeq.[1]reallywasgenerallyapplicable,whatwouldhappentoan objectreleasedatrestinsomeemptyregionofthesolarsystem? Answer,p.268 232 Chapter10Gravity

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g / Themoon'sacceleration is60 2 =3600timessmallerthan theapple's. Equation[1]wasthusausefulpieceofinformationwhichcould berelatedtothedataontheplanetssimplybecausetheplanets happenedtobegoinginnearlycircularorbits,butNewtonwanted tocombineitwithotherequationsandeliminate v algebraicallyin ordertondadeepertruth. Toeliminate v ,Newtonusedtheequation [2] v = circumference T = 2 r T Ofcoursethisequationwouldalsoonlybevalidforplanetsinnearly circularorbits.Pluggingthisintoeq.[1]toeliminate v gives [3] F = 4 2 mr T 2 Thisunfortunatelyhastheside-eectofbringingintheperiod, T whichweexpectonsimilarphysicalgroundswillnotoccurinthe nalanswer.That'swherethecircular-orbitcase, T / r 3 = 2 ,of Kepler'slawofperiodscomesin.Usingittoeliminate T givesa resultthatdependsonlyonthemassoftheplanetanditsdistance fromthesun: F / m=r 2 .[forceofthesunonaplanetofmass m atadistance r fromthesun;same proportionalityconstantforalltheplanets] SinceKepler'slawofperiodsisonlyaproportionality,thenal resultisaproportionalityratherthananequation,andthereisthis nopointinhangingontothefactorof4 2 Asanexample,thetwinplanets"UranusandNeptunehave nearlythesamemass,butNeptuneisabouttwiceasfarfromthe sunasUranus,sothesun'sgravitationalforceonNeptuneisabout fourtimessmaller. self-checkB Fillinthestepsleadingfromequation[3]to F / m = r 2 Answer,p. 269 Theforcesbetweenheavenlybodiesarethesametypeof forceasterrestrialgravity. OK,butwhatkindofforcewasit?Itprobablywasn'tmagnetic, sincemagneticforceshavenothingtodowithmass.Thencame Newton'sgreatinsight.Lyingunderanappletreeandlookingup atthemooninthesky,hesawanapplefall.Mightnottheearth alsoattractthemoonwiththesamekindofgravitationalforce? Themoonorbitstheearthinthesamewaythattheplanetsorbit thesun,somaybetheearth'sforceonthefallingapple,theearth's forceonthemoon,andthesun'sforceonaplanetwereallthesame typeofforce. Section10.2Newton'sLawofGravity 233

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Therewasaneasywaytotestthishypothesisnumerically.Ifit wastrue,thenwewouldexpectthegravitationalforcesexertedby theearthtofollowthesame F / m=r 2 ruleastheforcesexertedby thesun,butwithadierentconstantofproportionalityappropriate totheearth'sgravitationalstrength.Theissuearisesnowofhowto denethedistance, r ,betweentheearthandtheapple.Anapple inEnglandisclosertosomepartsoftheearththantoothers,but supposewetake r tobethedistancefromthecenteroftheearthto theapple,i.e.,theradiusoftheearth.Theissueofhowtomeasure r didnotariseintheanalysisoftheplanets'motionsbecausethe sunandplanetsaresosmallcomparedtothedistancesseparating them.Callingtheproportionalityconstant k ,wehave F earthonapple = km apple =r 2 earth F earthonmoon = km moon =d 2 earth-moon Newton'ssecondlawsays a = F=m ,so a apple = k=r 2 earth a moon = k=d 2 earth-moon TheGreekastronomerHipparchushadalreadyfound2000years beforethatthedistancefromtheearthtothemoonwasabout60 timestheradiusoftheearth,soifNewton'shypothesiswasright, theaccelerationofthemoonwouldhavetobe60 2 =3600timesless thantheaccelerationofthefallingapple. Applying a = v 2 =r totheaccelerationofthemoonyieldedan accelerationthatwasindeed3600timessmallerthan9.8m = s 2 ,and Newtonwasconvincedhehadunlockedthesecretofthemysterious forcethatkeptthemoonandplanetsintheirorbits. Newton'slawofgravity Theproportionality F / m=r 2 forthegravitationalforceonan objectofmass m onlyhasaconsistentproportionalityconstantfor variousobjectsiftheyarebeingactedonbythegravityofthesame object.Clearlythesun'sgravitationalstrengthisfargreaterthan theearth's,sincetheplanetsallorbitthesunanddonotexhibit anyverylargeaccelerationscausedbytheearthorbyoneanother. Whatpropertyofthesungivesititsgreatgravitationalstrength? Itsgreatvolume?Itsgreatmass?Itsgreattemperature?Newton reasonedthatiftheforcewasproportionaltothemassoftheobject beingactedon,thenitwouldalsomakesenseifthedetermining factorinthegravitationalstrengthoftheobjectexertingtheforce wasitsownmass.Assumingtherewerenootherfactorsaecting thegravitationalforce,thentheonlyotherthingneededtomake quantitativepredictionsofgravitationalforceswouldbeaproportionalityconstant.Newtoncalledthatproportionalityconstant G sohereisthecompleteformofthelawofgravityhehypothesized. 234 Chapter10Gravity

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i / Example3.ComputerenhancedimagesofPlutoand Charon,takenbytheHubble SpaceTelescope. h / Studentsoftenhavea hardtimeunderstandingthe physicalmeaningof G .It'sjust aproportionalityconstantthat tellsyouhowstronggravitational forcesare.Ifyoucouldchangeit, allthegravitationalforcesallover theuniversewouldgetstronger orweaker.Numerically,the gravitationalattractionbetween two1-kgmassesseparatedbya distanceof1mis6.67 10 )]TJ/F39 6.9738 Tf 6.227 0 Td [(11 N, andthisiswhat G isinSIunits. Newton'slawofgravity F = Gm 1 m 2 r 2 [gravitationalforcebetweenobjectsofmass m 1 and m 2 ,separatedbyadistance r ; r isnot theradiusofanything] Newtonconceivedofgravityasanattractionbetweenanytwo massesintheuniverse.Theconstant G tellsusthehowmany newtonstheattractiveforceisfortwo1-kgmassesseparatedbya distanceof1m.Theexperimentaldeterminationof G inordinary unitsasopposedtothespecial,nonmetric,unitsusedinastronomy isdescribedinsection10.5.Thisdicultmeasurementwasnot accomplisheduntillongafterNewton'sdeath. Theunitsof G example1 Whataretheunitsof G ? Solvingfor G inNewton'slawofgravitygives G = Fr 2 m 1 m 2 sotheunitsof G mustbeN m 2 = kg 2 .Fullyadornedwithunits,the valueof G is6.67 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(11 N m 2 = kg 2 Newton'sthirdlawexample2 IsNewton'slawofgravityconsistentwithNewton'sthirdlaw? Thethirdlawrequirestwothings.First, m 1 'sforceon m 2 should bethesameas m 2 'sforceon m 1 .Thisworksout,becausethe product m 1 m 2 givesthesameresultifweinterchangethelabels1 and2.Second,theforcesshouldbeinoppositedirections.This conditionisalsosatised,becauseNewton'slawofgravityrefers toanattraction:eachmasspullstheothertowarditself. PlutoandCharonexample3 Pluto'smoonCharonisunusuallylargeconsideringPluto'ssize, givingthemthecharacterofadoubleplanet.Theirmassesare 1.25 10 22 and1.9 x 19 21 kg,andtheiraveragedistancefromone anotheris1.96 10 4 km.Whatisthegravitationalforcebetween them? Ifwewanttousethevalueof G expressedinSImeter-kilogramsecondunits,wersthavetoconvertthedistanceto1.96 10 7 m.Theforceis 6.67 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(11 N m 2 = kg 2 )]TJ/F39 10.9091 Tf 5 -8.836 Td [(1.25 10 22 kg )]TJ/F39 10.9091 Tf 11.818 -8.836 Td [(1.9 10 21 kg )]TJ/F39 10.9091 Tf 5 -8.837 Td [(1.96 10 7 m 2 =4.1 10 18 N Section10.2Newton'sLawofGravity 235

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j / Theconicsectionsarethe curvesmadebycuttingthe surfaceofaninniteconewitha plane. k / Animaginarycannonable toshootcannonballsatveryhigh speedsisplacedontopofan imaginary,verytallmountain thatreachesupabovetheatmosphere.Dependingonthe speedatwhichtheballisred, itmayendupinatightlycurved ellipticalorbit,1,acircularorbit, 2,abiggerellipticalorbit,3,ora nearlystraighthyperbolicorbit,4. Theproportionalityto1 =r 2 inNewton'slawofgravitywasnot entirelyunexpected.Proportionalitiesto1 =r 2 arefoundinmany otherphenomenainwhichsomeeectspreadsoutfromapoint. Forinstance,theintensityofthelightfromacandleisproportional to1 =r 2 ,becauseatadistance r fromthecandle,thelighthasto bespreadoutoverthesurfaceofanimaginarysphereofarea4 r 2 Thesameistruefortheintensityofsoundfromarecracker,orthe intensityofgammaradiationemittedbytheChernobylreactor.It's important,however,torealizethatthisisonlyananalogy.Force doesnottravelthroughspaceassoundorlightdoes,andforceis notasubstancethatcanbespreadthickerorthinnerlikebutteron toast. AlthoughseveralofNewton'scontemporarieshadspeculated thattheforceofgravitymightbeproportionalto1 =r 2 ,noneof them,eventheoneswhohadlearnedNewton'slawsofmotion,had hadanyluckprovingthattheresultingorbitswouldbeellipses,as Keplerhadfoundempirically.Newtondidsucceedinprovingthat ellipticalorbitswouldresultfroma1 =r 2 force,butwepostponethe proofuntiltheendofthenextvolumeofthetextbookbecauseit canbeaccomplishedmuchmoreeasilyusingtheconceptsofenergy andangularmomentum. Newtonalsopredictedthatorbitsintheshapeofhyperbolas shouldbepossible,andhewasright.Somecomets,forinstance, orbitthesuninveryelongatedellipses,butotherspassthrough thesolarsystemonhyperbolicpaths,nevertoreturn.Justasthe trajectoryofafasterbaseballpitchisatterthanthatofamore slowlythrownball,sothecurvatureofaplanet'sorbitdependson itsspeed.Aspacecraftcanbelaunchedatrelativelylowspeed, resultinginacircularorbitabouttheearth,oritcanbelaunched atahigherspeed,givingamoregentlycurvedellipsethatreaches fartherfromtheearth,oritcanbelaunchedataveryhighspeed whichputsitinanevenlesscurvedhyperbolicorbit.Asyougo veryfaroutonahyperbola,itapproachesastraightline,i.e.,its curvatureeventuallybecomesnearlyzero. NewtonalsowasabletoprovethatKepler'ssecondlawsweepingoutequalareasinequaltimeintervalswasalogicalconsequence ofhislawofgravity.Newton'sversionoftheproofismoderately complicated,buttheproofbecomestrivialonceyouunderstandthe conceptofangularmomentum,whichwillbecoveredlaterinthe course.Theproofwillthereforebedeferreduntilsection5.7ofbook 2. self-checkC WhichofKepler'slawswoulditmakesensetoapplytohyperbolicorbits? Answer,p. 269 236 Chapter10Gravity

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. Solvedproblem:VisitingCerespage248,problem10 Solvedproblem:Geosynchronousorbitpage250,problem16 Solvedproblem:Why a equals g page248,problem11 Solvedproblem:IdaandDactylpage249,problem12 Solvedproblem:Anothersolarsystempage249,problem15 Solvedproblem:Weightlosspage250,problem19 Solvedproblem:Therecedingmoonpage250,problem17 DiscussionQuestions A HowcouldNewtonndthespeedofthemoontopluginto a = v 2 = r ? B Twoprojectilesofdifferentmassshotoutofgunsonthesurfaceof theearthatthesamespeedandanglewillfollowthesametrajectories, assumingthatairfrictionisnegligible.Youcanverifythisbythrowingtwo objectstogetherfromyourhandandseeingiftheyseparateorstayside byside.Whatcorrespondingfactwouldbetrueforsatellitesoftheearth havingdifferentmasses? C Whatiswrongwiththefollowingstatement?Acometinanelliptical orbitspeedsupasitapproachesthesun,becausethesun'sforceonitis increasing. D Whywoulditnotmakesensetoexpecttheearth'sgravitationalforce onabowlingballtobeinverselyproportionaltothesquareofthedistance betweentheirsurfacesratherthantheircenters? E Doestheearthaccelerateasaresultofthemoon'sgravitational forceonit?Supposetwoplanetswereboundtoeachothergravitationally thewaytheearthandmoonare,butthetwoplanetshadequalmasses. Whatwouldtheirmotionbelike? F Spacecraftnormallyoperatebyringtheirenginesonlyforafew minutesatatime,andaninterplanetaryprobewillspendmonthsoryears onitswaytoitsdestinationwithoutthrust.Supposeaspacecraftisina circularorbitaroundMars,anditthenbrieyresitsenginesinreverse, causingasuddendecreaseinspeed.Whatwillthisdotoitsorbit?What aboutaforwardthrust? 10.3ApparentWeightlessness Ifyouasksomebodyatthebusstopwhyastronautsareweightless, you'llprobablygetoneofthefollowingtwoincorrectanswers: They'reweightlessbecausethey'resofarfromtheearth. They'reweightlessbecausethey'removingsofast. Therstansweriswrong,becausethevastmajorityofastronautsnevergetmorethanathousandmilesfromtheearth'ssurface. Thereductioningravitycausedbytheiraltitudeissignicant,but not100%.ThesecondansweriswrongbecauseNewton'slawof Section10.3ApparentWeightlessness 237

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l / Gravityonlyappearsto pullstraightdownbecausethe nearperfectsymmetryofthe earthmakesthesidewayscomponentsofthetotalforceonan objectcancelalmostexactly.If thesymmetryisbroken,e.g.,by adensemineraldeposit,thetotal forceisalittleofftotheside. gravityonlydependsondistance,notspeed. Thecorrectansweristhatastronautsinorbitaroundtheearth arenotreallyweightlessatall.Theirweightlessnessisonlyapparent.Iftherewasnogravitationalforceonthespaceship,itwould obeyNewton'srstlawandmoveoonastraightline,ratherthan orbitingtheearth.Likewise,theastronautsinsidethespaceshipare inorbitjustlikethespaceshipitself,withtheearth'sgravitational forcecontinuallytwistingtheirvelocityvectorsaround.Thereason theyappeartobeweightlessisthattheyareinthesameorbitas thespaceship,soalthoughtheearth'sgravitycurvestheirtrajectory downtowardthedeck,thedeckdropsoutfromunderthematthe samerate. Apparentweightlessnesscanalsobeexperiencedonearth.Any timeyoujumpupintheair,youexperiencethesamekindofapparentweightlessnessthattheastronautsdo.Whileintheair,you canliftyourarmsmoreeasilythannormal,becausegravitydoesnot makethemfallanyfasterthantherestofyourbody,whichisfalling outfromunderthem.TheRussianairforcenowtakesrichforeign touristsupinabigcargoplaneandgivesthemthefeelingofweightlessnessforashortperiodoftimewhiletheplaneisnose-downand droppinglikearock. 10.4VectorAdditionofGravitationalForces Pickaoweronearthandyoumovethefartheststar. PaulDirac Whenyoustandontheground,whichpartoftheearthispulling downonyouwithitsgravitationalforce?Mostpeoplearetempted tosaythattheeectonlycomesfromthepartdirectlyunderyou, sincegravityalwayspullsstraightdown.Herearethreeobservations thatmighthelptochangeyourmind: Ifyoujumpupintheair,gravitydoesnotstopaectingyou justbecauseyouarenottouchingtheearth:gravityisanoncontactforce.Thatmeansyouarenotimmunefromthegravityofdistantpartsofourplanetjustbecauseyouarenot touchingthem. Gravitationaleectsarenotblockedbyinterveningmatter. Forinstance,inaneclipseofthemoon,theearthislinedup directlybetweenthesunandthemoon,butonlythesun'slight isblockedfromreachingthemoon,notitsgravitationalforce |ifthesun'sgravitationalforceonthemoonwasblockedin thissituation,astronomerswouldbeabletotellbecausethe moon'saccelerationwouldchangesuddenly.Amoresubtle butmoreeasilyobservableexampleisthatthetidesarecaused bythemoon'sgravity,andtidaleectscanoccurontheside 238 Chapter10Gravity

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oftheearthfacingawayfromthemoon.Thus,far-oparts oftheeartharenotpreventedfromattractingyouwiththeir gravityjustbecausethereisotherstubetweenyouandthem. Prospectorssometimessearchforundergrounddepositsofdense mineralsbymeasuringthedirectionofthelocalgravitational forces,i.e.,thedirectionthingsfallorthedirectionaplumb bobhangs.Forinstance,thegravitationalforcesintheregion tothewestofsuchadepositwouldpointalongalineslightly totheeastoftheearth'scenter.Justbecausethetotalgravitationalforceonyoupointsdown,thatdoesn'tmeanthat onlythepartsoftheearthdirectlybelowyouareattracting you.It'sjustthatthesidewayscomponentsofalltheforce vectorsactingonyoucomeveryclosetocancelingout. Acubiccentimeteroflavaintheearth'smantle,agrainofsilica insideMt.Kilimanjaro,andaeaonacatinParisareallattracting youwiththeirgravity.Whatyoufeelisthevectorsumofallthe gravitationalforcesexertedbyalltheatomsofourplanet,andfor thatmatterbyalltheatomsintheuniverse. WhenNewtontestedhistheoryofgravitybycomparingthe orbitalaccelerationofthemoontotheaccelerationofafallingapple onearth,heassumedhecouldcomputetheearth'sforceonthe appleusingthedistancefromtheappletotheearth'scenter.Was hewrong?Afterall,itisn'tjusttheearth'scenterattractingthe apple,it'sthewholeearth.Akilogramofdirtafewfeetunderhis backyardinEnglandwouldhaveamuchgreaterforceontheapple thanakilogramofmoltenrockdeepunderAustralia,thousandsof milesaway.There'sreallynoobviousreasonwhytheforceshould comeoutrightifyoujustpretendthattheearth'swholemassis concentratedatitscenter.Also,weknowthattheearthhassome partsthataremoredense,andsomepartsthatarelessdense.The solidcrust,onwhichwelive,isconsiderablylessdensethanthe moltenrockonwhichitoats.Byallrights,thecomputationofthe vectorsumofalltheforcesexertedbyalltheearth'spartsshould beahorrendousmess. Actually,Newtonhadsoundmathematicalreasonsfortreating theearth'smassasifitwasconcentratedatitscenter.First,althoughNewtonnodoubtsuspectedtheearth'sdensitywasnonuniform,heknewthatthedirectionofitstotalgravitationalforcewas verynearlytowardtheearth'scenter.Thatwasstrongevidence thatthedistributionofmasswasverysymmetric,sothatwecan thinkoftheearthasbeingmadeofmanylayers,likeanonion, witheachlayerhavingconstantdensitythroughout.Todaythere isfurtherevidenceforsymmetrybasedonmeasurementsofhowthe vibrationsfromearthquakesandnuclearexplosionstravelthrough theearth.Newtonthenconcentratedonthegravitationalforces Section10.4VectorAdditionofGravitationalForces 239

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m / Anobjectoutsideaspherical shellofmasswillfeelgravitational forcesfromeverypartoftheshell strongerforcesfromthecloser parts,andweakeronesfromthe partsfartheraway.Theshell theoremstatesthatthevector sumofalltheforcesisthesame asifallthemasshadbeen concentratedatthecenterofthe shell. exertedbyasinglesuchthinshell,andprovedthefollowingmathematicaltheorem,knownastheshelltheorem: Ifanobjectliesoutsideathin,sphericalshellofmass,then thevectorsumofallthegravitationalforcesexertedbyallthe partsoftheshellisthesameasiftheshell'smasshadbeen concentratedatitscenter.Iftheobjectliesinsidetheshell, thenallthegravitationalforcescanceloutexactly. Forterrestrialgravity,eachshellactsasthoughitsmasswasconcentratedattheearth'scenter,sothenalresultisthesameasif theearth'swholemasswasconcentratedatitscenter. Thesecondpartoftheshelltheorem,aboutthegravitational forcescancelinginsidetheshell,isalittlesurprising.Obviouslythe forceswouldallcanceloutifyouwereattheexactcenterofashell, butwhyshouldtheystillcanceloutperfectlyifyouareinsidethe shellbuto-center?Thewholeideamightseemacademic,sincewe don'tknowofanyhollowplanetsinoursolarsystemthatastronauts couldhopetovisit,butactuallyit'sausefulresultforunderstanding gravitywithintheearth,whichisanimportantissueingeology.It doesn'tmatterthattheearthisnotactuallyhollow.Inamineshaft atadepthof,say,2km,wecanusetheshelltheoremtotellusthat theoutermost2kmoftheearthhasnonetgravitationaleect,and thegravitationalforceisthesameaswhatwouldbeproducedifthe remaining,deeper,partsoftheearthwereallconcentratedatits center. self-checkD Supposeyou'reatthebottomofadeepmineshaft,whichmeansyou're stillquitefarfromthecenteroftheearth.Theshelltheoremsaysthat theshellofmassyou'vegoneinsideexertszerototalforceonyou. Discusswhichpartsoftheshellareattractingyouinwhichdirections, andhowstrongtheseforcesare.Explainwhyit'satleastplausiblethat theycancel. Answer,p.269 DiscussionQuestions A Ifyouholdanapple,doestheappleexertagravitationalforceon theearth?Isitmuchweakerthantheearth'sgravitationalforceonthe apple?Whydoesn'ttheearthseemtoaccelerateupwardwhenyoudrop theapple? B Whenastronautstravelfromtheearthtothemoon,howdoesthe gravitationalforceonthemchangeastheyprogress? C Howwouldthegravityintherst-oorlobbyofamassiveskyscraper comparewiththegravityinanopeneldoutsideofthecity? D Inafewbillionyears,thesunwillstartundergoingchangesthatwill eventuallyresultinitspufngupintoaredgiantstar.Nearthebeginning ofthisprocess,theearth'soceanswillboiloff,andbytheend,thesun willprobablyswallowtheearthcompletely.Asthesun'ssurfacestartsto getcloserandclosetotheearth,howwilltheearth'sorbitbeaffected? 240 Chapter10Gravity

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o / Asimpliedversionof Cavendish'sapparatus,viewed fromabove. 10.5WeighingtheEarth Let'slookmorecloselyattheapplicationofNewton'slawofgravity toobjectsontheearth'ssurface.Sincetheearth'sgravitational forceisthesameasifitsmasswasallconcentratedatitscenter, theforceonafallingobjectofmass m isgivenby F = GM earth m=r 2 earth Theobject'saccelerationequals F=m ,sotheobject'smasscancels outandwegetthesameaccelerationforallfallingobjects,aswe knewweshould: g = GM earth =r 2 earth n / Cavendish'sapparatus.Thetwolargeballsarexedinplace, buttherodfromwhichthetwosmallballshangisfreetotwistunderthe inuenceofthegravitationalforces. Newtonknewneitherthemassoftheearthnoranumericalvalue fortheconstant G .Butifsomeonecouldmeasure G ,thenitwould bepossibleforthersttimeinhistorytodeterminethemassofthe earth!Theonlywaytomeasure G istomeasurethegravitational forcebetweentwoobjectsofknownmass,butthat'sanexceedingly diculttask,becausetheforcebetweenanytwoobjectsofordinary sizeisextremelysmall.TheEnglishphysicistHenryCavendishwas thersttosucceed,usingtheapparatusshowninguresnando. Thetwolargerballswereleadspheres8inchesindiameter,andeach oneattractedthesmallballnearit.Thetwosmallballshungfrom theendsofahorizontalrod,whichitselfhungbyathinthread.The framefromwhichthelargerballshungcouldberotatedbyhand Section10.5WeighingtheEarth 241

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aboutaverticalaxis,sothatforinstancethelargeballontheright wouldpullitsneighboringsmallballtowardusandwhilethesmall ballontheleftwouldbepulledawayfromus.Thethreadfrom whichthesmallballshungwouldthusbetwistedthroughasmall angle,andbycalibratingthetwistofthethreadwithknownforces, theactualgravitationalforcecouldbedetermined.Cavendishset upthewholeapparatusinaroomofhishouse,nailingallthedoors shuttokeepaircurrentsfromdisturbingthedelicateapparatus. Theresultshadtobeobservedthroughtelescopesstuckthrough holesdrilledinthewalls.Cavendish'sexperimentprovidedtherst numericalvaluesfor G andforthemassoftheearth.Thepresently acceptedvalueof G is6.67 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(11 N m 2 = kg 2 Knowing G notonlyallowedthedeterminationoftheearth's massbutalsothoseofthesunandtheotherplanets.Forinstance, byobservingtheaccelerationofoneofJupiter'smoons,wecaninfer themassofJupiter.Thefollowingtablegivesthedistancesofthe planetsfromthesunandthemassesofthesunandplanets.Other dataaregiveninthebackofthebook. averagedistancefrom thesun,inunitsofthe earth'saveragedistance fromthesun mass,inunitsofthe earth'smass sun | 330,000 mercury 0.38 0.056 venus 0.72 0.82 earth 1 1 mars 1.5 0.11 jupiter 5.2 320 saturn 9.5 95 uranus 19 14 neptune 30 17 pluto 39 0.002 DiscussionQuestions A ItwouldhavebeendifcultforCavendishtostartdesigningan experimentwithoutatleastsomeideaoftheorderofmagnitudeof G Howcouldheestimateitinadvancetowithinafactorof10? B FillinthedetailsofhowonewoulddetermineJupiter'smassby observingtheaccelerationofoneofitsmoons.Whyisitonlynecessary toknowtheaccelerationofthemoon,nottheactualforceactingonit? Whydon'tweneedtoknowthemassofthemoon?Whataboutaplanet thathasnomoons,suchasVenushowcoulditsmassbefound? C Thegravitationalconstant G isverydifculttomeasureaccurately,andistheleastaccuratelyknownofallthefundamentalnumbers ofphysicssuchasthespeedoflight,themassoftheelectron,etc.But that'sinthemkssystem,basedonthemeterastheunitoflength,the kilogramastheunitofmass,andthesecondastheunitofdistance.Astronomerssometimesuseadifferentsystemofunits,inwhichtheunitof 242 Chapter10Gravity

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distance,calledtheastronomicalunitora.u.,istheradiusoftheearth's orbit,theunitofmassisthemassofthesun,andtheunitoftimeisthe yeari.e.,thetimerequiredfortheearthtoorbitthesun.Inthissystem ofunits, G hasaprecisenumericalvaluesimplyasamatterofdenition. Whatisit? 10.6 ? EvidenceforRepulsiveGravity Untilrecently,physiciststhoughttheyunderstoodgravityfairly well.EinsteinhadmodiedNewton'stheory,butcertaincharacteristricsofgravitationalforceswerermlyestablished.Forone thing,theywerealwaysattractive.Ifgravityalwaysattracts,then itislogicaltoaskwhytheuniversedoesn'tcollapse.Newtonhad answeredthisquestionbysayingthatiftheuniversewasinnitein alldirections,thenitwouldhavenogeometriccentertowardwhich itwouldcollapse;theforcesonanyparticularstarorplanetexertedbydistantpartsoftheuniversewouldtendtocanceloutby symmetry.Morecarefulcalculations,however,showthatNewton's universewouldhaveatendencytocollapseonsmallerscales:any partoftheuniversethathappenedtobeslightlymoredensethan averagewouldcontractfurther,andthiscontractionwouldresult instrongergravitationalforces,whichwouldcauseevenmorerapid contraction,andsoon. WhenEinsteinoverhauledgravity,thesameproblemrearedits uglyhead.LikeNewton,Einsteinwaspredisposedtobelieveina universethatwasstatic,soheaddedaspecialrepulsivetermtohis equations,intendedtopreventacollapse.Thistermwasnotassociatedwithanyattractionofmassformass,butrepresentedmerely anoveralltendencyforspaceitselftoexpandunlessrestrainedby thematterthatinhabitedit.ItturnsoutthatEinstein'ssolution, likeNewton's,isunstable.Furthermore,itwassoondiscovered observationallythattheuniversewasexpanding,andthiswasinterpretedbycreatingtheBigBangmodel,inwhichtheuniverse's currentexpansionistheaftermathofafantasticallyhotexplosion. 1 Anexpandinguniverse,unlikeastaticone,wascapableofbeingexplainedwithEinstein'sequations,withoutanyrepulsionterm.The universe'sexpansionwouldsimplyslowdownovertimeduetothe attractivegravitationalforces.Afterthesedevelopments,Einstein saidwoefullythataddingtherepulsiveterm,knownasthecosmologicalconstant,hadbeenthegreatestblunderofhislife. Thiswasthestateofthingsuntil1999,whenevidencebeganto turnupthattheuniverse'sexpansionhasbeenspeedinguprather thanslowingdown!Therstevidencecamefromusingatelescope asasortoftimemachine:lightfromadistantgalaxymayhave takenbillionsofyearstoreachus,soweareseeingitasitwasfar inthepast.Lookingbackintime,astronomerssawtheuniverse 1 Book3,section3.5,presentssomeoftheevidencefortheBigBang. Section10.6 ? EvidenceforRepulsiveGravity 243

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expandingatspeedsthatwarelower,ratherthanhigher.Atrst theyweremortied,sincethiswasexactlytheoppositeofwhathad beenexpected.Thestatisticalqualityofthedatawasalsonotgood enoughtoconstuteironcladproof,andtherewereworriesaboutsystematicerrors.Thecaseforanacceleratingexpansionhashowever beennaileddownbyhigh-precisionmappingofthedim,sky-wide afterglowoftheBigBang,knownasthecosmicmicrowavebackground.SometheoristshaveproposedrevivingEinstein'scosmologicalconstanttoaccountfortheacceleration,whileothersbelieve itisevidenceforamysteriousformofmatterwhichexhibitsgravitationalrepulsion.Thegenerictermforthisunknownstuisdark energy." Asof2008,mostoftheremainingdoubtabouttherepulsiveeffecthasbeendispelled.Duringthepastdecadeorso,astronomers considerthemselvestohaveenteredaneweraofhigh-precisioncosmology.Thecosmicmicrowavebackgroundmeasurements,forexample,havemeasuredtheageoftheuniversetobe13.7 0.2billion years,agurethatcouldpreviouslybestatedonlyasafuzzyrange from10to20billion.Weknowthatonly4%oftheuniverseis atoms,withanother23%consistingofunknownsubatomicparticles,and73%ofdarkenergy.It'smorethanalittleironictoknow aboutsomanythingswithsuchhighprecision,andyettoknow virtuallynothingabouttheirnature.Forinstance,weknowthat precisely96%oftheuniverseissomethingotherthanatoms,butwe knowpreciselynothingaboutwhatthatsomethingis. p / TheWMAPprobe'smapofthe cosmicmicrowavebackgroundis likeababypictureoftheuniverse. 244 Chapter10Gravity

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Summary SelectedVocabulary ellipse.......aattenedcircle;oneoftheconicsections conicsection...acurveformedbytheintersectionofaplane andaninnitecone hyperbola....anotherconicsection;itdoesnotcloseback onitself period.......thetimerequiredforaplanettocompleteone orbit;moregenerally,thetimeforonerepetitionofsomerepeatingmotion focus.......oneoftwospecialpointsinsideanellipse:the ellipseconsistsofallpointssuchthatthesum ofthedistancestothetwofociequalsacertain number;ahyperbolaalsohasafocus Notation G .........theconstantofproportionalityinNewton's lawofgravity;thegravitationalforceofattractionbetweentwo1-kgspheresatacenterto-centerdistanceof1m Summary Keplerdeducedthreeempiricallawsfromdataonthemotionof theplanets: Kepler'sellipticalorbitlaw: Theplanetsorbitthesuninellipticalorbitswiththesunatonefocus. Kepler'sequal-arealaw: Thelineconnectingaplanettothesun sweepsoutequalareasinequalamountsoftime. Kepler'slawofperiods: Thetimerequiredforaplanettoorbit thesunisproportionaltothelongaxisoftheellipseraisedto the3/2power.Theconstantofproportionalityisthesame foralltheplanets. Newtonwasabletondamorefundamentalexplanationforthese laws.Newton'slawofgravitystatesthatthemagnitudeofthe attractiveforcebetweenanytwoobjectsintheuniverseisgivenby F = Gm 1 m 2 =r 2 Weightlessnessofobjectsinorbitaroundtheearthisonlyapparent.Anastronautinsideaspaceshipissimplyfallingalongwith thespaceship.Sincethespaceshipisfallingoutfromundertheastronaut,itappearsasthoughtherewasnogravityacceleratingthe astronautdowntowardthedeck. Gravitationalforces,likeallotherforces,addlikevectors.A gravitationalforcesuchasweordinarilyfeelisthevectorsumofall Summary 245

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theforcesexertedbyallthepartsoftheearth.Asaconsequenceof this,Newtonprovedthe shelltheorem forgravitationalforces: Ifanobjectliesoutsideathin,uniformshellofmass,thenthe vectorsumofallthegravitationalforcesexertedbyallthepartsof theshellisthesameasifalltheshell'smasswasconcentratedatits center.Iftheobjectliesinsidetheshell,thenallthegravitational forcescanceloutexactly. 246 Chapter10Gravity

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Royhasamassof60kg.Lauriehasamassof65kg.They are1.5mapart. aWhatisthemagnitudeofthegravitationalforceoftheearthon Roy? bWhatisthemagnitudeofRoy'sgravitationalforceontheearth? cWhatisthemagnitudeofthegravitationalforcebetweenRoy andLaurie? dWhatisthemagnitudeofthegravitationalforcebetweenLaurie andthesun? p 2 Duringasolareclipse,themoon,earthandsunalllieon thesameline,withthemoonbetweentheearthandsun.Dene yourcoordinatessothattheearthandmoonlieatgreater x values thanthesun.Foreachforce,givethecorrectsignaswellasthe magnitude.aWhatforceisexertedonthemoonbythesun?b Onthemoonbytheearth?cOntheearthbythesun?dWhat totalforceisexertedonthesun?eOnthemoon?fOnthe earth? p 3 Supposethatonacertaindaythereisacrescentmoon,and youcantellbytheshapeofthecrescentthattheearth,sunand moonformatrianglewitha135 interiorangleatthemoon'scorner. Whatisthemagnitudeofthetotalgravitationalforceoftheearth andthesunonthemoon? Problem3. 4 HowhighabovetheEarth'ssurfacemustarocketbeinorder tohave1/100theweightitwouldhaveatthesurface?Expressyour answerinunitsoftheradiusoftheEarth. 5 ThestarLalande21185wasfoundin1996tohavetwoplanets inroughlycircularorbits,withperiodsof6and30years.Whatis theratioofthetwoplanets'orbitalradii? 6 InaStarTrekepisode,theEnterpriseisinacircularorbit aroundaplanetwhensomethinghappenstotheengines.Spock thentellsKirkthattheshipwillspiralintotheplanet'ssurface unlesstheycanxtheengines.Isthisscienticallycorrect?Why? 7 aSupposearotatingsphericalbodysuchasaplanethas Problems 247

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Problem8. aradius r andauniformdensity ,andthetimerequiredforone rotationis T .Atthesurfaceoftheplanet,theapparentacceleration ofafallingobjectisreducedbytheaccelerationofthegroundout fromunderit.Deriveanequationfortheapparentaccelerationof gravity, g ,attheequatorintermsof r T ,and G p bApplyingyourequationfroma,bywhatfractionisyourapparentweightreducedattheequatorcomparedtothepoles,duetothe Earth'srotation? p cUsingyourequationfroma,deriveanequationgivingthevalue of T forwhichtheapparentaccelerationofgravitybecomeszero, i.e.,objectscanspontaneouslydriftothesurfaceoftheplanet. Showthat T onlydependson ,andnoton r p dApplyingyourequationfromc,howlongwouldadayhaveto beinordertoreducetheapparentweightofobjectsattheequator oftheEarthtozero?[Answer:1.4hours] eObservationalastronomershaverecentlyfoundobjectstheycalled pulsars,whichemitburstsofradiationatregularintervalsofless thanasecond.Ifapulsaristobeinterpretedasarotatingsphere beamingoutanaturalsearchlight"thatsweepspasttheearthwith eachrotation,useyourequationfromctoshowthatitsdensity wouldhavetobemuchgreaterthanthatofordinarymatter. fAstrophysicistspredicteddecadesagothatcertainstarsthatused uptheirsourcesofenergycouldcollapse,formingaballofneutrons withthefantasticdensityof 10 17 kg = m 3 .Ifthisiswhatpulsars reallyare,useyourequationfromctoexplainwhynopulsarhas everbeenobservedthatasheswithaperiodoflessthan1msor so. 8 YouareconsideringgoingonaspacevoyagetoMars,inwhich yourroutewouldbehalfanellipse,tangenttotheEarth'sorbitat oneendandtangenttoMars'orbitattheother.Yourspacecraft's engineswillonlybeusedatthebeginningandend,notduringthe voyage.Howlongwouldtheoutwardlegofyourtriplast?Assume theorbitsofEarthandMarsarecircular. p 9 aIftheearthwasofuniformdensity,wouldyourweightbe increasedordecreasedatthebottomofamineshaft?Explain. bInreallife,objectsweighslightlymoreatthebottomofamine shaft.WhatdoesthatallowustoinferabouttheEarth? ? 10 Ceres,thelargestasteroidinoursolarsystem,isaspherical bodywithamass6000timeslessthantheearth's,andaradius whichis13timessmaller.Ifanastronautwhoweighs400Non earthisvisitingthesurfaceofCeres,whatisherweight? Solution,p.278 11 Prove,basedonNewton'slawsofmotionandNewton'slaw ofgravity,thatallfallingobjectshavethesameaccelerationifthey aredroppedatthesamelocationontheearthandifotherforces suchasfrictionareunimportant.Donotjustsay, g =9.8m = s 2 { it'sconstant."Youaresupposedtobe proving that g shouldbethe 248 Chapter10Gravity

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Problem12. samenumberforallobjects. Solution,p.278 12 ThegureshowsanimagefromtheGalileospaceprobe takenduringitsAugust1993ybyoftheasteroidIda.Astronomers weresurprisedwhenGalileodetectedasmallerobjectorbitingIda. Thissmallerobject,theonlyknownsatelliteofanasteroidinour solarsystem,waschristenedDactyl,afterthemythicalcreatures wholivedonMountIda,andwhoprotectedtheinfantZeus.For scale,IdaisaboutthesizeandshapeofOrangeCounty,andDactyl thesizeofacollegecampus.Galileowasunfortunatelyunableto measurethetime, T ,requiredforDactyltoorbitIda.Ifithad, astronomerswouldhavebeenabletomaketherstaccuratedeterminationofthemassanddensityofanasteroid.Findanequation forthedensity, ,ofIdaintermsofIda'sknownvolume, V ,the knownradius, r ,ofDactyl'sorbit,andthelamentablyunknown variable T .Thisisthesametechniquethatwasusedsuccessfully fordeterminingthemassesanddensitiesoftheplanetsthathave moons. Solution,p.278 13 Ifabulletisshotstraightupatahighenoughvelocity,itwill neverreturntotheearth.Thisisknownastheescapevelocity.We willdiscussescapevelocityusingtheconceptofenergyinthenext bookoftheseries,butitcanalsobegottenatusingstraightforward calculus.Inthisproblem,youwillanalyzethemotionofanobject ofmass m whoseinitialvelocityis exactly equaltoescapevelocity.Weassumethatitisstartingfromthesurfaceofaspherically symmetricplanetofmassMandradius b .Thetrickistoguessat thegeneralformofthesolution,andthendeterminethesolutionin moredetail.Assumeasistruethatthesolutionisoftheformr= kt p ,where r istheobject'sdistancefromthecenteroftheplanet attime t ,and k and p areconstants. aFindtheacceleration,anduseNewton'ssecondlawandNewton'slawofgravitytodetermine k and p .Youshouldndthatthe resultisindependentof m p bWhathappenstothevelocityas t approachesinnity? cDetermineescapevelocityfromtheEarth'ssurface. p R 14 Astronomershaverecentlyobservedstarsorbitingatvery highspeedsaroundanunknownobjectnearthecenterofourgalaxy. Forstarsorbitingatdistancesofabout10 14 mfromtheobject, theorbitalvelocitiesareabout10 6 m/s.Assumingtheorbitsare circular,estimatethemassoftheobject,inunitsofthemassof thesun,2 10 30 kg.Iftheobjectwasatightlypackedclusterof normalstars,itshouldbeaverybrightsourceoflight.Sinceno visiblelightisdetectedcomingfromit,itisinsteadbelievedtobe asupermassiveblackhole. 15 Astronomershavedetectedasolarsystemconsistingofthree planetsorbitingthestarUpsilonAndromedae.Theplanetshave beennamedb,c,andd.Planetb'saveragedistancefromthestar Problems 249

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is0.059A.U.,andplanetc'saveragedistanceis0.83A.U.,wherean astronomicalunitorA.U.isdenedasthedistancefromtheEarth tothesun.Fortechnicalreasons,itispossibletodeterminethe ratiosoftheplanets'masses,buttheirmassescannotpresentlybe determinedinabsoluteunits.Planetc'smassis3.0timesthatof planetb.Comparethestar'saveragegravitationalforceonplanet cwithitsaverageforceonplanetb.[BasedonaproblembyArnold Arons.] Solution,p.278 16 Somecommunicationssatellitesareinorbitscalledgeosynchronous:thesatellitetakesonedaytoorbittheearthfromwest toeast,sothatastheearthspins,thesatelliteremainsabovethe samepointontheequator.Whatissuchasatellite'saltitudeabove thesurfaceoftheearth? Solution,p.278 17 Asisdiscussedinmoredetailinsection5.1ofbook2,tidal interactionswiththeeartharecausingthemoon'sorbittogrow graduallylarger.Laserbeamsbouncedoofamirrorleftonthe moonbyastronautshaveallowedameasurementofthemoon'srate ofrecession,whichisabout1cmperyear.Thismeansthatthe gravitationalforceactingbetweenearthandmoonisdecreasing.By whatfractiondoestheforcedecreasewitheach27-dayorbit?[Hint: Ifyoutrytocalculatethetwoforcesandsubtract,yourcalculator willprobablygivearesultofzeroduetorounding.Instead,reason aboutthefractionalamountbywhichthequantity1 =r 2 willchange. Asawarm-up,youmaywishtoobservethepercentagechangein 1 =r 2 thatresultsfromchanging r from1to1.01.Basedonaproblem byArnoldArons.] Solution,p.279 18 Supposethatweinhabitedauniverseinwhich,insteadof Newton'slawofgravity,wehad F = k p m 1 m 2 =r 2 ,where k issome constantwithdierentunitsthan G .Theforceisstillattractive.However,weassumethat a = F=m andtherestofNewtonian physicsremainstrue,andweuse a = F=m todeneourmassscale, sothat,e.g.,amassof2kgisonewhichexhibitshalftheaccelerationwhenthesameforceisappliedtoitastoa1kgmass. aIsthisnewlawofgravityconsistentwithNewton'sthirdlaw? bSupposeyoulivedinsuchauniverse,andyoudroppedtwounequalmassessidebyside.Whatwouldhappen? cNumerically,supposea1.0-kgobjectfallswithanacceleration of10m = s 2 .Whatwouldbetheaccelerationofaraindropwitha massof0.1g?Wouldyouwanttogooutintherain? dIfafallingobjectbrokeintotwounequalpieceswhileitfell, whatwouldhappen? eInventalawofgravitythatresultsinbehaviorthatistheoppositeofwhatyoufoundinpartb.[BasedonaproblembyArnold Arons.] 19 aAcertainvilealiengangsterlivesonthesurfaceofan asteroid,wherehisweightis0.20N.Hedecidesheneedstolose 250 Chapter10Gravity

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Problem21. weightwithoutreducinghisconsumptionofprincesses,sohe'sgoing tomovetoadierentasteroidwherehisweightwillbe0.10N.The realestateagent'sdatabasehasasteroidslistedbymass,however, notbysurfacegravity.Assumingthatallasteroidsarespherical andhavethesamedensity,howshouldthemassofhisnewasteroid comparewiththatofhisoldone? bJupiter'smassis318timestheEarth's,anditsgravityisabout twiceEarth's.Isthisconsistentwiththeresultsofparta?Ifnot, howdoyouexplainthediscrepancy? Solution,p.279 20 Wherewouldanobjecthavetobelocatedsothatitwould experiencezerototalgravitationalforcefromtheearthandmoon? p 21 TheplanetUranushasamassof8.68 10 25 kgandaradius of2.56 10 4 km.ThegureshowstherelativesizesofUranusand Earth. aComputetheratio g U =g E ,where g U isthestrengthofthegravitationaleldatthesurfaceofUranusand g E isthecorresponding quantityatthesurfaceoftheEarth. p bWhatissurprisingaboutthisresult?Howdoyouexplainit? 22 TheInternationalSpaceStationorbitsatanaveragealtitude ofabout370kmabovesealevel.Computethevalueof g atthat altitude. Problems 251

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Appendix1:Exercises Exercise0A:ModelsandIdealization Equipment: coeelters rampsonepergroup ballsofvarioussizes stickytape vacuumpumpandguineaandfeather"apparatusone Themotionoffallingobjectshasbeenrecognizedsinceancienttimesasanimportantpieceof physics,butthemotionisinconvenientlyfast,soinoureverydayexperienceitcanbehardto tellexactlywhatobjectsaredoingwhentheyfall.Inthisexerciseyouwilluseseveraltechniques togetaroundthisproblemandstudythemotion.Yourgoalistoconstructascientic model of falling.Amodelmeansanexplanationthatmakestestablepredictions.Oftenmodelscontain simplicationsoridealizationsthatmakethemeasiertoworkwith,eventhoughtheyarenot strictlyrealistic. 1.Onemethodofmakingfallingeasiertoobserveistouseobjectslikefeathersthatweknow fromeverydayexperiencewillnotfallasfast.Youwillusecoeelters,instacksofvarious sizes,totestthefollowingtwohypothesesandseewhichoneistrue,orwhetherneitheristrue: Hypothesis1A:Whenanobjectisdropped,itrapidlyspeedsuptoacertainnaturalfalling speed,andthencontinuestofallatthatspeed.Thefallingspeedis proportional totheobject's weight.Aproportionalityisnotjustastatementthatifonethinggetsbigger,theotherdoes too.Itsaysthatifonebecomesthreetimesbigger,theotheralsogetsthreetimesbigger,etc. Hypothesis1B:Dierentobjectsfallthesameway,regardlessofweight. Testthesehypothesesanddiscussyourresultswithyourinstructor. 2.Asecondwaytoslowdowntheactionistoletaballrolldownaramp.Thesteeperthe ramp,theclosertofreefall.Basedonyourexperienceinpart1,writeahypothesisaboutwhat willhappenwhenyouraceaheavierballagainstalighterballdownthesameramp,starting thembothfromrest. Hypothesis: Showyourhypothesistoyourinstructor,andthentestit. Youhaveprobablyfoundthatfallingwasmorecomplicatedthanyouthought!Istheremore thanonefactorthataectsthemotionofafallingobject?Canyouimaginecertainidealized situationsthataresimpler?Trytoagreeverballywithyourgrouponaninformalmodelof fallingthatcanmakepredictionsabouttheexperimentsdescribedinparts3and4. 3.Youhavethreeballs:astandardcomparisonball"ofmediumweight,alightball,anda

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heavyball.Supposeyoustandonachairandadropthelightballsidebysidewiththe comparisonball,thenbdroptheheavyballsidebysidewiththecomparisonball,thenc jointhelightandheavyballstogetherwithstickytapeanddropthemsidebysidewiththe comparisonball. Useyourmodeltomakeaprediction: Testyourprediction. 4.Yourinstructorwillpumpnearlyalltheairoutofachambercontainingafeatheranda heavierobject,thenletthemfallsidebysideinthechamber. Useyourmodeltomakeaprediction: 253

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Exercise1A:ScalingAppliedtoLeaves Equipment: leavesofthreesizes,havingroughlysimilarproportionsoflength,width,andthickness example:bladesofgrass,largecusleaves,andagaveleaves balance graphpaperwithcentimetersquares 1.Eachgroupwillhaveoneleaf,andshouldmeasureitssurfaceareaandvolume,anddetermine itssurface-to-volumeratiosurfaceareadividedbyvolume.Forconsistency,everygroupshould useunitsofcm 2 andcm 3 ,andshouldonlyndtheareaofonesideoftheleaf.Theareacanbe foundbytracingtheareaoftheleafongraphpaperandcountingsquares.Thevolumecanbe foundbyweighingtheleafandassumingthatitsdensityis1g = cm 3 ,whichisnearlytruesince leavesaremostlywater. Writeyourresultsontheboardforcomparisonwiththeothergroups'numbers. 2.Boththesurfaceareaandthevolumearebiggerforbiggerleaves,butwhataboutthe surfacetovolumeratios?Whatimplicationswouldthishavefortheplants'abilitiestosurvive indierentenvironments? 254 Appendix1:Exercises

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Exercise2A:ChangingVelocity ThisexerciseinvolvesMichaelJohnson'sworld-record200-metersprintinthe1996Olympics. Thetablegivesthedistancehehascoveredatvarioustimes.Thedataaremadeup,exceptfor histotaltime.Eachgroupistondavalueof x= t betweentwospeciedinstants,withthe membersofthegroupcheckingeachother'sanswers.Wewillthencompareeveryone'sresults anddiscusshowthisrelatestovelocity. t s x m A10.200100.0000 B10.210100.0990 C10.300100.9912 D11.200110.0168 E19.320200.0000 group1:Find x= t usingpointsAandB. group2:Find x= t usingpointsAandC. group3:Find x= t usingpointsAandD. group4:Find x= t usingpointsAandE. 256 Appendix1:Exercises

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Exercise3A:ReasoningwithRatiosandPowers Equipment: ping-pongballsandpaddles two-metersticks Youhaveprobablybouncedapingpongballstraightupanddownintheair.Thetimebetween hitsisrelatedtotheheighttowhichyouhittheball.Ifyoutaketwiceasmuchtimebetween hits,howmanytimeshigherdoyouthinkyouwillhavetohittheball?Writedownyour hypothesis: Yourinstructorwillrstbeatoutatempoof240beatsperminutefourbeatspersecond, whichyoushouldtrytomatchwiththeping-pongball.Measuretheheighttowhichtheball rises: Nowtryitat120beatsperminute: Compareyourhypothesisandyourresultswiththerestoftheclass. 257

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Exercise4A:ForceandMotion Equipment: 2-meterpiecesofbutcherpaper woodblockswithhooks string massestoputontopoftheblockstoincreasefriction springscalespreferablycalibratedinNewtons Supposeapersonpushesacrate,slidingitacrosstheooratacertainspeed,andthenrepeats thesamethingbutatahigherspeed.Thisisessentiallythesituationyouwillactoutinthis exercise.Whatdoyouthinkisdierentaboutherforceonthecrateinthetwosituations? Discussthiswithyourgroupandwritedownyourhypothesis: 1.Firstyouwillmeasuretheamountoffrictionbetweenthewoodblockandthebutcherpaper whenthewoodandpapersurfacesareslippingovereachother.Theideaistoattachaspring scaletotheblockandthenslidethebutcherpaperundertheblockwhileusingthescaleto keeptheblockfrommovingwithit.Dependingontheamountofforceyourspringscalewas designedtomeasure,youmayneedtoputanextramassontopoftheblockinordertoincrease theamountoffriction.Itisagoodideatouselongpieceofstringtoattachtheblocktothe springscale,sinceotherwiseonetendstopullatanangleinsteadofdirectlyhorizontally. Firstmeasuretheamountoffrictionforcewhenslidingthebutcherpaperasslowlyaspossible: Nowmeasuretheamountoffrictionforceatasignicantlyhigherspeed,say1meterpersecond. Ifyoutrytogotoofast,themotionisjerky,anditisimpossibletogetanaccuratereading. Discussyourresults.Whyarewejustiedinassumingthatthestring'sforceontheblocki.e., thescalereadingisthesameamountasthepaper'sfrictionalforceontheblock? 2.Nowtrythesamethingbutwiththeblockmovingandthepaperstandingstill.Trytwo dierentspeeds. Doyourresultsagreewithyouroriginalhypothesis?Ifnot,discusswhat'sgoingon.Howdoes theblockknow"howfasttogo? 258 Appendix1:Exercises

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Exercise4B:Interactions Equipment: neodymiumdiscmagnets/group compass triple-armbalance/group clampand50-cmverticalrodforholdingbalanceup string tape scissors Yourgoalinthisexerciseistocomparetheforcestwomagnetsexertoneachother,i.e.,to comparemagnetA'sforceonmagnetBtomagnetB'sforceonmagnetA.MagnetBwillbe madeoutoftwoofthesmalldiscmagnetsputtogether,soitistwiceasstrongasmagnetA. 1.Notethatthesemagnetsareextremelystrong!Beingcarefulnottopinchyourskin,puttwo discmagnetstogethertomakemagnetB. 2.Familiarizeyourselfwithhowthemagnetsbehave.InadditiontomagnetsAandB,there aretwoothermagnetsthatcancomeintoplay.Thecompassneedleitselfisamagnet,andthe planetearthisamagnet.Ordinarilythecompassneedletwistsaroundundertheinuenceof theearth,butthediscmagnetsareverystrongcloseup,soifyoubringthemwithinafewcm ofthecompass,thecompassisessentiallyjustrespondingtothem.Investigatehowdierent partsofmagnetsAandBinteractwiththecompass,andlabelthemappropriately.Investigate howmagnetsAandBcanattractorrepeloneanother. 3.Youarereadytoformahypothesisaboutthefollowingsituation.Supposewesetuptwo balancesasshowninthegure.Themagnetsarenottouching.Thetopmagnetishangingfrom ahookunderneaththepan,givingthesameresultasifitwasontopofthepan.Makesureit ishangingunderthe center ofthepan.Youwillwanttomakesurethemagnetsarepullingon eachother,notpushingeachotheraway,sothatthetopmagnetwillstayinoneplace. Thebalanceswillnotshowthemagnets'trueweights,becausethemagnetsareexertingforces oneachother.Thetopbalancewillreadahighernumberthanitwouldwithoutanymagnetic forces,andthebottombalancewillhavealowerthannormalreading.Thedierencebetween eachmagnet'strueweightandthereadingonthebalancegivesameasureofhowstronglythe magnetisbeingpushedorpulledbytheothermagnet. Howdoyouthinktheamountofpushingorpullingexperiencedbythetwomagnetswillcompare?Inotherwords,whichreadingwillchangemore,orwilltheychangebythesameamount? 259

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Writedownahypothesis: Beforegoingontopart4,discussyourhypothesiswithyourinstructor. 4.Nowsetuptheexperimentdescribedabovewithtwobalances.Sinceweareinterestedin thechangseinthescalereadingscausedbythemagneticforces,youwillneedtotakeatotalof fourscalereadings:onepairwiththebalancesseparatedandonepairwiththemagnetsclose togetherasshowninthegureabove. Whenthebalancesaretogetherandthemagneticforcesareacting,itisnotpossibletogetboth balancestoreachequilibriumatthesametime,becauseslidingtheweightsononebalancecan causeitsmagnettomoveupordown,tippingtheotherbalance.Therefore,whileyoutakea readingfromonebalance,youneedtoimmobilizetheotherinthehorizontalpositionbytaping itstipsoitpointsexactlyatthezeromark. Youwillalsoprobablyndthatasyouslidetheweights,thepointerswingssuddenlytothe oppositeside,butyoucannevergetittobestableinthemiddlezeroposition.Trybringing thepointermanuallytothezeropositionandthenreleasingit.Ifitswingsup,you'retoolow, andifitswingsdown,you'retoohigh.Searchforthedividinglinebetweenthetoo-lowregion andthetoo-highregion. Ifthechangesinthescalereadingsareverysmallsayafewgramsorless,youneedtoget themagnetsclosertogether.Itshouldbepossibletogetthescalereadingstochangebylarge amountsupto10or20g. 260 Appendix1:Exercises

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Exercise5A:Friction Equipment: 2-meterpiecesofbutcherpaper woodblockswithhooks string massestoputontopoftheblockstoincreasefriction springscalespreferablycalibratedinNewtons 1.Usingthesameequipmentasinexercise4A,testthestatementthatkineticfrictionis approximatelyindependentofvelocity. 2.Testthestatementthatkineticfrictionisindependentofsurfacearea. 261

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Exercise8A:VectorsandMotion Eachdiagramonpage263showsthemotionofanobjectinan x )]TJ/F20 10.9091 Tf 11.253 0 Td [(y plane.Eachdotisone locationoftheobjectatonemomentintime.Thetimeintervalfromonedottothenextis alwaysthesame,soyoucanthinkofthevectorthatconnectsonedottothenextasa v vector, andsubtracttond v vectors. 1.Supposetheobjectindiagram1ismovingfromthetoplefttothebottomright.Deduce whateveryoucanabouttheforceactingonit.Doestheforcealwayshavethesamemagnitude? Thesamedirection? Inventaphysicalsituationthatthisdiagramcouldrepresent. Whatifyoureinterpretthediagram,andreversetheobject'sdirectionofmotion? 2.Whatcanyoudeduceabouttheforcethatisactingindiagram2? Inventaphysicalsituationthatdiagram2couldrepresent. 3.Whatcanyoudeduceabouttheforcethatisactingindiagram3? Inventaphysicalsituation. 262 Appendix1:Exercises

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Exercise10A:TheShellTheorem Thisexerciseisanapproximatenumericaltestoftheshelltheorem.Therearesevenmasses A-G,eachbeingonekilogram.MassesA-E,eachonemeterfromthecenter,formashapelike twoEgyptianpyramidsjoinedattheirbases;thisisaroughapproximationtoasix-kilogram sphericalshellofmass.MassGisvemetersfromthecenterofthemaingroup.Theclasswill divideintosixgroupsandsplituptheworkrequiredinordertocalculatethevectorsumofthe sixgravitationalforcesexertedonmassG.Dependingonthesizeoftheclass,morethanone groupmaybeassignedtodealwiththecontributionofthesamemasstothetotalforce,and theredundantgroupscancheckeachother'sresults. 1.Discussasaclasswhatcanbedonetosimplifythetaskofcalculatingthevectorsum,and howtoorganizethingssothateachgroupcanworkinparallelwiththeothers. 2.Eachgroupshouldwriteitsresultsontheboardinunitsofpiconewtons,retainingsix signicantguresofprecision. 3.Theclasswilldeterminethevectorsumandcomparewiththeresultthatwouldbeobtained withtheshelltheorem. 264 Appendix1:Exercises

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Appendix2:PhotoCredits Exceptasspecicallynotedbeloworinaparentheticalcreditinthecaptionofagure,alltheillustrationsin thisbookareundermyowncopyright,andarecopyleftlicensedunderthesamelicenseastherestofthebook. Insomecasesit'sclearfromthedatethatthegureispublicdomain,butIdon'tknowthenameoftheartist orphotographer;Iwouldbegratefultoanyonewhocouldhelpmetogivepropercredit.Ihaveassumedthat imagesthatcomefromU.S.governmentwebpagesarecopyright-free,sinceproductsoffederalagenciesfallinto thepublicdomain.I'veincludedsomepublic-domainpaintings;photographicreproductionsofthemarenot copyrightableintheU.S.BridgemanArtLibrary,Ltd.v.CorelCorp.,36F.Supp.2d191,S.D.N.Y.1999. WhenPSSCPhysics"isgivenasacredit,itindicatesthatthegureisfromthersteditionofthetextbook entitledPhysics,bythePhysicalScienceStudyCommittee.Theearlyeditionsofthesebooksneverhadtheir copyrightsrenewed,andarenowthereforeinthepublicdomain.Thereisalsoablanketpermissiongivenin thelaterPSSCCollegePhysicsedition,whichstatesonthecopyrightpagethatThematerialstakenfromthe originalandsecondeditionsandtheAdvancedTopicsofPSSCPHYSICSincludedinthistextwillbeavailable toallpublishersforuseinEnglishafterDecember31,1970,andintranslationsafterDecember31,1975." CreditstoMillikanandGalerefertothetextbooksPracticalPhysicsandElementsofPhysics. Botharepublicdomain.The1927versiondidnothaveitscopyrightrenewed.Sinceispossiblethatsomeof theillustrationsinthe1927versionhadtheircopyrightsrenewedandarestillundercopyright,Ihaveonlyused themwhenitwasclearthattheywereoriginallytakenfrompublicdomainsources. Inafewcases,Ihavemadeuseofimagesunderthefairusedoctrine.However,Iamnotalawyer,andthelaws onfairusearevague,soyoushouldnotassumethatit'slegalforyoutousetheseimages.Inparticular,fairuse lawmaygiveyoulessleewaythanitgivesme,becauseI'musingtheimagesforeducationalpurposes,andgiving thebookawayforfree.Likewise,ifthephotocreditsayscourtesyof...,"thatmeansthecopyrightownergave mepermissiontouseit,butthatdoesn'tmeanyouhavepermissiontouseit. Cover Moon: LoewyandPuiseux,1894. Contents Ballerina: RickDikeman,1981,GFDL1.2,fromthe WikipediaarticleonballetretouchedbyB.Crowell. Contents Bee,motorcyclist: seebelow. 19 Mars ClimateOrbiter: NASA/JPL/CIT. 43 Bee: WikipediauserFir0002,GFDLlicensed. 62 AlbertEinstein: publicdomain. 63 E.Colibacteria: EricErbe,digitalcolorizationbyChristopherPooley,bothofUSDA,ARS, EMU.Apublic-domainproductoftheAgriculturalResearchService.. 70 Trapeze: CalvertLitho.Co.,Detroit, ca.1890. 73 Gymnasticswheel: CopyrightHansGenten,Aachen,Germany.Thecopyrightholderofthisle allowsanyonetouseitforanypurpose,providedthatthisremarkisreferencedorcopied.". 73 Highjumper: DuniaYoung. 81 Rocketsled: U.S.AirForce,publicdomainworkoftheU.S.Government. 81 Aristotle: FrancescoHayez,1811. 81 Shanghai: AgnieszkaBojczuk,GFDL1.2. 81 AngelStadium: U.S.MarineCorps, StaSgt.ChadMcMeen,publicdomainworkoftheU.S.Government. 81 JetsoverNewYork: U.S.AirForce, Tech.Sgt.SeanMateoWhite,publicdomainworkoftheU.S.Government. 90 Tuna'smigration: Modied fromagureinBlocketal. 91 Galileo'strial: CristianoBanti. 111 InternationalSpaceStation: NASA. 111 Weightlessastronauts: NASA. 110 SpaceShipOne: courtesyofScaledCompositesLLC. 97 Gravitymap: USNavy,EuropeanSpaceAgency,D.Sandwell,andW.Smith. 123 Newton: GodfreyKneller, 1702. 146 Spaceshuttlelaunch: NASA. 147 Swimmer: AdrianPingstoneWikipediauserArpingstone, publicdomain. 157 Locomotive: LocomotiveCyclopediaofAmericanPractice,1922,publicdomain. 158 Hummer: WikimediacommonsuserBull-Doser,publicdomain. 158 Prius: WikimediacommonsuserIFCAR, publicdomain. 162 GoldenGateBridge: WikipediauserDschwen,GFDLlicensed. 168 Footballplayer andoldlady: HazelAbaya. 175 Ringtoss: ClarenceWhite,1899. 187 AerialphotoofMondavivineyards: NASA. 199 Gallopinghorse: EadweardMuybridge,1878. 205 Sled: ModiedfromMillikanandGale,1920. 213 Hangingboy: MillikanandGale,1927. 214 Hurricanetrack: Publicdomain,NASAandWikipedia userNilfanion. 219 Motorcyclist: WikipediauserFir0002,GFDLlicensed. 216 Craney: Wikipediauser Pinzo,publicdomain. 224 Spacecolony: NASA. 230 TychoBrahe: publicdomain. 235 PlutoandCharon: HubbleSpaceTelescope,STSCi. 229 Saturn: Voyager2team,NASA. 251 Uranus: Voyager2team,NASA. 251 Earth: Apollo11,NASA. 244 WMAP: NASA.

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Appendix3:HintsandSolutions AnswerstoSelf-Checks AnswerstoSelf-ChecksforChapter0 Page21,self-checkA: Ifonlyhehasthespecialpowers,thenhisresultscanneverbe reproduced. Page22,self-checkB: Theywouldhavehadtoweightherays,orcheckforalossofweight intheobjectfromwhichtheywerehaveemitted.Fortechnicalreasons,thiswasnotameasurementtheycouldactuallydo,hencetheopportunityfordisagreement. Page29,self-checkC: Adictionarymightdenestrong"aspossessingpowerfulmuscles," butthat'snotanoperationaldenition,becauseitdoesn'tsayhowtomeasurestrengthnumerically.Onepossibleoperationaldenitionwouldbethenumberofpoundsapersoncanbench press. Page32,self-checkD: Amicrosecondis1000timeslongerthanananosecond,soitwould seemlike1000seconds,orabout20minutes. Page33,self-checkE: Exponentshavetodowithmultiplication,notaddition.Therstline shouldbe100timeslongerthanthesecond,notjusttwiceaslong. Page37,self-checkF: Thevariousestimatesdierby5to10million.TheCIA'sestimate includesaridiculousnumberofgratuitoussignicantgures.DoestheCIAunderstandthat everyday,peopleinarebornin,diein,immigrateto,andemigratefromNigeria? Page37,self-checkG: 4;2;2 AnswerstoSelf-ChecksforChapter1 Page44,self-checkA: 1yd 2 ft = 1yd 2 =9ft 2 1yd 3 ft = 1yd 3 =27ft 3 AnswerstoSelf-ChecksforChapter2 Page73,self-checkA: Coastingonabikeandcoastingonskatesgiveone-dimensionalcenterof-massmotion,butrunningandpedalingrequiremovingbodypartsupanddown,whichmakes thecenterofmassmoveupanddown.Theonlyexampleofrigid-bodymotioniscoastingon skates.Coastingonabikeisnotrigid-bodymotion,becausethewheelstwist. Page73,self-checkB: Byshiftinghisweightaround,hecancausethecenterofmassnotto coincidewiththegeometriccenterofwhewheel. Page74,self-checkC: apointintime;timeintheabstractsense;atimeinterval

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Page75,self-checkD: Zero,becausetheafter"andbefore"valuesof x arethesame. Page82,self-checkE: Theeectonlyoccursduringblasto,whentheirvelocityis changing.Oncetherocketenginesstopring,theirvelocitystopschanging,andtheynolonger feelanyeect.Itisonlyanobservableeectofyourmotionrelativetotheair. AnswerstoSelf-ChecksforChapter3 Page93,self-checkA: Itsspeedincreasesatasteadyrate,sointhenextseconditwilltravel 19cm. AnswerstoSelf-ChecksforChapter4 Page134,self-checkA: Thecaseof =0representsanobjectfallinginavacuum,i.e., thereisnodensityofair.Theterminalvelocitywouldbeinnite.Physically,weknowthatan objectfallinginavacuumwouldneverstopspeedingup,sincetherewouldbenoforceofair frictiontocanceltheforceofgravity.The4-cmballwouldhaveamassthatwasgreaterbya factorof4 4 4,butitscross-sectionalareawouldbegreaterbyafactorof4 4.Itsterminal velocitywouldbegreaterbyafactorof p 4 3 = 4 2 =2.Itisn'tofanygeneralimportance.It's justanexampleofonephysicalsituation.Youshouldnotmemorizeit. Page136,self-checkB: Thisismotion,notforce.Thisisadescriptionofhowthe subisabletogetthewatertoproduceaforwardforceonit.Thesubrunsoutofenergy, notforce. AnswerstoSelf-ChecksforChapter5 Page147,self-checkA: Thesprinterpushesbackwardagainsttheground,andbyNewton's thirdlaw,thegroundpushesforwardonher.Laterintherace,sheisnolongeraccelerating, buttheground'sforwardforceisneededinordertocanceloutthebackwardforces,suchasair friction. Page154,self-checkB: It'skineticfriction,becauseheruniformisslidingoverthedirt. It'sstaticfriction,becauseeventhoughthetwosurfacesaremovingrelativetothelandscape, they'renotslippingovereachother.Onlykineticfrictioncreatesheat,aswhenyourub yourhandstogether.Ifyoumoveyourhandsupanddowntogetherwithoutslidingthemacross eachother,noheatisproducedbythestaticfriction. Page155,self-checkC: Frictionlessicecancertainlymakeanormalforce,sinceotherwisea hockeypuckwouldsinkintotheice.Frictionisnotpossiblewithoutanormalforce,however: wecanseethisfromtheequation,orfromcommonsense,e.g.,whileslidingdownaropeyou donotgetanyfrictionunlessyougriptherope. Page156,self-checkD: Normalforcesarealwaysperpendiculartothesurfaceofcontact, whichmeansrightorleftinthisgure.Normalforcesarerepulsive,sothecli'sforceonthe feetistotheright,i.e.,awayfromthecli.Frictionalforcesarealwaysparalleltothesurface ofcontact,whichmeansrightorleftinthisgure.Staticfrictionalforcesareinthedirection thatwouldtendtokeepthesurfacesfromslippingovereachother.Ifthewheelwasgoingto slip,itssurfacewouldbemovingtotheleft,sothestaticfrictionalforceonthewheelmustbe inthedirectionthatwouldpreventthis,i.e.,totheright.Thismakessense,becauseitisthe 267

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staticfrictionalforcethatacceleratesthedragster.Normalforcesarealwaysperpendicular tothesurfaceofcontact.Inthisdiagram,thatmeanseitherupandtotheleftordownandto theright.Normalforcesarerepulsive,sotheballispushingthebatawayfromitself.Therefore theball'sforceisdownandtotherightonthisdiagram. AnswerstoSelf-ChecksforChapter6 Page177,self-checkA: Thewindincreasestheball'soverallspeed.Ifyouthinkaboutit intermsofoverallspeed,it'snotsoobviousthattheincreasedspeedisexactlysucientto compensateforthegreaterdistance.However,itbecomesmuchsimplerifyouthinkaboutthe forwardmotionandthesidewaysmotionastwoseparatethings.Supposetheballisinitially movingatonemeterpersecond.Evenifitpicksupsomesidewaysmotionfromthewind,it's stillgettingclosertothewallbyonemetereverysecond. AnswerstoSelf-ChecksforChapter7 Page189,self-checkA:v = r = t Page189,self-checkB: Page193,self-checkC:A )]TJ/F17 10.9091 Tf 9.662 0 Td [(B isequivalentto A + )]TJ/F17 10.9091 Tf 8.484 0 Td [(B ,whichcanbecalculatedgraphically byreversing B toform )]TJ/F17 10.9091 Tf 8.485 0 Td [(B ,andthenaddingitto A AnswerstoSelf-ChecksforChapter8 Page202,self-checkA: Itisspeedingup,becausethenalvelocityvectorhasthegreater magnitude.Theresultwouldbezero,whichwouldmakesense.Speedingupproduced a v vectorinthesamedirectionasthemotion.Slowingdownwouldhavegivena v that bointedbackward. Page203,self-checkB: Aswehavealreadyseen,theprojectilehas a x =0and a y = )]TJ/F20 10.9091 Tf 8.485 0 Td [(g ,so theaccelerationvectorispointingstraightdown. AnswerstoSelf-ChecksforChapter9 Page217,self-checkA: Uniform.Theyhavethesamemotionasthedrumitself,which isrotatingasonesolidpiece.Nopartofthedrumcanberotatingatadierentspeedfromany otherpart.2Nonuniform.Gravityspeedsituponthewaydownandslowsitdownonthe wayup. AnswerstoSelf-ChecksforChapter10 Page232,self-checkA: Itwouldjuststaywhereitwas.Plugging v =0intoeq.[1]wouldgive F =0,soitwouldnotacceleratefromrest,andwouldneverfallintothesun.Noastronomer hadeverobservedanobjectthatdidthat! 268 Appendix3:HintsandSolutions

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Page233,self-checkB: F / mr=T 2 / mr= r 3 = 2 2 / mr=r 3 = m=r 2 Page236,self-checkC: Theequal-arealawmakesequallygoodsenseinthecaseofahyperbolicorbitandobservationsverifyit.TheellipticalorbitlawhadtobegeneralizedbyNewton toincludehyperbolas.Thelawofperiodsdoesn'tmakesenseinthecaseofahyperbolicorbit, becauseahyperbolaneverclosesbackonitself,sothemotionneverrepeats. Page240,self-checkD: Aboveyouthereisasmallpartoftheshell,comprisingonlyatiny fractionoftheearth'smass.Thispartpullsyouup,whilethewholeremainderoftheshellpulls youdown.However,thepartaboveyouisextremelyclose,soitmakessensethatitsforceon youwouldbefaroutofproportiontoitssmallmass. SolutionstoSelectedHomeworkProblems SolutionsforChapter0 Page40,problem6: 134mg 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 g 1mg 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 kg 1g =1.34 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 kg Page41,problem8: aLet'sdo10.0gand1000g.Thearithmeticmeanis505grams.It comesouttobe0.505kg,whichisconsistent.bThegeometricmeancomesouttobe100 gor0.1kg,whichisconsistent.cIfwemultiplymetersbymeters,wegetsquaremeters. Multiplyinggramsbygramsshouldgivesquaregrams!Thissoundsstrange,butitmakessense. Takingthesquarerootofsquaregramsg 2 givesgramsagain.dNo.Thesuperdupermean oftwoquantitieswithunitsofgramswouldn'tevenbesomethingwithunitsofgrams!Related tothisshortcomingisthefactthatthesuperdupermeanwouldfailthekindofconsistencytest carriedoutinthersttwopartsoftheproblem. SolutionsforChapter1 Page61,problem10: 1mm 2 1cm 10mm 2 =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 cm 2 Page61,problem11: Thebiggerscopehasadiameterthat'stentimesgreater.Areascales asthesquareofthelineardimensions,soitslight-gatheringpowerisahundredtimesgreater 10. Page61,problem12: SincetheydierbytwostepsontheRichterscale,theenergyofthe biggerquakeis10000timesgreater.Thewaveformsahemisphere,andthesurfaceareaofthe hemisphereoverwhichtheenergyisspreadisproportionaltothesquareofitsradius.Ifthe amountofvibrationwasthesame,thenthesurfaceareasmuchbeintheratioof10000:1,which meansthattheratiooftheradiiis100:1. Page62,problem17: Theconeofmixedginandvermouthisthesameshapeastheconeof vermouth,butitslineardimensionsaredoubled,soitsvolumeis8timesgreater.Theratioof 269

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gintovermouthis7to1. Page62,problem19: Scalingdownthelineardimensionsbyafactorof1/10reducesthe volumebyafactorof = 10 3 =1 = 1000,soifthewholecubeisaliter,eachsmalloneisone milliliter. Page63,problem20: aThey'realldenedintermsoftheratioofsideofatriangleto another.Forinstance,thetangentisthelengthoftheoppositesideoverthelengthofthe adjacentside.Dividingmetersbymetersgivesaunitlessresult,sothetangent,aswellasthe othertrigfunctions,isunitless.bThetangentfunctiongivesaunitlessresult,sotheunitson theright-handsidehadbettercancelout.Theydo,becausethetopofthefractionhasunitsof meterssquared,andsodoesthebottom. Page63,problem21: Let'sestimatetheGreatWall'smass,andthengureouthowmany bricksthatwouldrepresent.ThewallisfamousbecauseitcoversprettymuchallofChina's northernborder,solet'ssayit's1000kmlong.Frompictures,itlookslikeit'sabout10mhigh and10mwide,sothetotalvolumewouldbe10 6 m 10m 10m=10 8 m 3 .Ifasinglebrick hasavolumeof1liter,or10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 m 3 ,thenthisrepresentsabout10 11 bricks.Ifonepersoncan lay10bricksinanhourtakingintoaccountallthepreparation,etc.,thenthiswouldbe10 10 man-hours. SolutionsforChapter2 Page89,problem4: 1light-year= v t = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(3 10 8 m = s year = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(3 10 8 m = s year 365days 1year 24hours 1day 3600s 1hour =9.5 10 15 m Page89,problem5: Velocityisrelative,sohavingtoleantellsyounothingaboutthetrain's velocity.FullertonismovingatahugespeedrelativetoBeijing,butthatdoesn'tproduceany noticeableeectineithercity.Thefactthatyouhavetoleantellsyouthatthetrainis changing itsspeed,butitdoesn'ttellyouwhatthetrain'scurrentspeedis. Page89,problem7: Tothepersonridingthemovingbike,bugAissimplygoingincircles. Theonlydierencebetweenthemotionsofthetwowheelsisthatoneistravelingthroughspace, butmotionisrelative,sothisdoesn'thaveanyeectonthebugs.It'sequallyhardforeachof them. Page90,problem10: Inonesecond,theshipmoves v meterstotheeast,andtheperson moves v metersnorthrelativetothedeck.Relativetothewater,hetracesthediagonalofa trianglewhoselengthisgivenbythePythagoreantheorem, v 2 + v 2 1 = 2= p 2 v .Relativeto thewater,heismovingata45-degreeanglebetweennorthandeast. SolutionsforChapter3 Page117,problem14: 270 Appendix3:HintsandSolutions

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Page117,problem15: Taking g tobe10m = s 2 ,thebulletloses10m/sofspeedeverysecond, soitwilltake10stocometoastop,andthenanother10stocomebackdown,foratotalof 20s. Page117,problem16: x = 1 2 at 2 ,soforaxedvalueof x ,wehave t / 1 = p a .Decreasing a byafactorof3meansthat t willincreasebyafactorof p 3=1.7.Thegivenpieceofdata, 3,onlyhasonesigg,butroundingthenalresultotoonesigg,giving2ratherthen1.7, wouldbealittletoosevere.Asdiscussedinsection0.10,siggsareonlyaruleofthumb,and whenindoubt,youcanchangetheinputdatatoseehowmuchtheoutputwouldhavechanged. TheratioofthegravitationaleldsonEarthandMarsmustbeintherangefrom2.5to3.5, sinceotherwisethegivendatawouldnothavebeenroundedoto3.Usingthisrangeofinputs, thepossiblerangeofvaluesforthenalresultbecomes1.6to1.9.Thenaldigitinthe1.7 isthereforealittleuncertain,butit'snotcompletegarbage.Itcarriesusefulinformation,and shouldnotbethrownout. Page117,problem17: v = d x d t =10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 t 2 a = d v d t = )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 t = )]TJ/F15 10.9091 Tf 8.484 0 Td [(18m = s 2 Page118,problem18: aSolvingfor x = 1 2 at 2 for a ,wend a =2 x=t 2 =5.51m = s 2 b v = p 2 a x =66.6m/s.cTheactualcar'snalvelocityislessthanthatoftheidealized constant-accelerationcar.Iftherealcarandtheidealizedcarcoveredthequartermileinthe sametimebuttherealcarwasmovingmoreslowlyattheendthantheidealizedone,thereal carmusthavebeengoingfasterthantheidealizedcaratthebeginningoftherace.Therealcar apparentlyhasagreateraccelerationatthebeginning,andlessaccelerationattheend.This makesense,becauseeverycarhassomemaximumspeed,whichisthespeedbeyondwhichit cannotaccelerate. Page118,problem19: Sincethelinesareatintervalsofonem/sandonesecond,eachbox representsonemeter.From t =0to t =2s,theareaunderthecurverepresentsapositive x of6m.Thetrianglehashalftheareaofthe2 6rectangleittsinside.After t =2s,the areaabovethecurverepresentsnegative x .Toget )]TJ/F15 10.9091 Tf 8.484 0 Td [(6mworthofarea,weneedtogooutto t =6s,atwhichpointthetriangleundertheaxishasawidthof4sandaheightof3m/s,for anareaof6mhalfof3 4. Page118,problem20: aWechooseacoordinatesystemwithpositivepointingtotheright. Somepeoplemightexpectthattheballwouldslowdownonceitwasonthemoregentleramp. 271

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Thismaybetrueifthereissignicantfriction,butGalileo'sexperimentswithinclinedplanes showedthatwhenfrictionisnegligible,aballrollingonaramphasconstantacceleration,not constantspeed.Thespeedstopsincreasingasquicklyoncetheballisonthemoregentleslope, butitstillkeepsonincreasing.Thea-tgraphcanbedrawnbyinspectingtheslopeofthev-t graph. bTheballwillrollbackdown,sothesecondhalfofthemotionisthesameasinparta.In therstrisinghalfofthemotion,thevelocityisnegative,sincethemotionisintheopposite directioncomparedtothepositive x axis.Theaccelerationisagainfoundbyinspectingthe slopeofthev-tgraph. Page118,problem21: Thisisacasewhereit'sprobablyeasiesttodrawtheacceleration graphrst.Whiletheballisintheairbc,de,etc.,theonlyforceactingonitisgravity,so itmusthavethesame,constantaccelerationduringeachhop.Choosingacoordinatesystem wherethepositive x axispointsup,thisbecomesanegativeaccelerationforceintheopposite directioncomparedtotheaxis.Duringtheshorttimesbetweenhopswhentheballisincontact withthegroundcd,ef,etc.,itexperiencesalargeacceleration,whichturnsarounditsvelocity veryrapidly.Theseshortpositiveaccelerationsprobablyaren'tconstant,butit'shardtoknow howthey'dreallylook.Wejustidealizethemasconstantaccelerations.Similarly,thehand's forceontheballduringthetimeabisprobablynotconstant,butwecandrawitthatway, sincewedon'tknowhowtodrawitmorerealistically.Sinceouraccelerationgraphconsists ofconstant-accelerationsegments,thevelocitygraphmustconsistoflinesegments,andthe positiongraphmustconsistofparabolas.Onthe x graph,Ichosezerotobetheheightofthe centeroftheballabovetheoorwhentheballisjustlyingontheoor.Whentheballis touchingtheoorandcompressed,asinintervalcd,itscenterisbelowthislevel,soits x is negative. 272 Appendix3:HintsandSolutions

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Page118,problem22: Wehave v 2 f =2 a x ,sothedistanceisproportionaltothesquareof thevelocity.Togetuptohalfthespeed,theballneeds1/4thedistance,i.e., L= 4. SolutionsforChapter4 Page142,problem7: a = v= t ,andalso a = F=m ,so t = v a = m v F = kgm = s )]TJ/F15 10.9091 Tf 10.909 0 Td [(20m = s 3000N =10s Page143,problem10: aThisisameasureofthebox'sresistancetoachangeinitsstate ofmotion,soitmeasuresthebox'smass.Theexperimentwouldcomeoutthesameinlunar gravity. bThisisameasureofhowmuchgravitationalforceitfeels,soit'sameasureofweight.In lunargravity,theboxwouldmakeasoftersoundwhenithit. cAsinparta,thisisameasureofitsresistancetoachangeinitsstateofmotion:itsmass. Gravityisn'tinvolvedatall. SolutionsforChapter5 Page170,problem14: a topspring'srightwardforceonconnector ...connector'sleftwardforceontopspring bottomspring'srightwardforceonconnector ...connector'sleftwardforceonbottomspring hand'sleftwardforceonconnector ...connector'srightwardforceonhand 273

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Lookingatthethreeforcesontheconnector,weseethatthehand'sforcemustbedoublethe forceofeitherspring.Thevalueof x )]TJ/F20 10.9091 Tf 10.511 0 Td [(x o isthesameforbothspringsandforthearrangement asawhole,sothespringconstantmustbe2 k .Thiscorrespondstoastierspringmoreforce toproducethesameextension. bForcesinwhichtheleftspringparticipates: hand'sleftwardforceonleftspring ...leftspring'srightwardforceonhand rightspring'srightwardforceonleftspring ...leftspring'sleftwardforceonrightspring Forcesinwhichtherightspringparticipates: leftspring'sleftwardforceonrightspring ...rightspring'srightwardforceonleftspring wall'srightwardforceonrightspring ...rightspring'sleftwardforceonwall Sincetheleftspringisn'taccelerating,thetotalforceonitmustbezero,sothetwoforcesacting onitmustbeequalinmagnitude.Thesameappliestothetwoforcesactingontherightspring. TheforcesbetweenthetwospringsareconnectedbyNewton'sthirdlaw,soalleightofthese forcesmustbeequalinmagnitude.Sincethevalueof x )]TJ/F20 10.9091 Tf 10.267 0 Td [(x o forthewholesetupisdoublewhat itisforeitherspringindividually,thespringconstantofthewholesetupmustbe k= 2,which correspondstoalessstispring. Page170,problem16: aSpringconstantsinparalleladd,sothespringconstanthastobe proportionaltothecross-sectionalarea.Twospringsinseriesgivehalfthespringconstant,three springsinseriesgive1/3,andsoon,sothespringconstanthastobeinverselyproportional tothelength.Summarizing,wehave k / A=L .bWiththeYoung'smodulus,wehave k = A=L E .ThespringconstanthasunitsofN = m,sotheunitsof E wouldhavetobeN = m 2 Page171,problem18: aTheswimmer'saccelerationiscausedbythewater'sforceonthe swimmer,andtheswimmermakesabackwardforceonthewater,whichacceleratesthewater backward.bTheclub'snormalforceontheballacceleratestheball,andtheballmakesa backwardnormalforceontheclub,whichdeceleratestheclub.cThebowstring'snormalforce acceleratesthearrow,andthearrowalsomakesabackwardnormalforceonthestring.This forceonthestringcausesthestringtoacceleratelessrapidlythanitwouldifthebow'sforce wastheonlyoneactingonit.dThetracks'backwardfrictionalforceslowsthelocomotive down.Thelocomotive'sforwardfrictionalforcecausesthewholeplanetearthtoaccelerateby atinyamount,whichistoosmalltomeasurebecausetheearth'smassissogreat. Page171,problem20: Theperson'snormalforceontheboxispairedwiththebox'snormal forceontheperson.Thedirt'sfrictionalforceontheboxpairswiththebox'sfrictionalforce onthedirt.Theearth'sgravitationalforceontheboxmatchesthebox'sgravitationalforceon theearth. Page172,problem26: aAliterofwaterhasamassof1.0kg.Themassisthesamein allthreelocations.Massindicateshowmuchanobjectresistsachangeinitsmotion.Ithas nothingtodowithgravity.bThetermweight"referstotheforceofgravityonanobject. Thebottle'sweightonearthis F W = mg =9.8N.Itsweightonthemoonisaboutonesixth thatvalue,anditsweightininterstellarspaceiszero. 274 Appendix3:HintsandSolutions

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SolutionsforChapter6 Page184,problem5: aTheeasieststrategyistondthetimespentaloft,andthennd therange.Theverticalmotionandthehorizontalmotionareindependent.Theverticalmotion hasacceleration )]TJ/F20 10.9091 Tf 8.485 0 Td [(g ,andthecannonballspendsenoughtimeintheairtoreverseitsvertical velocitycomponentcompletely,sowehave v y = v yf )]TJ/F20 10.9091 Tf 10.909 0 Td [(vyi = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 v sin Thetimespentaloftistherefore t = v y =a y =2 v sin =g Duringthistime,thehorizontaldistancetraveledis R = v x t =2 v 2 sin cos =g bTherangebecomeszeroatboth =0andat =90 .The =0casegiveszerorange becausetheballhitsthegroundassoonasitleavesthemouthofthecannon.A90-degreeangle giveszerorangebecausethecannonballhasnohorizontalmotion. SolutionsforChapter8 Page212,problem8: Wewanttondoutaboutthevelocityvector v BG ofthebulletrelative totheground,soweneedtoaddAnnie'svelocityrelativetotheground v AG tothebullet's velocityvector v BA relativetoher.Lettingthepositive x axisbeeastand y north,wehave v BA x =mi = hrcos45 =100mi = hr v BA y =mi = hrsin45 =100mi = hr and v AG x =0 v AG y =30mi = hr. Thebullet'svelocityrelativetothegroundthereforehascomponents v BG x =100mi = hr and v BG y =130mi = hr. Itsspeedonimpactwiththeanimalisthemagnitudeofthisvector j v BG j = p mi = hr 2 +mi = hr 2 =160mi = hr 275

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roundedoto2signicantgures. Page212,problem9: Sinceitsvelocityvectorisconstant,ithaszeroacceleration,andthe sumoftheforcevectorsactingonitmustbezero.Therearethreeforcesactingontheplane: thrust,lift,andgravity.Wearegiventhersttwo,andifwecanndthethirdwecaninferits mass.Thesumofthe y componentsoftheforcesiszero,so 0= F thrust y + F lift y + F W y = j F thrust j sin + j F lift j cos )]TJ/F20 10.9091 Tf 10.909 0 Td [(mg Themassis m = j F thrust j sin + j F lift j cos =g =6.9 10 4 kg Page212,problem10: aSincethewagonhasnoacceleration,thetotalforcesinboththe xandydirectionsmustbezero.Therearethreeforcesactingonthewagon: F T F W ,andthe normalforcefromtheground, F N .Ifwepickacoordinatesystemwithxbeinghorizontalandy vertical,thentheanglesoftheseforcesmeasuredcounterclockwisefromthexaxisare90 )]TJ/F20 10.9091 Tf 10.742 0 Td [( 270 ,and90 + ,respectively.Wehave F x total = j F T j cos )]TJ/F20 10.9091 Tf 10.909 0 Td [( + j F W j cos + j F N j cos + F y total = j F T j sin )]TJ/F20 10.9091 Tf 10.909 0 Td [( + j F W j sin + j F N j sin + whichsimpliesto 0= j F T j sin )-222(j F N j sin 0= j F T j cos )-222(j F W j + j F N j cos Thenormalforceisaquantitythatwearenotgivenanddonotwithtond,soweshould chooseittoeliminate.Solvingtherstequationfor j F N j =sin = sin j F T j ,weeliminate j F N j fromthesecondequation, 0= j F T j cos )-222(j F W j + j F T j sin cos = sin andsolvefor j F T j ,nding j F T j = j F W j cos +sin cos = sin Multiplyingboththetopandthebottomofthefractionbysin ,andusingthetrigidentityfor sin + givesthedesiredresult, j F T j = sin sin + j F W j bThecaseof =0,i.e.,pullingstraightuponthewagon,resultsin j F T j = j F W j :wesimply supportthewagonanditglidesuptheslopelikeachair-liftonaskislope.Inthecaseof =180 )]TJ/F20 10.9091 Tf 10.99 0 Td [( j F T j becomesinnite.Physicallythisisbecausewearepullingdirectlyintothe ground,sonoamountofforcewillsuce. 276 Appendix3:HintsandSolutions

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Page213,problem11: aIftherewasnofriction,theangleofreposewouldbezero,sothe coecientofstaticfriction, s ,willdenitelymatter.Wealsomakeupsymbols m and g for theangleoftheslope,themassoftheobject,andtheaccelerationofgravity.Theforcesform atrianglejustliketheoneinsection8.3,butinsteadofaforceappliedbyanexternalobject, wehavestaticfriction,whichislessthan s j F N j .Asinthatexample, j F s j = mg sin ,and j F s j < s j F N j ,so mg sin < s j F N j Fromthesametriangle,wehave j F N j = mg cos ,so mg sin < s mg cos Rearranging, < tan )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 s bBoth m and g canceledout,sotheangleofreposewouldbethesameonanasteroid. SolutionsforChapter9 Page226,problem5: Eachcyclisthasaradialaccelerationof v 2 =r =5m = s 2 .Thetangential accelerationsofcyclistsAandBare375N = 75kg=5m = s 2 Page227,problem6: aTheinwardnormalforcemustbesucienttoproducecircular motion,so j F N j = mv 2 =r Wearesearchingfortheminimumspeed,whichisthespeedatwhichthestaticfrictionforceis justbarelyabletocanceloutthedownwardgravitationalforce.Themaximumforceofstatic frictionis j F s j = s j F N j andthiscancelsthegravitationalforce,so j F s j = mg Solvingthesethreeequationsfor v gives v = r gr s bGreaterbyafactorof p 3. Page227,problem7: Theinwardforcemustbesuppliedbytheinwardcomponentofthe normalforce, j F N j sin = mv 2 =r 277

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Theupwardcomponentofthenormalforcemustcancelthedownwardforceofgravity, j F N j cos = mg Eliminating j F N j andsolvingfor ,wend =tan )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 v 2 gr SolutionsforChapter10 Page248,problem10: Newton'slawofgravitytellsusthatherweightwillbe6000times smallerbecauseoftheasteroid'ssmallermass,but13 2 =169timesgreaterbecauseofitssmaller radius.Puttingthesetwofactorstogethergivesareductioninweightbyafactorof6000/169, soherweightwillbeN = =11N. Page248,problem11: Newton'slawofgravitysays F = Gm 1 m 2 =r 2 ,andNewton'ssecond lawsays F = m 2 a ,so Gm 1 m 2 =r 2 = m 2 a .Since m 2 cancels, a isindependentof m 2 Page249,problem12: Newton'ssecondlawgives F = m D a D where F isIda'sforceonDactyl.UsingNewton'suniversallawofgravity,F= Gm I m D =r 2 ,and theequation a = v 2 =r forcircularmotion,wend Gm I m D =r 2 = m D v 2 =r Dactyl'smasscancelsout,giving Gm I =r 2 = v 2 =r Dactyl'svelocityequalsthecircumferenceofitsorbitdividedbythetimeforoneorbit: v = 2 r=T .Insertingthisintheaboveequationandsolvingfor m I ,wend m I = 4 2 r 3 GT 2 soIda'sdensityis = m I =V = 4 2 r 3 GVT 2 Page249,problem15: Newton'slawofgravitydependsontheinversesquareofthedistance, soifthetwoplanets'masseshadbeenequal,thenthefactorof0.83 = 0.059=14indistancewould havecausedtheforceonplanetctobe14 2 =2.0 10 2 timesweaker.However,planetc'smass is3.0timesgreater,sotheforceonitisonlysmallerbyafactorof2.0 10 2 = 3.0=65. Page250,problem16: Thereasoningisreminiscentofsection10.2.FromNewton'ssecond lawwehave F = ma = mv 2 =r = m r=T 2 =r =4 2 mr=T 2 278 Appendix3:HintsandSolutions

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andNewton'slawofgravitygives F = GMm=r 2 ,where M isthemassoftheearth.Setting theseexpressionsequaltoeachother,wehave 4 2 mr=T 2 = GMm=r 2 whichgives r = GMT 2 4 2 1 = 3 =4.22 10 4 km. Thisisthedistancefromthecenteroftheearth,sotondthealtitude,weneedtosubtract theradiusoftheearth.Thealtitudeis3.58 10 4 km. Page250,problem17: Anyfractionalchangein r resultsindoublethatamountoffractional changein1 =r 2 .Forexample,raising r by1%causes1 =r 2 togodownbyverynearly2%.The fractionalchangein1 =r 2 isactually 2 = 27cm 3.84 10 5 km 1km 10 5 cm =2 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(12 Page250,problem19: aTheasteroid'smassdependsonthecubeofitsradius,andfor agivenmassthesurfacegravitydependson r )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 .Theresultisthatsurfacegravityisdirectly proportionaltoradius.Halfthegravitymeanshalftheradius,oroneeighththemass.b Toagreewitha,Earth'smasswouldhavetobe1/8Jupiter's.Weassumedsphericalshapes andequaldensity.Bothplanetsareatleastroughlyspherical,sotheonlywayoutofthe contradictionisifJupiter'sdensityissignicantlylessthanEarth's. 279

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Index acceleration,95 asavector,202 constant,107 denition,102 negative,98 alchemy,21 area,105 operationaldenition,43 scalingof,45 areaunderacurve areaundera-tgraph,107 underv-tgraph,105 astrology,21 Bacon,Francis,25 calculus dierential,86 fundamentaltheoremof,113 integral,113 inventionbyNewton,86 Leibnitznotation,86 withvectors,206 cathoderays,23 centerofmass,70 motionof,71 center-of-massmotion,71 centi-metricprex,28 circularmotion,215 nonuniform,217 uniform,217 cockroaches,53 coecientofkineticfriction,155 coecientofstaticfriction,155 component dened,179 conversionsofunits,33 coordinatesystem dened,76 Copernicus,80 Darwin,24 deltanotation,74 derivative,86 second,113 DialoguesConcerningtheTwoNewSciences, 46 dynamics,66 elephant,55 energy distinguishedfromforce,135 fallingobjects,91 Feynman,94 Feynman,Richard,94 force analysisofforces,158 AristotelianversusNewtonian,124 asavector,205 attractive,151 contact,126 distinguishedfromenergy,135 frictional,153 gravitational,153 net,127 noncontact,126 normal,153 oblique,151 positiveandnegativesignsof,127 repulsive,151 transmission,161 forces classicationof,150 frameofreference dened,76 inertialornoninertial,138 FrenchRevolution,28 friction uid,157 kinetic,153,154 static,153,154 GalileoGalilei,45 gammarays,22 grandjete,71 graphing,78 graphs ofpositionversustime,76 velocityversustime,85

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highjump,73 Hooke'slaw,164 inertia principleof,80 integral,113 Kepler,230 Kepler'slaws,231 ellipticalorbitlaw,231 equal-arealaw,231 lawofperiods,231,233 kilo-metricprex,28 kilogram,30 kinematics,66 Laplace,22 Leibnitz,86 light,22 magnitudeofavector dened,188 matter,22 mega-metricprex,28 metermetricunit,30 metricsystem,27 prexes,28 micro-metricprex,28 microwaves,22 milli-metricprex,28 mksunits,30 model scientic,154 models,71 motion rigid-body,69 typesof,69 Muybridge,Eadweard,199 nano-metricprex,28 Newton rstlawofmotion,127 secondlawofmotion,131 Newton'slawsofmotion inthreedimensions,181 Newton'sthirdlaw,146 Newton,Isaac,27 denitionoftime,30 operationaldenitions,29 order-of-magnitudeestimates,57 parabola motionofprojectileon,180 Pauliexclusionprinciple,24 period ofuniformcircularmotion,222 photon,149 physics,22 POFOSTITO,147 Pope,46 projectiles,180 pulley,164 radialcomponent dened,224 radiowaves,22 reductionism,24 Renaissance,19 rotation,69 salamanders,53 scalar dened,188 scaling,45 appliedtobiology,53 scienticmethod,20 secondunit,29 SIunits,30 signicantgures,35 simplemachine dened,164 slamdunk,71 springconstant,163 Stanford,Leland,199 strain,163 Swift,Jonathan,45 tension,162 time duration,74 pointin,74 transmissionofforces,161 unitvectors,194 units,conversionof,33 vector,66 acceleration,202 addition,188 Index 281

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dened,188 force,205 magnitudeof,188 velocity,200 velocity additionofvelocities,83 asavector,200 denition,77 negative,83 vertebra,56 volume operationaldenition,43 scalingof,45 weightforce dened,126 relationshiptomass,132 weightlessness biologicaleects,110 x-rays,22 Young'smodulus,170 282 Index

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MathematicalReview Algebra Quadraticequation: Thesolutionsof ax 2 + bx + c =0 are x = )]TJ/F10 6.9738 Tf 6.227 0 Td [(b p b 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 ac 2 a Logarithmsandexponentials: ln ab =ln a +ln b e a + b = e a e b ln e x = e ln x = x ln a b = b ln a Geometry,area,andvolume areaofatriangleofbase b andheight h = 1 2 bh circumferenceofacircleofradius r =2 r areaofacircleofradius r = r 2 surfaceareaofasphereofradius r =4 r 2 volumeofasphereofradius r = 4 3 r 3 Trigonometrywitharighttriangle sin = o=h cos = a=h tan = o=a Pythagoreantheorem: h 2 = a 2 + o 2 Trigonometrywithanytriangle LawofSines: sin A = sin B = sin C LawofCosines: C 2 = A 2 + B 2 )]TJ/F15 9.9626 Tf 9.962 0 Td [(2 AB cos Propertiesofthederivativeandintegralfor studentsincalculus-basedcourses Let f and g befunctionsof x ,andlet c beaconstant. Linearityofthederivative: d d x cf = c d f d x d d x f + g = d f d x + d g d x Thechainrule: d d x f g x = f 0 g x g 0 x Derivativesofproductsandquotients: d d x fg = d f d x g + d g d x f d d x f g = f 0 g )]TJ/F20 9.9626 Tf 11.159 6.739 Td [(fg 0 g 2 Somederivatives: d d x x m = mx m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,exceptfor m =0 d d x sin x =cos x d d x cos x = )]TJ/F15 9.9626 Tf 9.409 0 Td [(sin x d d x e x = e x d d x ln x = 1 x Thefundamentaltheoremofcalculus: Z d f d x d x = f Linearityoftheintegral: Z cf x d x = c Z f x d x Z [ f x + g x ]= Z f x d x + Z g x d x Integrationbyparts: Z f d g = fg )]TJ/F26 9.9626 Tf 9.962 13.56 Td [(Z g d f Index 283

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TrigTable sin cos tan sin cos tan sin cos tan 0 0.0001.0000.00030 0.5000.8660.57760 0.8660.5001.732 1 0.0171.0000.01731 0.5150.8570.60161 0.8750.4851.804 2 0.0350.9990.03532 0.5300.8480.62562 0.8830.4691.881 3 0.0520.9990.05233 0.5450.8390.64963 0.8910.4541.963 4 0.0700.9980.07034 0.5590.8290.67564 0.8990.4382.050 5 0.0870.9960.08735 0.5740.8190.70065 0.9060.4232.145 6 0.1050.9950.10536 0.5880.8090.72766 0.9140.4072.246 7 0.1220.9930.12337 0.6020.7990.75467 0.9210.3912.356 8 0.1390.9900.14138 0.6160.7880.78168 0.9270.3752.475 9 0.1560.9880.15839 0.6290.7770.81069 0.9340.3582.605 10 0.1740.9850.17640 0.6430.7660.83970 0.9400.3422.747 11 0.1910.9820.19441 0.6560.7550.86971 0.9460.3262.904 12 0.2080.9780.21342 0.6690.7430.90072 0.9510.3093.078 13 0.2250.9740.23143 0.6820.7310.93373 0.9560.2923.271 14 0.2420.9700.24944 0.6950.7190.96674 0.9610.2763.487 15 0.2590.9660.26845 0.7070.7071.00075 0.9660.2593.732 16 0.2760.9610.28746 0.7190.6951.03676 0.9700.2424.011 17 0.2920.9560.30647 0.7310.6821.07277 0.9740.2254.331 18 0.3090.9510.32548 0.7430.6691.11178 0.9780.2084.705 19 0.3260.9460.34449 0.7550.6561.15079 0.9820.1915.145 20 0.3420.9400.36450 0.7660.6431.19280 0.9850.1745.671 21 0.3580.9340.38451 0.7770.6291.23581 0.9880.1566.314 22 0.3750.9270.40452 0.7880.6161.28082 0.9900.1397.115 23 0.3910.9210.42453 0.7990.6021.32783 0.9930.1228.144 24 0.4070.9140.44554 0.8090.5881.37684 0.9950.1059.514 25 0.4230.9060.46655 0.8190.5741.42885 0.9960.08711.430 26 0.4380.8990.48856 0.8290.5591.48386 0.9980.07014.301 27 0.4540.8910.51057 0.8390.5451.54087 0.9990.05219.081 28 0.4690.8830.53258 0.8480.5301.60088 0.9990.03528.636 29 0.4850.8750.55459 0.8570.5151.66489 1.0000.01757.290 90 1.0000.000 1 284 Index

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Index 285

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286 Index

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Index 287

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288 Index

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Index 289

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UsefulData MetricPrexes M-mega-10 6 k-kilo-10 3 m-milli-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(3 -Greekmumicro-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(6 n-nano-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(9 p-pico-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(12 f-femto-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(15 Centi-,10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(2 ,isusedonlyinthecentimeter. TheGreekAlphabet Aalpha Nnu Bbeta xi )-1633(gamma oOomicron delta pi Eepsilon Prho Zzeta sigma Heta Ttau theta Yupsilon Iiota phi Kkappa Xchi lambda psi Mmu omega FundamentalConstants gravitationalconstant G =6.67 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(11 N m 2 = kg 2 speedoflight c =3.00 10 8 m/s SubatomicParticles particlemasskgradiusfm electron9.109 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(31 0.01 proton1.673 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 neutron1.675 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 Theradiiofprotonsandneutronscanonlybegivenapproximately,sincetheyhavefuzzysurfaces.Forcomparison,a typicalatomisaboutamillionfminradius. NotationandUnits quantityunitsymbol distancemeter,m x x timesecond,s t t masskilogram,kg m densitykg = m 3 aream 2 squaremetersA volumem 3 cubicmetersV velocitym/s v accelerationm = s 2 a gravitationaleldJ = kg morm = s 2 g forcenewton,1N=1kg m = s 2 F pressure1Pa=1N = m 2 P energyjoule,J E powerwatt,1W=1J/s P momentumkg m = s p angularmomentumkg m 2 = sorJ s L torqueN m periods T Conversions Nonmetricunitsintermsofmetricones: 1inch=25.4mmbydenition 1pound-force=4.5newtonsofforce kg g =2.2pounds-force 1scienticcalorie=4.18J 1kcal=4.18 10 3 J 1gallon=3.78 10 3 cm 3 1horsepower=746W Whenspeakingoffoodenergy,thewordCalorie"isused tomean1kcal,i.e.,1000calories.Inwriting,thecapitalC maybeusedtoindicate1Calorie=1000calories. RelationshipsamongU.S.units: 1footft=12inches 1yardyd=3feet 1milemi=5280feet Earth,Moon,andSun bodymasskgradiuskmradiusoforbitkm earth5.97 10 24 6.4 10 3 1.49 10 8 moon7.35 10 22 1.7 10 3 3.84 10 5 sun1.99 10 30 7.0 10 5 | 290 Index

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Index 291