工程基础力学 Ⅱ 动力学 课件汇 李晨亮 Chapter 7 Theorem of Moment-Chapter 10 Vibrations

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)(a)Example

7-10RθCr（+）αSolution

：roundwheelmakeplanemotiononthesurface,theexternalforcesincludegravity

mg,thenormalreactionforceofthearcsurfaceFNandfrictionF.

Assumingangleθinacounterclockwisedirectionispositive,takingthetan