Engineering Basic Mechanics Ⅱ Dynamics 工程基础力学 Ⅱ 动力学 课件 Chapter 6 Theorem of Momentum

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,Ifthealgebraicsumoftheprojectionoftheexternalforcesactingonthesystemofparticlesononeaxisisequaltozero,theprojectionofvelocityofthemasscenterontheaxisremainsconstant（thecenterofmassmakesinertialmotionalongtheaxis）Theconclusionaboveiscalledtheconservationlawofthemotionofthemasscenterforasystemofparticles.§6.4Theoremofmotionofthecenterofmass

Example

6-4ObjectsAandBhavemassm1andm2,respectively,andareconnectedbyanon-extendableropewrappedaroundapulleyC.Thetwoobjectsslidealongthesmoothinclinedsurfaceofaright-angledprism.ThebottomsurfaceDEoftheprismisplacedonasmoothhorizontalsurface,asshowninthisFigure.Determinethedisplacementoftheprismalongthehorizontalplane,whentheobjectAfallsdownaheighth=10cm.Letthemassoftheprismbem=4m1=16m2,themassoftheropeandpulleycanbeneglected,andthesystemisatrestattheinitialinstant.§6.4TheoremofmotionofthecenterofmassExample

6-4Solution:Thewholesystemistakenastheobjectofstudy.Theexternalforcesonthesystemarethegravitationalforcesm1g,m2g,mgandthenormalreactionforce

.Becausethesystemisatrestattheinitialinstant,thecenterofmassofthesystemisconstantinthehorizontalxdirection,i.e.,§6.4TheoremofmotionofthecenterofmassExample

6-4Letthedisplacementoftheprismalongthehorizontalplanebes,thenthetransversecoordinateofthecenterofmassofthesystemafterthemovementisThetransversecoordinateofthecenterofmassofthesystembeforethemovementofthetrigonometricprismis§6.4TheoremofmotionofthecenterofmassABxφyxOExample

6-5Asshowninfigure,thefulcrumofthependulumBisfixedontheslideblockAwhichcantranslatealongasmoothhorizontalstraighttrack，assumingthemassofA,BaremA,

mB

respectively.Whenthemotionstarts,determinethetrajectoryequationofsimplependulumB.§6.4TheoremofmotionofthecenterofmassmBgmAgExample

6-5ABxφyxOSolution：choosethesystemastheobjecttobeinvestigated，itsmotioncanbedeterminethroughthetwogeneralizedcoordinates:thecoordinatexoftheslideblockAandtheangleφof

theswingingpendulum.Becausethesystemisnotaffectedbytheexternalforceinthexdirection,andisinitiallyatrest,themomentumofthesystemconservesinthexdirection,thecoordinatexCofthemasscentershouldmaintainaconstantxC0.ThenWeobtainThecoordinateofthesimpl