工程基礎(chǔ)力學 Ⅱ 動力學 課件全套 李晨亮 Chapter 1 Kinematics of a Particle-Chapter 10 Vibrations

Chapter1

KinematicsofaParticle§1.1Motionequationofaparticle§1.2

Velocityandaccelerationofaparticle

Maincontents1.1MotionequationofaparticleThreetypicalmotionequation(1)Motionequationusing

vector(2)Motionequationusingrectangularcoordinate1.1Motionequationofaparticle(3)MotionequationusingnaturalcoordinateThreetypicalmotionequationThecrankofellipticcompassescanrotatearoundfixedaxisO,theendAishingedwithBC；ThepointsBandCcanmovealongtheverticalslidingchutes,respectively.FindthetrajectoryequationofaarbitrarypointMonBC.KnownExample11.1MotionequationofaparticleSolution:Consideranarbitraryposition，thecoordinateofMcanbeexpressedasfollowing:Eliminate

intheaboveequations,thetrajectoryequationofMcanbeobtainedas:

1.1Motionequationofaparticle1.2Velocityandaccelerationofaparticle(1)DefinitionforvelocityandaccelerationofaparticleusingvectorDisplacement：Velocity：Acceleration：(2)Definitionforvelocityandaccelerationofaparticleontherectangularcoordinate(3)

ProjectionforvelocityandaccelerationofaparticleonthenaturalaxesTangentNormalplaneOsculatingplanePrincipalnormalSubnormal(a)Naturalcoordinatesystem1.2Velocityandaccelerationofaparticle(3)

Projectionforvelocityandaccelerationofaparticleonthenaturalaxes(b)Velocityofaparticle(c)AccelerationofaparticleThefirstcomponentrepresentsthechangerateofspeedmagnitudefortheparticle,notedas()TangentialaccelerationThesecondcomponentrepresentsthechangerateofspeeddirectionfortheparticle,notedas()Normalacceleration1.2Velocityandaccelerationofaparticle(3)

Projectionforvelocityandaccelerationofaparticleonthenaturalaxes◆Tangentialacceleration◆NormalaccelerationTangentialacceleration

representsthechangerateofspeedmagnitudetotime,itsalgebraicvalueisequaltothefirstderivativeofthealgebraicvalueofvelocitytotime,orthesecondderivativeofcurvilinearcoordinatetotime,itisalongthetangentoftrajectory.

1.2Velocityandaccelerationofaparticle(1)Thevectormethodisusedtodeduceformula；(2)Therectangularcoordinateandnaturalcoordinatemethodsareusedtocalculate：Theadvantageofnaturalcoordinatemethodistheclearphysicalmeaningandmoresimplethanrectangularcoordinatemethod.Thedisadvantageisthetrajectorymustbeknown,whichlimitstheapplicability.Theadvantageofrectangularcoordinatemethodisthewideapplicability(whichcanonlybeusedwhenthetrajectoryisunknown).Thedisadvantageismorecomplexthannaturalcoordinatemethod.SummaryBothofthetwomethodsareneedtosolvesomeproblems

TheEnd

Chapter2FundamentalKinematicsofaRigidBody§2.1Translationalmotionofarigidbody§2.2Rotationofarigidbodyaboutafixedaxis§2.3Velocityandaccelerationofapointin

arigidbodyrotatingaboutafixedaxis

Maincontents1.

DefinitionThedirectionofthelinelinkingarbitrarytwopointsintherigidbodyneverchangesduringitsmotion.2.1Translationalmotionofarigidbody2.

FeaturesDifferentiate:Allparticlesinarigidbodywithtranslationalmotionhavethesametrajectories.Arbitrarytwoparticlesinarigidbodywithtranslationalmotionhavethesamevelocitiesandaccelerationsinthesameinstant.Themotionregularitiesofalltheparticlesinarigidbodywithtranslationalmotionarecompletelysame,sothetranslationofarigidbodycanbesimplifiedtothemotionofaparticleinit.2.1TranslationalmotionofarigidbodyExamplesoftranslationalmotion

2.1Translationalmotionofarigidbody1.

DefinitionTherearetwofixedpointsduringthemovementofarigidbody,itiscalledtherotationaboutafixedaxis.Theaxisisafixedlinethroughthetwofixedpoints.2.2Rotationofarigidbodyaboutafixedaxis2.FeaturesThedistancesbetweeneverypointintherigidbodyandthefixedaxisremainconstants.Everypointintherigidbodywhichisnotonthisaxismovesalongacircularpathinaplaneperpendiculartothisfixedaxis.3.Equationofrotationφ

isanalgebraicquantity.Signdefinitionofφ:followsright-handrule.φ

isthemonotropiccontinuousfunctionoftimet,whentherigidbodyrotates.

2.2Rotationofarigidbodyaboutafixedaxis4.Angularvelocityandangularaccelerationω

isanalgebraicquantitySigndefinitionofω:followsright-handrule.Unit：(1)Angularvelocity:

Inordertodescribethespeedanddirectionoftherotationofarigidbody,theangularvelocityisdefinedas,2.2Rotationofarigidbodyaboutafixedaxis4.Angularvelocityandangularacceleration(2)Angularacceleration:

Inordertodescribethespeedofangularvelocitychangedwithtimeoftherotationofarigidbody，theangularaccelerationisdefinedas,α

isanalgebraicquantitySigndefinitionofα:followsright-handrule.Unit：Arigidbodyhasacceleratedrotationwhenαand

havethesamesigns,deceleratedrotationwhenα

and

haveoppositesigns.

=const,uniformrotation.

α

=const,rotationwithconstantangularacceleration.2.2Rotationofarigidbodyaboutafixedaxis2.3Velocityandaccelerationofapointinarigidbodyrotatingaboutafixedaxis1.Themotionequationdefinedbycurvilinearcoordinate

2.Velocityofapoint

Differentiatetheexpressionabovewithrespecttotimet,gives

∵,

,∴itcanbeobtainedas,Directionofvelocity:alongthetangentline,pointtotherotationdirection.3.AccelerationofapointNormalacceleration:Tangentialacceleration:TheaccelerationofpointM:Thedirectionofacceleration:2.3Velocityandaccelerationofapointinarigidbodyrotatingaboutafixedaxis4.Thedistributionregularitiesofthevelocityandaccelerationintherotatedrigidbody(1)Ateverytime,thevelocityandaccelerationofapointareproportionaltoR.(2)Ateverytime,thedirectionsofthevelocityandaccelerationofapointareperpendiculartoR.Theangle

betweentheaccelerationofanypointanditsradiusisidentical.2.3Velocityandaccelerationofapointinarigidbodyrotatingaboutafixedaxis

TheEnd

Chapter3ComplexMotionofParticle(orPoint)

§3.1Basicconceptofcomplexmotionofparticle

§

3.2Velocitycompositiontheoremofparticle§

3.3Accelerationcompositiontheoremwhenthetransportmotionistranslation§

3.4Accelerationcompositiontheoremwhenthetransportmotionisrotation

Maincontents1.

Whatiscomplexmotionofparticle?Motionisrelative.Amotionrelativetoareferenceobjectcanbecomposedofseveralsimplemotionsrelativetootherreferenceobjects.Themotioniscalled

complexmotion.2.ProblemstosolvebytheoryofcomplexmotionofparticleAcomplexmotioncanbedecomposedintotwosimplemotions.Thevaluesofcomplexmotioncanbecomposedbythoseoftwosimplemotions.Therelationsofthemotionofeverycomponentinthemovingmechanism.Therelationoftwomovingobjectswithoutdirectiveconnection.(1)AmovingpointApointintheresearchingobject.(2)Tworeferencesystems(3)Three

kindsof

motionsApoint,tworeferencesystems,andthreekindsofmotionsFixedreferencesystem:Areferencesystemfixedtotheearthground.Movingreferencesystem:

Areferencesystemfixedtoamovingobjectrelativetotheearthground.Absolutemotion:Motionofthemovingpointrelativetothefixedreferencesystem.Relativemotion:

Motionofthemovingpointrelativetothemovingreferencesystem.Transportmotion:Motionofthemovingreferencesystemrelativetothefixedreferencesystem.

3.1BasicconceptofcomplexmotionofparticleAbsolutemotionRelativemotionTransportmotionBothofabsolutemotionandrelativemotionaremotionsofaparticle.Transportmotionismotionofreferenceobject,actuallymotionofarigidbody.

3.1BasicconceptofcomplexmotionofparticleCorrespondingtoabsolutemotion:AbsolutetrajectoryAbsolute

velocityAbsoluteaccelerationCorrespondingtorelativemotion:

RelativetrajectoryRelativevelocityRelativeaccelerationThereisn’ttrajectoryfortransportmotion,becauseitisn’taparticle,butarigidbody.Correspondingtotransportmotion:TransportvelocityTransportaccelerationTransportvelocity

and

transportacceleration

arethevelocityandaccelerationofthepointinthemovingreferencesystemcoincidingwiththemovingpoint(transportpoint)

relativetothefixedreferencesystematanyinstantoftime.

3.1BasicconceptofcomplexmotionofparticleExample

3-1Crankrockermechanism,thecrankOAisconnectedtothesleevebypinA,andthesleeveissetontherockerO1B.WhenthecrankrotatesaroundtheOaxiswithangularvelocityω,therockerO1BisdriventoswingaroundtheO1axisthroughthesleeve.AnalyzethemotionoftheApoint.

3.1BasicconceptofcomplexmotionofparticleSolution：Movingreferencesystem－O1x'y'，fixedtorockingbarO1B.2.Motionanalysis.Movingpoint－pin

A

onthesleeve.y'x'1.Choosethemovingpoint,movingreferencesystemandfixedreferencesystem.Fixedreferencesystem－Fixedtotheground.Absolutemotion－CircularmotionwiththecentreO.Relativemotion－ThestraightlinemotionalongO1B.Transportmotion－RotationofrockingbarabouttheaxisO1.

3.1BasicconceptofcomplexmotionofparticleHowtoselectthemovingpointandmovingsystem1.Themovingsystemcanberegardedasaninfiniterigidbody,andthebasicmotionoftherigidbodyistranslationalandfixed-axisrotation.Therefore,themovingsystemisgenerallytakenasthecoordinatesystemoftranslationalmotionorfixed-axisrotation.2.Themovingpointandthemovingreferencecannotbechosenonthesameobject,otherwisetherelativemotionofthemovingpointwithrespecttothemovingreferencewilldisappear.3.Themovingpointmustalwaysbethesamepointinthesystem,andstudyitsmotionatdifferentmoments.Itisnotallowedtotakeapointatoneinstantandanotherpointasthemovingpointatthenextinstant.1.TheoremAtanyinstantoftime,theabsolutevelocityofamovingpointisequaltothegeometricsumofitsrelativevelocityandtransportvelocity.Thisisthe

velocitycompositiontheoremofpoint.

Theabsolutevelocityofamovingpointcanbedeterminedbythediagonallineoftheparallelogramcomposedbyitstransportvelocityandrelativevelocity.

Thisisthe

parallelogramofvelocity.

3.2Velocitycompositiontheoremofparticle

moveto

2.Provement

3.2VelocitycompositiontheoremofparticleExample

3-2

Thequick-returnmechanismofplanerisshowninthefigure.TheendAofacrankOAisarticulatedwithaslideblock.ThecrankOArotatesaroundthefixedaxisOwiththeuniformangularvelocityω.Theslideblockslidesontherockingbar,whichisdriventoswingaboutthefixedaxisO1.ThelengthofthecrankOA=r，OO1=l.Findtheangularvelocityω1oftherockingbarwhenthecrankmovestothehorizontalposition.

3.2VelocitycompositiontheoremofparticleSolution：Movingreferencesystem－O1x'y'，fixedtorockingbarO1B.2.Motionanalysis.Movingpoint－pin

A

onthesleeve.y'x'1.Choosethemovingpoint,movingreferencesystemandfixedreferencesystem.Fixedreferencesystem－Fixedtotheground.Absolutemotion－CircularmotionwiththecentreO.Relativemotion－ThestraightlinemotionalongO1B.Transportmotion－RotationofrockingbarabouttheaxisO1.

3.2Velocitycompositiontheoremofparticle3.VelocityanalysisvavevrAbsolutevelocityva：va＝OA·ω

＝rω,

Direction:verticaltoOA，plumbedupwardsTransportvelocity

ve：ve

istheunknownquantity,andneedtobesolvedDirection:verticaltoO1BRelativevelocityvr：themagnitudeisunknownDirection:alongtherockingbarO1B

Accordingtothevelocitycompositiontheoremofapoint

3.2Velocitycompositiontheoremofparticle∵∴Supposetheangularvelocityoftherockingbaratthemomentisω1,yieldsSovavevr

3.2Velocitycompositiontheoremofparticle1.Relativeandabsolutederivativeofvector●MOxyzisafixedcoordinatesystem,andO1x1y1z1isamotioncoordinatesystem,theradiusvectorofthemovingpointMinthemotionsystemisWetakethetimederivativeinthefixedsystemtoobtainThisistheabsoluterateofchangeofthevectorr1Takethederivativeofr1withrespecttotimeinthemotionsystemtoobtainThisistherelativerateofchangeofthevectorr13.3Accelerationcompositiontheoremwhenthetransportmotionistranslation2.Threekindsofaccelerations(1)Absoluteacceleration(2)Relativeacceleration3.3Accelerationcompositiontheoremwhenthetransportmotionistranslation●M2.Threekindsofaccelerations(3)Transportacceleration3.3Accelerationcompositiontheoremwhenthetransportmotionistranslation●M3.AccelerationcompositiontheoremWhenthemotionsystemistranslatingmotion,andi1,j1,k1

areconstantvectors,andtheirmagnitudesanddirectionsareconstant,sotheirtimederivativesareallzero,wecangetAccelerationcompositiontheoremwhenthetransportmotionistranslation3.3Accelerationcompositiontheoremwhenthetransportmotionistranslation●MExample

3-3

Aplanemechanismshowninthefigure，thecrankOA=r，rotatesuniformlywithangularvelocityω0.SleeveAcanslidsalongthebarBC.BC=DE，且BD=CE=l.FindtheangularvelocityandangularaccelerationofBDatthemomentshowninthefigure.ABCDEOω0ωαSolution：Choosethemovingpoint,movingreferencesystemandfixedreferencesystemMovingreferencesystem－Cx′y′，fixedtothebar

BC.2.MotionanalysisTransportmotion－translationMovingpoint－slideblock

A.Fixedreferencesystem－

fixedtothebase.ABCDEOω0ωαx'y'Absolutemotion－CircularmotionwithcentreORelativemotion－straightlinemotionalongBCABCDEOω0ωαvBvevavr3.VelocityanalysisyieldsSotheangularvelocityof

BDAbsolute

velocity

va：va=ω0r，verticalto

OA

downwards.

Transportvelocity

ve：ve=

vB，verticalto

BDrightdownwands.

Relativevelocity

vr：magnitudeunknown，along

BCleftEmployingthetheoremofcompositionofvelocities4.AccelerationanalysisAbsoluteacceleration

aa：aa=ωor

，along

OA，pointtoOTransportaccelerationae:tangentialcomponentaet：sametoaBt,magnitude

unknown,verticaltoDB,

supposedownwardsRelativeacceleration

ar：magnitude

unknown,along

BC,

supposetoleftnormalcomponentaen：aen

=aBn=

ω2l

=ωo2r2

/l,alongDB,

pointtoDaaarABCDEOω0ωα

Projecttoaxisy,

yieldsyieldsApplyingthecompositiontheoremofaccelerationsSotheangularaccelerationof

BD:

aaarABCDEOωαyAfixedcoordinatesystemOxyzandmotioncoordinatesystemOx1y1z1,letthemovingpointMmoveinthemotionsystemOx1y1z1,andthemotionsystemOx1y1z1rotatesaboutthez-axisofthefixedsystemwithangularvelocityωandangularaccelerationε●MBasedonthepreviousproofofthevelocitycompositiontheorem,wehave

TherelativevelocityandrelativeaccelerationofthemovingpointM3.4AccelerationcompositiontheoremwhenthetransportmotionisrotationAndthen

Basedonthevelocitycompositiontheorem：AccordingtothePoissonformula：3.4Accelerationcompositiontheoremwhenthetransportmotionisrotation

Coriolisacceleration:Thisistheaccelerationcompositiontheoremwhenthetransportmotionisrotation.3.4AccelerationcompositiontheoremwhenthetransportmotionisrotationExample

3-4Thequick-returnmechanismofplanerisshowninthefigure.TheendAofacrankOAisarticulatedwithaslideblock.ThecrankOArotatesaroundthefixedaxisOwiththeuniformangularvelocityω.Theslideblockslidesontherockingbar,whichisdriventoswingaboutthefixedaxisO1.ThelengthofthecrankOA=r，OO1=l.Findtheangularaccelerationα1oftherockingbarwhenthecrankmovestothehorizontalposition.

Basedonthe

velocityanalysisobtainedfromlastclass,weknowthatSolution：Choosethemovingpoint,movingreferencesystemandfixedreferencesystem.Movingreferencesystem－O1x1y1，fixedtorockingbarO1B.Movingpoint－slideblock

A.vavevry1x1Fixedreferencesystem－Fixedtothe

base2.AccelerationanalysisAbsoluteacceleration

aa：

aa

=ω2r

，along

OA，pointto

O.Relative

acceleration

ar：magnitude

is

unknown

，suppose

it

is

along

O1B

upwards.

Tangential

component

aet：magnitude

is

unknown，

vertical

to

O1B，supposerightdownwardsTransport

acceleration:Normal

component

：

along

O1A，point

to

O1Coriolis

acceleration

aC：verticaltoO1B，showninthefigurex'y'O1Oφωω1ABaaaraCProjectitto

O1x'yieldsTheangularaccelerationofrockingbar:α1Applyingtheaccelerationcompositiontheoremx'y'O1Oφωω1ABaaaraCor

TheEnd

Chapter4

PlanarMotionofaRigidBody§

4.1Basicconceptanddecompositionofrigidbodyplanarmotion

Maincontents§4.2

Velocityofanypointinaplanarmotion§4.3

Accelerationofanypointinaplanarmotion1.Whatisplanarmotionofarigidbody?Thedistancebetweenanypointinarigidbodyandafixedplanealwayskeepsunchangedduringitsmotion.Thismotionofrigidbodyiscalled

planarmotionofarigidbody.4.1Basicconceptanddecompositionofrigidbodyplanarmotion2.SimplificationofaplanarmotionTheplanarmotionofarigidbodycanbesimplifiedtoamotionofaplanegraphintheplaneitselfwithoutconsideringitsthickness.

(a)Connectingrodmotion(b)Simplificationofconnectingrodmotion4.1Basicconceptanddecompositionofrigidbodyplanarmotion3.EquationsofplanarmotionSTodeterminethemotionofaplanegraph,choosethefixedreferencesystemOxy,anarbitrarypointO'intheplanegraphS,anarbitrarylinesegmentO'M.Todeterminetheplanarmotionofarigidbody,onlythepositionofthelinesegmentO'Minthisgraphisneededtobedetermined.EquationsofplanarmotionAplanemotioncanberegardedasthecompositionofa

translation

androtation．4.1Basicconceptanddecompositionofrigidbodyplanarmotion4.Planarmotioncanbedecomposedintotranslationandrotation

Aplanemotionofarigidbodycanbedecomposedintoa

translationwithabasicpointanda

rotation

aboutanaxisthatpassesthroughthebasicpoint.Thevelocityandaccelerationofthe

translation

withabasicpoint

intheplanegraphdependson

theselectionof

thebasicpoint,however,theangularvelocityandaccelerationoftherotationabouttheselectedbasicpoint

doesn’tdependon

thechoiceofthebasicpoint．4.1BasicconceptanddecompositionofrigidbodyplanarmotionAThevelocityofpointAintheplanegraphSis,andtherotationalvelocityoftheplanegraphis.SelectAasthebasicpoint;ThemovingreferencesystemattachedtopointA;Thetransportmotionistranslationwiththebasicpoint

A;Therelativemotionisrotationaboutthebasicpoint

A.(1)Basicpointmethod

·BDeterminethevelocityofpointBintheplanegraph.4.2VelocityofanypointinaplanarmotionABTheorem：Forplanarmotionofarigidbody,thevelocityofanypointinthegraphcanbeobtainedasthevectorsumofthevelocityofthebasicpointandtherelativerotationalvelocitywithrespecttothebasicpoint.4.2Velocityofanypointinaplanarmotion

isverticaltothelinkofABallthetime,sotheprojectionofonABisvanish.Thevelocityprojectiontheorem:thevelocityprojectionsofanytwopointinaplanegraphonthelinelinkingthesetwopointsareidentical．(2)VelocityprojectiontheoremAB4.2Velocityofanypointinaplanarmotiona.Background

Ifapointwhosevelocityiszeroisselectedasthebasicpoint,theprocessoffindingthevelocityofanypointwillbegreatlysimplified.Therefore,itisnaturaltoaskifsuchapointexistsinanyinstant.Ifitdoesexist,howtofindsuchapoint?b.InstantaneouscenterofvelocityAtanyinstant,itmustexistasolepointwhosevelocityiszerointheplanegraphoritsexpandingarea，whichiscalledtheinstantaneousvelocitycenterofthisplanegraphatthisinstant.Foraplanegraph,itsinstantaneousvelocitycenteralwaysexistsuniquely.

(3)Instantaneouscenterofvelocitymethod4.2Velocityofanypointinaplanarmotionc.InstantaneouscenterofvelocitymethodConsideraplanegraph.TheinstantaneousvelocitycenterisP,andtheangularvelocityoftheplanegraphis.SelectinstantaneousvelocitycenterPisabasicpoint,thevelocityofanarbitrarypointAintheplanegraph:4.2Velocityofanypointinaplanarmotiond.MethodstodeterminetheinstantaneousvelocitycenterPA(1)Whenthevelocityofapointandtheangularvelocity

oftheplanegraphareknown,theinstantaneousvelocitycenter(pointP)canbedetermined,

pointPisinthedirectionofthelineformedbyrotatingthethrough90ointhedirectionof

aroundpointA．4.2Velocityofanypointinaplanarmotion(2)Whenaplanegraphrollsalongafixedsurfacewithoutslipping,thecontactpointPbetweenthegraphandthefixedsurfacewillbetheinstantaneousvelocitycenter.

(3)WhenthedirectionsofthevelocitiesattwopointsAandBinagraphareknown,andisnotparallelto,drawlinesfromAandBperpendiculartorespectively,andthecrosspointPofthesetwolineswillbetheinstantaneousvelocitycenter.ABP4.2Velocityofanypointinaplanarmotion(4)WhenthevelocitiesoftwopointsAandBaregivenatanyinstant,and.Therearethreecases:ABP

Whenandpointtothesamedirection,but.DrawtheextensionlineofAB,thelinkinglineoftheendingsofand,thecrosspointofthesetwolineswillbetheinstantaneousvelocitycenter.Therotationdirectionofcanbedetermined,anditsmagnitudeis:◆ω

◆Whenandhaveoppositedirections,drawthelinkinglineoftheendingsofand,andthelineconnectingAB.Thecrosspointofthesetwolineswillbetheinstantaneousvelocitycenter.Therotationdirectionofcanbedetermined,anditsmagnitudeis:

ω4.2VelocityofanypointinaplanarmotionBPAB(5)ThevelocitiesoftwopointsAandBpointtothesamedirectionatanyinstant,,,buttheyarenotperpendiculartolineAB.Inthiscase,theinstantaneousvelocitycenterisindefinitelyfaraway,andtheangularvelocity

=0,i.e.allpointinthefigurehavethesamevelocityatthisinstantoftime.Suchamotioniscalledinstantaneoustranslation,buttheiraccelerationsarenotequal.

When,

theinstantaneousvelocitycenterisindefinitelyfaraway.Theplanegraphhasinstantaneoustranslation,=0,allpointsinthegraphhavethesamevelocityatthisinstantoftime,buttheiraccelerationsarenotequal.◆ω4.2VelocityofanypointinaplanarmotionAABAAttheinstant,theangularvelocityofthegraphis,angularaccelerationis,accelerationofapointAis

.DeterminetheaccelerationofanarbitrarypointBinthegraph.·

4.3

AccelerationofanypointinaplanarmotionBA1.

：

4.3

AccelerationofanypointinaplanarmotionBAB(1)thetangentialacceleration

(2)thenormalacceleration2.

：hastwocomponents:

4.3

AccelerationofanypointinaplanarmotionTheabsoluteaccelerationofpointB:Theorem:

Theaccelerationofanarbitrarypointisequaltothevectorsumofaccelerationofthebasicpoint,thetangentialandnormalaccelerationsoftheplanegraphrotatingaboutthebasicpoint.

4.3

Accelerationofanypointinaplanarmotionω1ⅠⅡO1OABCAnexternaltoothingplanetgearmechanismshowninthefigure.ThelinkingbarO1O=l，rotatesaboutaxisO1withauniformangularvelocityω1.ThebiggergearIIisfixed,theplanetgearIofradiusrrollsalongthegearIIwithoutsliding.AandBaretwopointsontheedgegearI,showninthefigure.Findtheaccelerationsof