工程基础力学 Ⅱ 动力学 课件 Chapter 5 Dynamics of Particle、Chapter 6 Theorem of Momentum

Chapter5

DynamicsofParticleMainContents§5.1Newton’sLawsofMotion§5.2DifferentialEquationsofmotionofaParticle

§5.3ThetwotypesofbasicproblemsofparticlekineticsKineticsofaParticle

Instatics,westudytheforceactingonabody,andstudytheproblemsofequilibriumofbodiesthatareactedonforces,butdon’tstudythemotionofabodyactedonunbalancedforces;Inkinematics,weonlystudythemotionofabodyfromthegeometricalaspects,anddonotaccountfortherelationsbetweenkinematicsofaparticleandtheforcethatactsontheparticle;Inkinetics,wewilldeterminetherelationsbetweenkinematicsofaparticleandtheforcethatactsontheparticle.Comparedwiththestaticsandkinematics,kineticsisthestudyofthemoregeneralruleofthemechanicalmotionofamaterialbody.Inkinetics,weusuallyusetwomodelsofmechanics:particleandparticles.Whentheshapeandsizeofabodyisnotsignificant;arigidbodywithtranslationalmotionmaybedefinedasaparticle,theparticleconcentrateallmassoftherigidbody,andlieinthecenterofmassoftherigidbody;Sometimes,thebodymaybemodeledasaparticle,andtherotationalmotionofthebodycanbeignored;Aparticle:aparticlehasamassbutnegligiblesizeandshape.Specifically,atwhatpointcanabodybeabstractedandsimplifiedasaparticle?Forexample,instudyingthemotionoftheeartharoundthesun,itispermissibletoconsidertheearthasaparticle.KineticsofaParticleAsystemofparticles:asystemcomposedoffiniteorinfiniteparticlescontactedeachother.Ifabodycannotbestudiedasaparticle,itmustbeaccountedasasystemofparticles.Theconceptofasystemofparticlesisveryuniversal,itincludesarigidbody,adeformablebody,andasystemcomposedofmanyparticlesandbodies.

KineticsofaParticle

Kineticscanbedividedintotwoparts:kineticsofaparticleandkineticsofasystemofparticles(includekineticsofrigidbodies).

Inthischapterwediscusskineticsofaparticle,thatistodeterminetherelationsbetweenkinematicsofaparticleandtheforcethatactsontheparticle.Asfundamentalsofothertheoryofkinetics,kineticsofaparticlebasedonthefundamentalsofNewton’sthreelaws.Thischapteremphasizehowtosolvethetwotypesofproblemsofparticlekineticsapplyingbasicequationsofkineticsandusingmethodsofdifferentialandintegralcalculus.KineticsofaParticle§5.1Newton’sLawsofMotionNewtonfirstlaw(inertialaw):

Intheabsenceofappliedforces,aparticleoriginallyatrestormovingwithconstantspeedinastraightlinewillremainatrestorcontinuetomovewithconstantspeedinastraightline.Inertia:apropertyofmatterbywhichitcontinuesinitsexistingstateofrestoruniformmotioninastraightline,unlessthatstateischangedbyanexternalforce.Therefore,thislawisalsocalledtheinertialaw.Newtonsecondlaw：Ifaparticleissubjectedtoaforce,theparticlewillbeaccelerated.Theaccelerationoftheparticlewillbeinthedirectionoftheforce,andthemagnitudeoftheaccelerationwillbeproportionaltothemagnitudeoftheforceandinverselyproportionaltothemassoftheparticle.Newtonsecondlawmaybeexpressedmathematicallyasfollows.§5.1Newton’sLawsofMotionorTheaboveformulaisthebasicequationforsolvingdynamicproblems,whichiscalledthebasicdynamicequation.Thisformuladiscribestherelationshipbetweenthemotionofaparticleandtheforcesactingontheparticle.

Newtonthirdlaw(thelawofactionandreaction):Foreveryaction,thereisanequalandoppositereaction;thatis,theforcesofinteractionbetweentwoparticlesareequalinmagnitudeandoppositeindirection,andcollinear.Itshouldbenotedthatthefirsttwolawsinthefundamentalequationsofdynamicsonlyapplyininertialcoordinates.Newtonthirdlawhasnothingtodowiththeselectionofcoordinatesystems,anditappliestoallcoordinatesystems.§5.1Newton’sLawsofMotionMechanicalsystemofunits

Inmechanics，weusuallyuseInternationalsystemofunits(SI).IntheSIsystem，allunitsaredividedintothreecategoriesbaseunits,derivedunitsandauxiliaryunits.ThebasedimensionsintheSIsystemaremass,lengthandtime,andthebaseunitsarekilogram(kg),meter(m)andsecond(s).Aunitofforceisderivedunits,beingcalledNewton(N).OneNewtonforcemakeonekilogramofmassgenerateonemeter/second2ofaccelerate,thatisRadianisanauxiliaryunit,canbeusedtoformderivedunits,forexampleangularvelocityunitandangularaccelerationunit,andsoon.§5.1Newton’sLawsofMotionInengineering,weoftenuseengineeringsystemofunits.Thebasedimensionsinengineeringsystemofunitsareforcelengthandtime,andthebaseunitsarekilogramforce(kgf),meter(m)andsecond(s).Massunitisderivedunits,whenonekilogramforcemakeabodygenerateonemeter/second2ofaccelerate,themassofthebodyisoneengineeringunitmass.Thatis§5.1Newton’sLawsofMotionWhenonekilogramforce(9.80665Newton)generateacceleration,themassis9.80665kilogram，hence1massofengineeringunitmass=9.80665kilogram≈9.8kilogramOnekilogramforce（kgf）isthegravityactingonabodythathasonekilogramofmassatalatitudeofofthesea.Hence§5.1Newton’sLawsofMotionThefundamentalequationsofmotionarerepresentedasequationsindifferentialform,knownasthedifferentialequationsofmotionofaparticle.1.VectorformWhenaparticlemovesinanarbitraryspatialcurve,itspositionisrepresentedbythevectordiameter

derivedfromanarbitraryspatialfixedpointO,asshowninthefigure.§5.2DifferentialEquationsofmotionofaParticle

2.Formsinrectangularcoordinates§5.2DifferentialEquationsofmotionofaParticle

3.Formsinpathn-tcoordinates§5.2DifferentialEquationsofmotionofaParticle

Applyingthedifferentialequationsofmotionofaparticlewecansolvetwotypesofproblemsofparticlekinetics.§5.3ThetwotypesofbasicproblemsofparticlekineticsThefirstbasicproblem：knowingthemotionofaparticle,todeterminetheforceactingontheparticle.Thatis,knowingequationsofmotionofaparticle,thesecondderivativewithrespecttotimeofthepositionvectoriscalledtheacceleration,whichissubstitutedintothefundamentalequationsofmotionofaparticle,weobtaintheforceactingontheparticle.Thesecondbasicproblem：knowingtheforceactingonaparticle,todeterminethemotionoftheparticle.(forexample,todeterminethevelocity,trajectoryandequationsofmotionofaparticleandsoon).Inthefundamentalequationsofmotionofaparticle,knowingtheforceactingonaparticle,wecanobtaintheaccelerationofthemotionoftheparticle,todeterminethevelocity,trajectory,equationsofmotionofaparticlebytheaccelerationistheintegralcalculationproblem。Example

5-1

§5.3ThetwotypesofbasicproblemsofparticlekineticsThefollowingexamplesarehowtosolvetwotypesofproblemsofparticlekineticsbyapplyingthedifferentialequationsofmotionofaparticle.，Example

5-1Solution:Firstly,choosetheparticleMastheobjectofstudy,theparticleMdoesplumb-rectilinearmotion,choosethetrajectorylineastherectangularcoordinateaxis,andthedownwarddirectionispositive.ThenputtheparticleMonthegeneralpositionofthemotiontodrawitsforcediagram.TheforcesontheparticleinthispositionaregravityPanddielectricresistanceR.ThenthedifferentialequationofmotionoftheparticleMinrectangularcoordinateiswherePxandRxaretheprojectionsofPandRontheOxaxis,respectively.wehave§5.3ThetwotypesofbasicproblemsofparticlekineticsFromtheknownequationofmotionofthemassExample

5-1ThedifferentialequationofmotioncanthenbewrittenasFromtheknownequationofmotionoftheparticlewegetso,therewas§5.3ThetwotypesofbasicproblemsofparticlekineticsExample

5-2Aparticleofmassmunderahorizontalforce

movesalongthehorizontallinefromrest.Determinetheequationofmotionoftheparticle.when

，aswellasThusgetting

Solution:Fortheproblem,forceisknown,andweneedtosolvemotion.Theforceisadiscontinuoustimefunction.Theobjectofstudyisaparticle,whichisrectilinearlymoving,andtheequationispresentedalongthedirectionofmotion.

§5.3ThetwotypesofbasicproblemsofparticlekineticsExample

5-2Thus,theequationofmotionoftheparticleiswhen

,thevelocityoftheparticle

,positionoftheparticle

thesearetheinitialconditionsat.when

,

,so

Fromtheinitialconditionsat

，ThusTheforceinthisproblemisadiscontinuousfunctionoftime,sotheanalysisshouldbesegmented,whilepayingattentiontotheinitialconditionsofeachsegment.§5.3ThetwotypesofbasicproblemsofparticlekineticsExample5-3

Solution:Taketheinitialpositionoftheobjectastheorigin,andbuildthecoordinatesystemalongthedirectionoftheobject'smotion.Motionanalysis:rectilinearmotion.

§5.3ThetwotypesofbasicproblemsofparticlekineticsEstablishdifferentialequationsofmotion:Example

5-3useSubstitutingintothedifferentialequationofmotionyieldsthusso

§5.3ThetwotypesofbasicproblemsofparticlekineticsthusExample

5-3Integrateoncemoreandget

soConsideringthatx=0att=0,wegetC3=9

§5.3ThetwotypesofbasicproblemsofparticlekineticsxyOABφβωExample

5-4Acrank-guidemechanismisshowninfigure.TheangularvelocityofthecrankOAisconstant,lengthOA=r,lengthoftheconnectingrodisAB=l.Ifλ=r/lissmaller，thepointOistheoriginofthecoordinatesystem,theequationofmotionofsliderBmaybewrittenapproximatelyasfollows:Themassofthesliderisdenotedbynotationm,

NeglectfrictionandthemassoftheconnectingrodAB，whenand,respectively,determinetheforceactingattheconnectingrodAB.§5.3ThetwotypesofbasicproblemsofparticlekineticsxBβxyOABφβωExample

5-4SoSolution：ConsideringthesliderB.Whenφ=ωt,thefree-bodydiagram(FBD)ofthesliderBshowninfigure,wheretherod(AB)istwoforcemember.WritingtheequationofmotionofthesliderBKnown

when，andWeobtainthepullactedontherodAB§5.3ThetwotypesofbasicproblemsofparticlekineticsABrodundercompression.Example

5-4When

WehaveHence

§5.3ThetwotypesofbasicproblemsofparticlekineticsOlθExample

5-5Aconicalpendulumasshowninfigure.themassof

asmallballism=0.1kg,thelengthof

theropeisl=0.3m,θ=60º,thelineOCisvertical.Ifmotionofthesmallballistheuniformcircumferentialmotiononthehorizontalplane,determinethevelocityofthesmallballvandthetensileforceoftheropeF.

§5.3ThetwotypesofbasicproblemsofparticlekineticsCThetensionoftheropeisequaltothemagnitudeofthetensionF.OlθExample

5-5Solution:Choosethesmallballastheobjecttobeinvestigated.Theweightoftheballismg,thetensileforceFalongtheropeWritethedifferentialequationsofmotioninthenaturalaxis,weobtainSubstituting

intotheequationsaboveWeobtain§5.3Thetwotypesofbasicproblemsofparticlekinetics

TheEnd

Chapter6

TheoremofMomentum§6.1Momentumofaparticleandasystemofparticles§6.2Impulseofaforce§6.3Theoremofmomentum§6.4TheoremofmotionofthecenterofmassMainContentsProblemforkineticsofaparticle:establishthedifferentialequationsofmotionofaparticleandsolvethem.Problemforkineticsofasystemofparticles:

theoretically,wecanwritedown3ndifferentialequationsforasystemofnparticlesandthensolvethem.Practicalproblemsare:1.combiningandsolvingdifferentialequations(performingtheintegraloperation）isverydifficult.2.Inagreatnumberofproblemsweonlyneedtoinvestigatethemotionofthewholesystemofparticleswithoutknowingthemotionofeveryparticle.TheoremofmomentumFromthischapter,

wewilldiscussothermethodsofsolvingkineticproblems,andfirstlywewilldiscussthegeneraltheoremsofkinetics:（1）theoremofmomentum（2）theoremofmomentofmomentum（3）theoremofkineticenergy（4）someotherstheoremsderivedfromthem.TheoremofmomentumUndercertainconditions,byusingthesetheoremstosolvekineticproblemsisveryconvenient.Theypossessconcisemathematicalformsandclearphysicalsignificance;

theyshowtherelationshipbetweentwokindsofquantities,onekindisthequantitiesrelatedtomotioncharacteristics(momentum,momentofmomentum,kineticsenergyandsoon)，theotherkindisthequantitiesrelatedtotheforces(impulse,momentofaforce,workandsoon),andstudythoroughlythemechanicalmotionofobjectsfromthedifferentsides.Notes：AsNewton’slaw,theyonlyapplytotheinertialcoordinatesystem.Theycanallbederivedfromthebasicequationofkinetics.Theoremofmomentum

Inthischapterwewillinvestigatetheoremofmomentumofaparticleorasystemofparticles，andestablishtherelationshipbetweenthechangeofmomentumandtheimpulseofaforce,andstudyanotherimportantformoftheoremofmomentum——theoremofmotionofthecenterofmass.Theoremofmomentum1）Themomentumofaparticleisavectorquantitythatactsinthesamedirectionasthevelocityvector

1.Momentumofaparticle

Theproductofthemassofaparticleanditsvelocityiscalledthemomentumofaparticle.2）TheinternationalunitofmomentumisThevectorformofmomentumofaparticleTheprojectionformofmomentuminspacerectangularcoordinates§6.1Momentumofaparticleandparticles2.Momentumofasystemofparticles

whereisthetotalmassofthesystem,definingthepositionvectorofthecenterofmassCashenceThemomentumofasystemofparticlesisequaltothemassofthesystemtimesthevelocityofthecenterofmassofthesystem,thedirectionofmomentumisthesameasthedirectionofthevelocityofthecenterofmass.Thetotalmomentumofasystemofparticlesisequaltothevectorsumofthemomentaoftheindividualparticles§6.1MomentumofaparticleandparticlesArigidbodyconsistsofaninfinitenumberofmasses,inwhichthedistancebetweenanytwomassesremainsconstantandthecenterofmassisadefinitepointwithintherigidbody.Forarigidbodywithuniformlydistributedmass,thecenterofmassisalsoitsgeometriccenter.TheprojectionformulaofmomentumofasystemofparticlesinrectangularcoordinateOxyzare§6.1MomentumofaparticleandparticlesOExample1Thehomogeneousrodoflengthandmassrotatesintheverticalplaneaboutpoint,therodhastheangular,

determinethemomentumoftherod.ThevelocityofthemasscenteroftherodSolution：ThemomentumoftherodDirectionwiththesameas§6.1MomentumofaparticleandparticlesShownasthefigure,homogeneouswheelrotatesaboutthecenter,nomatterhowbigthevelocityandmass,becausethecenterofmassdoesnotmove,themomentumisalwayszero.Example2Shownasthefigure,homogeneousrollerhasmassandthevelocityoftherollercenter,

HencethemomentumisExample3§6.1MomentumofaparticleandparticlesExample

6-1ThewheelAweighsW,thehomogeneousrodABweighsP,andtherodlengthl.ThevelocityofthecenterAofthewheelatthepositionshowninFigureisv,andtheangleofinclinationofABis45°.Determinethemomentumofthesystematthisinstant.Solution:ThepointIistheinstantaneouscenteroftheABrod,thentheangularvelocityoftheABrodisThespeedofthecenterofmassofABrodis§6.1MomentumofaparticleandparticlesExample

6-1HorizontalmomentumofABrodVerticalmomentumofABrodTotalmomentumofABrod§6.1Momentumofaparticleandparticles1.ImpulseofaforceTheproductofaforceactingonabodyandtheactiontimeisimpulseofaforce.

1）Force

isaconstantvector：2）Force

isavariablevector：TheelementaryimpulseofaforceImpulseofaforce

inafinitetimeintervalTheinternationalunitofimpulse§6.2ImpulseofaforceTheprojectionformofimpulseinx,y,zaxesrespectivelyTheprojectionformofforceinrectangularcoordinate§6.2ImpulseofaforceAconcurrentforcesystemconsistingofnforcesactingonabody,theresultantforceis,thentheimpulseoftheresultantforceoftheconcurrentforcesysteminatimeintervalTheimpulseofaresultantforceinaconcurrentforcesystemisequaltothevectorsumoftheimpulsesofallcomponentforces.§6.2Impulseofaforce2.TheimpulseofaresultantforceThedifferentialofthemomentumofaparticleisequaltothevectorsumoftheelementaryimpulsesofallforcesactingontheparticle,whichiscalledthedifferentialformmomentumtheoremoftheparticle.orUsingtheconceptofmomentumtodescribethemotionofaparticle,thebasicequationofparticledynamicscanbeexpressedinanotherform1.Theoremofmomentumforoneparticle

Differentialform:§6.3TheoremofmomentumThechangeofmomentumofaparticleinafinitetimeintervalisequaltothevectorsumoftheimpulsesofallforcesactingontheparticleduringthistimeinterval.Thisisthemomentumtheoremintheintegralformoftheparticle.Integralform：2.Theoremofmomentumforasystemofparticles1）Differentialform：Assumingasystemofparticlesconsistsofnparticles，thearbitraryithparticlehasthemassandthevelocity,

theresultantexternalforcerepresentstheeffectbetweentheithparticleandadjacentbodiesorparticlesnotincludedwithinthesystem,

theresultantinternalforceisdeterminedfromtheforceswhichtheotherparticlesexertontheithparticle.§6.3Theoremofmomentum(1)Intheformula,thesecondtermontherightisthesumoftheinternalforcesoftheparticlesystem,representingthevectorsumoftheinteractionforcesbetweennparticlesintheparticlesystem.AstheoremofmomentumforaparticleThereareatotalofnsuchequations,addingthenequationsrespectivelyatbothends,weobtainSincetheinternalforcesactingbetweenparticlesoccurinequalbutoppositecollinearpairsandthereforecancelout,thevectorsumofimpulseoftheinternalforcesisequaltozero.§6.3TheoremofmomentumIntheformula,thefirsttermontherightrepresentsthevectorsumofallexternalforcesactingontheparticlesystemAs,

thedifferentialformoftheoremofmomentumforasystemofparticlesThedifferentialformoftheoremofmomentumforasystemofparticles：thefirstorderderivativewithrespecttotimeofmomentumofasystemofparticlesisequaltothevectorsumoftheexternalforcesactingonthesystem.

Theprojectionforminrectangularcoordinatesystem§6.3TheoremofmomentumMultiplybothsidesoftheformulaby,andthenintegratetimewithinthetimeinterval[t1,t2].Assumingthatthetwoinstantaneousst1,t2,andthemomentumoftheparticlesystemare,thereisanintegralformofthemomentumtheoremoftheparticlesystem:Thefirstorderderivativewithrespecttotimeoftheprojectionaboutoneaxisofmomentumofasystemofparticlesisequaltoalgebraicsumofprojectionofallexternalforcesalongthisaxisactingonthesystemofparticles.2)Integralform:§6.3TheoremofmomentumTheintegralformoftheoremofmomentumforasystemofparticles：thechangeinmomentumofasystemofparticlesduringafinitetimeintervalisequaltothevectorsumoftheimpulsesofallexternalforcesactingonthesystemduringthesametimeinterval.Theprojectionforminrectangularcoordinatesystem:Theprojectingincrementofmomentumofasystemofparticlesonanaxisduringafinitetimeintervalisequaltoalgebraicsumofprojectionoftheimpulsesofallexternalforcesactingonthesystemonthesameaxisduringthesametimeinterval.§6.3Theoremofmomentum（1）Ifthevectorsumoftheexternalforcesactingonasystemofparticlesiszero,themomentumofthesystemisconserved.ThatisIf

then

Hence

（2）Iftheprojectionofthevectorsumoftheexternalforcesonacoordinateaxisactingonasystemofparticlesiszero,theprojectionofthemomentumofthesystemonthecoordinateaxisisconserved.ThatisIf

Then,3.TheconservationlawofthelinearmomentumofasystemofparticlesTheaboveconclusioniscalledtheconservationlawofthelinearmomentumofasystemofparticlesTheinternalforcescanchangethemomentumoftheindividualparticlesofthesystem,theycannotchangethetotalmomentumofthesystem,onlytheexternalforcescanchangethetotalmomentumofthesystem.§6.3TheoremofmomentumExample

6-2ThereareobjectsAandBonahorizontalsurface,mAis2kg,mBis1kg.LetAmovewithacertainspeedandhitBwhichwasatrest,asshowninFig.6-3.AndthenAandBimpacttogetherandmoveforward,whichstopaftert=2s.LetthecoefficientofkineticfrictionbetweenA,Bandtheplanebef=1/4.DeterminethevelocityofAbeforetheimpactandtheimpulseoftheinteractionbetweenAandBfromtheimpacttotherestofAandB.

§6.3TheoremofmomentumExample

6-2Solution:TakethesystemcomposedofAandBastheobjectofstudy,andwritethemomentumtheoremalongthehorizontaldirection:

§6.3TheoremofmomentumExample

6-2WithBastheobjectofstudy,theimpactforceofAonBistransformedintotheexternalforceF.Writethemomentumtheoremalongthehorizontaldirection:Theimpactimpulsealongthehorizontaldirectionis§6.3TheoremofmomentumStepstosolvetheproblem：1.Selecttheresearchobjectandestablishthecoordinatesystem.2.Doforceanalysisandmotionanalysis.3.Applytheuniversaltheoremofdynamics.§6.3TheoremofmomentumABφExample

6-3BlockAcanslidefreelyalongthesmoothhorizontalplane,itsmassismA;themassoftheballBismB,theballwashingedtotheblockwithathinrod,asshowninfigure.Assumingthelengthoftherodisl,neglectingmass,

thesystemwasinitiallyatrest,andhadtheinitialpendulumangleφ0;letitfree,

thethinrodswingsapproximatelywiththelaw(kistheknownconstant),determinethemaximumvelocityofblockA.§6.3TheoremofmomentumABφExample

6-3TheangularvelocityoftherodisChoosetheblockandthesmallballastheobjecttobeinvestigated,thegravityandthereactionofthehorizontalplaneareinverticaldirection.Theexternalforceactingonthesystemiszeroinhorizontaldirection,thenthemomentuminhorizontaldirectionconserves.Solution：when,

itsabsolutevalueismaximum,

atthistimethereshouldbe,Thereshouldbe

,

thatis§6.3TheoremofmomentumTherefore,whenthethinrodisvertical,theballhasthemaximumhorizontalvelocityrelativetotheblock,withavalueofExample

6-3ABφWhenthevelocityvristotheleft，theblockshouldhavetheabsolutevelocitytotheright，assumingisv,theabsolutevelocityvalueoftheballtotheleftisva=vr－v.Accordingtothemomentumconservationcondition,thereisWorkingoutthevelocityoftheblock§6.3TheoremofmomentumExample

6-3ABφWhen,thereis.Atthistimetheballrelativetotheblockhasthemaximumvelocitytotheleftkφ0l,Wecanobtainthemaximumvelocityoftheblocktotheleft§6.3Theoremofmomentum1.ThecenterofmassProjectionformsofthepositionofthemasscenterinrectangularcoordinate§6.4Theoremofmotionofthecenterofmassor2.Theoremofmotionofthecenterofmass

ApplyingthedifferentialformoftheoremofmomentumForsystemofparticlesofconstantmassTheproductofthemassofthesystemofparticlesandtheaccelerationofthemasscenterisequaltothevectorsumofallexternalforcesactingonthesystem.Thatistheoremofmotionofthecenterofmass.Theinternalforcesofthesystemofparticlesdonotaffectthemotionofthecenterofmass,onlytheexternalforcescanchangethemotionofthecenterofmass.§6.4Theoremofmotionofthecenterofmass（1）if

isalwaysequalto0，then

constantTheprojectionformsoftheoremofmotionofthemasscenterinrectangularcoordinate：theproductoftheprojectionofthemassofthesystemofparticlesandtheaccelerationofthemasscenterononecoordinateaxisisequaltothealgebraicsumofprojectionofallexternalforcesactingonthesystemofparticlesonthesameaxis.

3.TheconservationlawofthemotionofthemasscenterforasystemofparticlesIfthevectorsumoftheexternalforcesactingonthesystemofparticlesisequaltozero,thecenterofmassmakesinertialmotion.§6.4TheoremofmotionofthecenterofmassIfthevelocityprojectionisequaltozeroatthebeginning,thecoordinatesofthecenterofmassalongthisaxisremainconstant.（2）if,thenisalwaysequaltoC,Ifthealgebraicsumoftheprojectionoftheexternalforcesactingonthesystem