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Chapter7TheoremofMomentofMomentumMainContents§7.1Momentofmomentumofaparticleandasystemofparticles§7.2Momentofinertiaofarigidbodywithrespecttotheaxis§7.3Momentofmomentumtheorem§7.4Differentialequationsfortherotationofarigidbodyaroundafixed-axis§7.5Momentofmomentumtheoremforasystemwithrespecttoitscenterofmass§7.6Differentialequationsofplanemotionofarigidbody1.Forexamplewhenasymmetricalcircularwheelrotatesaroundaunmovingcenterofmass,nomatterhowfasttheroundwheelrotates,nomatterwhatchangestherotatingstatehave,itsmomentumisalwaysequaltozero,somomentumcannotcharacterizeormeasurethemotion.2.Theoremofmomentumandtheoremofmotionofthecenterofmassdiscussedtherelationshipbetweenprincipalvectoroftheexternalforcesystemandthemotionchangeofasystemofparticles,butdidnotdiscusstheinfluenceoftheprincipalmomentoftheexternalforcesystemonthemotionchangeofthesystemofparticles.MomentofmomentumtheoremTheoremofmomentumdescribedinthepreviouschaptercannotcompletelydescribethemotionstateofasystemofparticles.Therefore,wemusthavenewconcepttodescribethesimilarmotion.Momentofmomentumtheoremisthetheoryofthedescriptionofparticlesrelativetoapoint(orafixedaxis)orthecenterofmassmotion.Assumingaparticleinaninstanthasthemomentum
,thepositionoftheparticlerelativetopointisrespectedthroughpositionvector,asshowninfigure.§7.1Momentofmomentumofaparticleandasystemofparticles1.Momentofmomentumofaparticle
Themomentofmomentumofaparticleaboutpointisdefinedasthe“moment”oftheparticle’smomentumaboutpoint,thatisEstablisharectangularcoordinatesystembyafixedpointastheorigin,thecoordinateofaparticleis,thentheanalysisofprojectiontypeofthepositionvectorandthevelocityoftheparticleare:ThemomentofmomentumofaparticlewithrespecttoapointOcanbewrittenasadeterminantform:
Themomentofmomentumofaparticlewithrespecttoafixedpointisavector,
thevectorisperpendiculartotheplaneformedbythepositionvectorandthevelocity,itsmagnitudeisequaltotheareaofparallelogramcomposedofthepositionvectorandthemomentum,itssenseisgovernedbytheright-handrule,
andthemomentofmomentumoftheparticlewithrespecttoafixedpointisanpositioningvector,whichshouldbedrawnonthecenterofmomentO.§7.1MomentofmomentumofaparticleandasystemofparticlesThemomentofmomentumofaparticlewithrespecttothepointOisprojectedtotherectangularcoordinateaxis,
accordingtotherelationshipbetweenthemomentofthevectoraboutthepointandthemomentabouttheaxisthroughthepointweknown,
momentsofmomentumoftheparticleabouteachcoordinateaxisthroughthepointOarerespectively:ThatisTheprojectionofmomentofmomentumaboutafixedpointinanyaxisthroughthepointisequaltomomentofmomentumabouttheaxis.Momentofmomentumaboutanaxisisanalgebraicquantity,theregulationofitssymbolisthesameastheregulationofthesymbolofmomentofforceaboutanaxis,afterprovidingthepositiveoftheaxis,bytheright-handruletodeterminethepositivedirection.TheunitofmomentofmomentuminSIunitsis
or§7.1Momentofmomentumofaparticleandasystemofparticles§7.1Momentofmomentumofaparticleandasystemofparticles2.MomentofmomentumofasystemofparticlesAndthereisThevectorsumofmomentofmomentumofalltheparticlesinasystemaboutpointiscalledthemomentofmomentumofthesystemofparticlesaboutthepoint,thatisThescalarsumofmomentofmomentumofallparticlesinasystemaboutanyaxisiscalledthemomentofmomentumofthesystemofparticlesabouttheaxis.Theprojectionofmomentofmomentumofasystemofparticlesaboutpointintherectangularcoordinateaxisthroughthepointisthemomentofmomentumofthesystemofparticlesabouttheaxisthroughthepoint:wheredenotesthemomentummomentoftheithparticleinthesystemforthepointO.3.Calculationofmomentofmomentumofseveralkindsofrigidbody(1)momentofmomentumofarigidbodyintranslationalmotionwithrespecttoafixedpointCalculationofmomentofmomentumofarigidbodyintranslationalmotionissimilartocalculationformulaofmomentofmomentumofaparticle,whenwecalculatemomentofmomentumofarigidbodyintranslationalmotion,therigidbodycanberegardedasaparticle,whichhasthewholemassoftherigidbodyintranslationalmotion,locatedinthecenterofmassoftherigidbody,andmovingwiththecenterofmassoftherigidbody.§7.1Momentofmomentumofaparticleandasystemofparticles3.Calculationofmomentofmomentumofseveralkindsofrigidbody(2)momentofmomentumofarigidbodyinfixed-axisrotationwithrespecttotheaxisofrotation:
Themomentofmomentumoftheentirerigidbodytothez-axisisMomentofmomentumofarigidbodyinfixed-axisrotationwithrespecttotheaxisofrotationisequaltotheproductofthemassmomentofinertiaoftherigidbodyabouttheaxisanditstheangularvelocity.Lettherigidbodyrotatearoundafixedaxiswithangularvelocity.Themassofthethmassontherigidbodyis,thedistancefromthemasstothez-axisis,andthevelocityofthemassiswhere,
isdefinedasthemassmomentofinertiaoftherigidbodyaboutthez-axis.§7.1Momentofmomentumofaparticleandasystemofparticles§7.2Momentofinertiaofarigidbodywithrespecttoanaxis1.Conceptofthemassmomentofinertia(1)definition:thesumoftheproductofeachparticlemassofabodyandthesquareofeachparticletoanaxisdistanceiscalledthemassmomentofinertiaoftherigidbodyabouttheaxis.Forarigidbodyofcontinuousmassdistribution,then(2)Calculationofthemassmomentofinertiaofsimpleshapedbody(a)ahomogeneousslenderrodAssuminglineardensityofarodis,
consideringmicro-segment,thenthemassofthemicro-segmentis,
thusthemassmomentofinertiaoftherodaboutz-axisisMassoftherodis,
then(b)homogeneousthincircularringAssumingmassofacircularringis,thedistancebetweenmassandthecentralaxisisequaltoradius,thusthemassmomentofinertiaofthecircularringaboutthecentralaxisis(c)ahomogeneousdiskAssumingradiusofthediskis,
massis,Thecircularplateisdividedintoaninfinitenumberofconcentricthinrings,theradiusofanyringisandthewidthis.Themassofthethinringiswhere,isthemassperunitareaofthehomogeneouscircularplate,sotherotationalinertiaofthecircularplatetothecentralaxisis§7.2Momentofinertiaofarigidbodywithrespecttoanaxis(d)homogeneousrectangularplate2.RadiusofgyrationRadiusofgyrationisdefinedasthus3.Theparallel-axistheoremTheorem:themassmomentofinertiaofarigidbodywithrespecttoanyaxisisequaltothemassmomentofinertiaoftherigidbodywithrespecttoaparallelaxisthroughthemasscenterofthebodyplustheproductofthemassofthebodyandthesquareofthedistancebetweenthetwoaxes.Thatis§7.2Momentofinertiaofarigidbodywithrespecttoanaxis§7.2MomentofinertiaofarigidbodywithrespecttoanaxisExample
7-1Figureshowsahomogeneousslenderrodofmassandlength.Determinethemassmomentofinertiaoftherodabouttheaxisthatpassesthoughthemasscenterandisperpendiculartotherodaxis.Solution:themassmomentofinertiaofthehomogeneousslenderrodaboutthez-axisthatpassesthroughitsleftendandisperpendiculartotherodaxisisUsingtheparallel-axistheorem,themassmomentofinertiaabouttheaxisisOCExample
7-2Thependulumissimplifiedasfollows.Weknownmassofhomogeneousslenderrodisandmassofhomogeneousdiskis,lengthofrodis,diameterofdiskis.Determinethemassmomentofinertiaofthependulumaboutthehorizontalaxisthatpassesthroughthesuspensionpoint.Solution:themassmomentofinertiaofthependulumaboutthehorizontalaxisOiswhereAssumingisthemassmomentofinertiaofthediskaboutthecenterC,then
Thus
§7.2MomentofinertiaofarigidbodywithrespecttoanaxisThefirstderivativeofmomentofmomentumwithrespecttotime1.MomentofmomentumtheoremofaparticleAssumingmomentofmomentumofaparticleaboutafixedpointis,themomentoftheforceaboutthesamepointis,asshowninfigureAccordingtotheoremofmomentumofaparticleandHencetheaboveequationbecomessinceHenceweobtain§7.3MomentofmomentumtheoremMomentofmomentumofaparticle:thefirstderivativeofmomentofmomentumofaparticleaboutafixedpointwithrespecttotimeisequaltothemomentaboutthesamepointoftheresultantforceactingontheparticle.Makingaprojectionoftheaboveequationontherectangularcoordinateaxiswhichtakesthecenterofmomentfortheorigin,andnotingtheprojectionofthemomentofmomentumandforceaboutapointonanaxisisequaltomomentofmomentumandforceabouttheaxis,weobtain:2.MomentofmomentumtheoremofasystemofparticlesWeassumeasystemofparticlesthatisaclosedsystemofparticles,thearbitraryithparticleissubjectedtoaresultantinternalforceandaresultantexternalforceaccordingtomomentofmomentumofaparticleweobtain§7.3MomentofmomentumtheoremTherearensameequations,addedtogetherSincetheinternalforcesoccurinequalbutoppositecollinear,thefirsttermontherightsideoftheaboveequationTheleftsideoftheaboveequationhence§7.3MomentofmomentumtheoremMomentofmomentumtheoremofasystemofparticles:thetime–derivativeofmomentofmomentumofasystemofparticlesaboutafixedpointisequaltothevectorsumofthemomentsoftheexternalforcesactingonthesystemaboutthesamepoint.TheprojectionformulaisItmustbepointedoutthat,theabovetheoremofmomentofmomentumexpressionformisonlyapplicabletoafixedpointorafixedaxis.Forageneralmovingpointormovingaxis,thetheoremofmomentofmomentumhasmorecomplicatedexpressions.3.Conservationlawofmomentofmomentum(1)Ifthemomentoftheforceactingontheparticleaboutafixedpointiszero,themomentofmomentumoftheparticleaboutthepointisconstant,thatis(2)Ifthemomentoftheforceactingontheparticleaboutafixedaxisiszero,themomentofmomentumoftheparticleabouttheaxisisconstant,thatis§7.3MomentofmomentumtheoremExample
7-3Asthepictureshows,asmoothballofmassmisplacedinsideafixedcirculartubeofradiusR.Theballisgivenaninitialsmallperturbation,anddeterminethelawofmotionofthesmallball.§7.3MomentofmomentumtheoremSolution:Thetrajectoryoftheballisaknowncirculararc,sothenaturalmethodcanbeusedtodescribethemotionoftheball.Thevelocityoftheballisalwaysalongthetangentdirectionofthearc,soitissuitabletoapplythemomentummomenttheoremtosolvetheproblem.First,thesmallballischosenastheobjectofstudy.TheballisplacedinageneralpositionofmotionwiththeforceofgravitymgandthereactionforceNofthetube,withthedirectionofpointingtothecenterO.ApplyingthemomentummomenttheoremaboutpointO(i.e.,abouttheaxispassingthroughpointOandperpendiculartotheplaneofthecirculartube),wehaveorExample
7-3§7.3MomentofmomentumtheoremConsiderorExample
7-3Substitutingtheaboveequation,yieldsThisisthedifferentialequationofmotionoftheball.Thelawofmotionoftheballisdescribedbythevariableθ.Consideringthatθissmallwhensmallmoving,sosinθ≈θ,andthentheequationcanbesimplifiedasItcanbeseenthattheballdoessimpleharmonicmotion.Thearbitraryconstantsθandαintheequationcanbedeterminedbytheinitialconditionsofmotion.TheSolutionofthisdifferentialequationis§7.3MomentofmomentumtheoremMExample
7-4Windlassofblastfurnacewhichtransportsore,showninfigure.TheradiusofdrumisR,themassism1,thedrumrotatesaboutaxisO.Thetotalmassofthecarandtheoreism2.ThemomentofcoupleactingonthedrumisM,themassmomentofinertiaofthedrumabouttherotatingaxisisJ,dipangleofthetrackisθ.Neglectthemassoftheropeandvariousfriction,determinetheacceleration
aofthecar.§7.3MomentofmomentumtheoremExample
7-4§7.3MomentofmomentumtheoremMSolution:consideringthesystemofboththecarandthedrum,consideringthecarasaparticle.Clockwiseispositive.Themomentofmomentumofasystemofparticlesaboutaxisisand
,ThemomentoftheexternalforceofthesystemisTheexternalforcesactingonthesystemofparticlesincludecouple,gravity;reactionforceofbearingand
constraintforceoftrackactingonthecar.Themomentofforceaboutaxisiszero.Decomposeintoandalongthetrackandvertically,andoffseteachother.since
,we
obtainExample
7-4ApplyingmomentofmomentumofasystemofparticlesaboutaxisO,wehaveIf,then,theaccelerationofthecarupalongtheslope.§7.3MomentofmomentumtheoremMOA
Example
7-5Trytousemomentofmomentumtheoremtoderivethedifferentialequationofmotionofsimplependulum(mathematicalpendulum).§7.3MomentofmomentumtheoremOA
,Example
7-5Solution:consideringthependulumasaparticleAmovingin
thearc,themassofthependulumism,thelengthofthecycloidisl.AssuminginanytransienttheparticleAhavethevelocityv
,theangleofthecycloid
OAandtheplumbline
is
.Choosethefixedaxis
zwhichisthroughsuspensionpointOandperpendiculartotheplaneofmotion
asmomentaxis,applyingmomentofmomentumtheoremofaparticleabouttheaxis.SincemomentofmomentumandmomentofforceareThusweobtainSimplyit,weobtaindifferentialequationofmotionofthependulum.§7.3MomentofmomentumtheoremzaallABzaaθθllABExample
7-6SmallballAandBare
connectedtothestring.Themassofeveryballism,neglecttheothercomponentmassandfriction,thesystemrotatesfreelyaroundaxisz,theinitialangularvelocityofthesystemisω0.Whenthestringisbroken,theangleofeachbarandtheplumblineisθ,determinetheangularvelocityω
ofthesystem.§7.3MomentofmomentumtheoremzaallABzaaθθllABExample
7-6Solution:themomentsofthegravityactingonthesystemandreactionforceofbearingabouttherotatingaxisarezero,soconservationofmomentofmomentumofthesystemabouttheaxis.Whenθ=0,momentofmomentumWhenθ≠0,momentofmomentumBecauseLz1=Lz2,weobtain§7.3MomentofmomentumtheoremorororAssumingtheforcesactingonarigidbodywhichrotatesaroundafixed-axisincludetheactiveforcesand
thereactionforcesofbearingshowninfigure,theseforcesareallexternalforces.Themassmomentofinertiaoftherigidbodyabouttheaxisis,theangularvelocityis,momentofmomentumaboutaxisis.Ifneglectfrictionofbearing,momentsofreactionforcesofbearingaboutaxisarezero,accordingtomomentofmomentumtheoremofthesystemofparticlesaboutaxiswehaveTheaboveequationsarecalleddifferentialequationsfortherotationofarigidbodyaroundafixed-axis.§7.4Differentialequationsfortherotationofarigidbodyaroundafixed-axisExample
7-7Themagnitudeofthemassmomentofinertiaoftherigidbodyshowswhetheritisdifficultoreasyfortherotationalstateofarigidbodytobechanged,thatis:themassmomentofinertiaisameasureofarigidbody’sinertiaconcerningitsrotationalmotion.RαOShowninfigure,weknowntheradiusofpulleyisR,themassmomentofinertiaisJ,belttensionswhichdrivepulleyareF1andF2.Determinetheangularaccelerationofpulleyα
.§7.4Differentialequationsfortherotationofarigidbodyaroundafixed-axisExample
7-7RαOSolution:accordingtodifferentialequationsforrotationofarigidbodyaroundafixed-axiswehavehence
Fromtheaboveequationwesee,onlywhenthefixedpulleyrotatesataconstantspeedor(includingstatic)ataunconstantspeed,butneglectingthemassmomentofinertiaofthepulley,belttensionwhichcrossthefixedpulleyisequal.§7.4Differentialequationsfortherotationofarigidbodyaroundafixed-axisOCbExample
7-8Compoundpendulumcomposesofarigidbodyrotatingaroundthehorizontalaxis.Weknownthemassofcompoundpendulumism,thedistancebetweencenterofgravityCandtherotatingaxisOisOC=b,themassmomentofinertiaofcompoundpendulumabouttherotatingaxisOisJO.WhenswingingstartstheslipanglebetweenOCand
theplumb
lineis
0,andinitialangularvelocityofcompoundpendulumiszero,determinetheslightswinglawofcompoundpendulum.Neglectbearingfrictionandairresistance.§7.4Differentialequationsfortherotationofarigidbodyaroundafixed-axisOCbF1F2mgExample
7-8Solution:forceasshowninfigure.Assumingangleinthecounterclockwisedirectionispositive.Whensmallangleispositive,themomentofgravityaboutpointisnegative.Accordingtodifferentialequationsfortherotationofarigidbodyaroundafixed-axiswehavehenceWhencompoundpendulumswingsslightly,makingsin
≈
.Thenafterlinearizingtheaboveequation,weobtaindifferentialequationofcompoundpendulumwhichswingsslightly.Thisisthestandarddifferentialequationofsimpleharmonicmotion.Wecanseemicro-amplitudevibrationofcompoundpendulumisalsosimpleharmonicmotion.§7.4Differentialequationsfortherotationofarigidbodyaroundafixed-axisExample
7-8OCbF1F2mgConsideringtheinitialconditionsofthemotionofcompoundpendulum:whent=0ThenmotionlawofcompoundpendulumcanbewrittenasSwingingfrequencyω0
andperiodTisrespectivelyUsingtherelationship(b)wecandeterminethemassmomentofinertiaoftherigidbody.Therefore,weputtherigidbodyintoacompoundpendulumandmeasureitsperiodTofswingbyusingtest,thenusingequation(b)wedeterminethemassmomentofinertia§7.4Differentialequationsfortherotationofarigidbodyaroundafixed-axisMomentofmomentumtheoremexpressedaboveisonlyapplicabletofixedpointorfixedaxisintheinertiareferencesystem,thenwhencenterofmomentmoves,howtoapplymomentofmomentumtheorem?Furtherstudiesshowedthat,undercertainconditions,theformofmomentofmomentumtheoremremainsthesame.Oneofthemostimportantcaseis:inthetranslationalcoordinatesystemmovingwiththecenterofmass,takingcenterofmassascenterofmoment,thentheformofmomentofmomentumtheoremremainsthesame.Takingmass
centerCastheorigin,amovingreferencesystemshowninfigure.Inthemovingreferencesystem,therelativeradiusvectorofanymassis,
relativevelocityis.§7.5MomentofmomentumtheoremforasystemwithrespecttoitscenterofmassMomentofmomentumofthesystemwithrespecttothemasscenterCisInfact,momentofmomentumofthesystemaboutthemasscentercalculatedthroughtherelativevelocityoftheparticleorthoughtheabsolutevelocitytheresultisequal,
thatisThepositionvectorofaparticle,aboutfixedpointOis,
theabsolutevelocityis,
thenmomentofmomentumofthesystemaboutfixedpointOisThefigureshows§7.5MomentofmomentumtheoremforasystemwithrespecttoitscenterofmassThusAccordingtotheoremofcompositionofvelocities,wehaveBycalculationformulaofmomentumofasystemofparticlesWheremis
thetotalmassofthesystem,
isvelocityofthemasscenterC.Substitutingtheabovetwoequations,momentofmomentumofthesystemaboutfixedpointOcanbewrittenasThelasttermofaboveequationis,
accordingtotheformulaofmasscentercoordinateispositionvectorofmasscenterCaboutmovingsystem.Cistheoriginofthemovingsystem,
obviously,
thatis,
thenthemiddletermofaboveequationiszero,
and
§7.5MomentofmomentumtheoremforasystemwithrespecttoitscenterofmassTheaboveequationshows,momentofmomentumofasystemofparticlesaboutanypointOisequaltomomentofmomentumwhichfocusesonmasscenterofthesystemaboutpointOplusmomentofmomentumofthesystemaboutmasscenterC.(vectorsum)MomentofmomentumtheoremforasystemofparticlesaboutfixedpointOcanbewrittenasExpandingtheaboveequationinbrackets,
notingtherightside,thustheaboveequationcanbewrittenasthus
Thentheaboveequationbecomes§7.5MomentofmomentumtheoremforasystemwithrespecttoitscenterofmassTherightsideofaboveequationistheprincipalmomentofexternalforceaboutcenterofmass.ThenweobtainThefirstorderderivativeabouttimeofmomentofmomentumofasystemofparticlesaboutmasscenterisequaltotheprincipalmomentofexternalforceactingonthesystemofparticlesaboutmasscenter.Thatismomentofmomentumtheoremforasystemwithrespecttoitscenterofmass.Thetheoremintheformisthesameasmomentofmomentumofasystemofparticleswithrespecttofixedpoint.§7.5Momentofmomentumtheoremforasystemwithrespecttoitscenterofmass§7.6DifferentialequationsofplanemotionofarigidbodyThepositionofrigidbodyinplanemotioncanbedeterminedbypositionofthebasepointandrotationangleofrigidbodyaroundbasepoint.ChoosemasscenterCasbasepoint,
showninfigure,itsordinatesare.AssumingDisanypointontherigidbody,
theangleofCDandx-axisis,thenpositionofrigidbodycanbedeterminedbyand.Motionofrigidbodyisdecomposedintotranslationwiththemasscenterandrotationaroundthemasscenter.ShowninfigureistranslationreferencesystemfixedtomasscenterC,
themotionofrigidbodyinplanemotionwithrespecttothemovingsystemisrotationaroundmasscenterC,
thenmomentofmomentumofrigidbodyaboutmasscenterisisthemassmomentofinertiaofarigidbodywithrespecttoanaxiswhichpassesthroughthecenterofmassandisverticaltothemotionplane,
istheangularvelocity.Assumingtheexternalforcesactingontherigidbodycanbesimplifiedasaplaneforcesystemtothemovingplaneofthemasscenter,
thenapplyingtheoremofmotionofthecenterofmassandmomentofmomentumtheoremwithrespecttothecenterofmass,weobtainisthemassofrigidbody,
isaccelerationofthemasscenter,
isangularvelocityoftherigidbody.TheaboveequationcanbewrittenasTheabovetwoequationsarecalleddifferentialequationsofplanemotionofrigidbody.§7.6DifferentialequationsofplanemotionofarigidbodyThisistheprojectionexpressionofthedifferentialequationofplanemotionofarigidbodyinarectangularcoordinatesystem.§7.6DifferentialequationsofplanemotionofarigidbodyMCrxExample
7-9Ahomogeneousroundwheelofradiusrandmassmrollsalongahorizontalline,showninfigure.AssumingradiusofgyrationofwheelisρC,momentofcoupleactingonthewheelisM.Determinetheaccelerationofthecenterofwheel.Assumingthecoefficientofthestaticslidingfrictionofthewheelonthegroundisfs,whatconditionsmustmomentofcoupleMmeet,thewheeldoesn’tslide?§7.6DifferentialequationsofplanemotionofarigidbodyaC=
rαMCrxαExample
7-9Solution:accordingtodifferentialequationsofplanemotionofarigidbody,wecanwritethefollowingthreeequations:Mandαinaclockwisedirectionispositive.sinceaCy=0,thenaCx=aC.Accordingtotheconditionofroundwheelrollingwithoutsliding,wehave§7.6DifferentialequationsofplanemotionofarigidbodyExample
7-9Simultaneoussolution,weobtain:Inordertomakeroundwheelfromstaticrollswithoutsliding,theremustbeF≤fsFN,orF≤fsmg.Thenweobtaintheconditionofroundwheelrollingwithoutsliding§7.6DifferentialequationsofplanemotionofarigidbodyMCrxαRθCExample
7-10Afterahomogeneousroundwheelofradiusrandmassmsubjectedtoaslightdisturbance,itrollsbackandforthinacirculararcofradiusR,showninfigure.Assumingthesurfaceisroughenough,roundwheelrollswithoutsliding.DeterminethelawofmotionofthemasscenterC.§7.6Differentialequationsofplanemotionofarigidbody(b)(c)(
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