Part1MechanicsChapter4FundamentalsofRigidBodyMechnics4-1KinematicsofRotationsaboutaFixedAxis4-2Fundamentalsofthedynamicsofrigidbodyrotationaboutafixedaxis4-3CalculatingtheRotationalInertia4-4Applicationofthelawofrotation4-5KineticEnergyandWorkinRotationalMotion4-6AngularMomentumofaRigidBodyandConservationofAngularMomentumContents:24-1KinematicsofRotationsaboutaFixedAxis4.1.1RigidBodyArigidbodyisabodythatallitspartslocktogetherandwithoutanychangeinitsshape.TranslationalMotionIfabodyismovingsuchthatalinedrawnbetweenanytwoofitsinternalpointsremainsparalleltoitself,thebodyistranslating.4.1.2BasicmotionofaRigidBody4-1KinematicsofRotationsaboutaFixedAxisRotationalMotionIfabodyismovingsuchthatalinedrawnbetweenanytwoofitsinternalpointsdoesnotparalleltoitself,thebodyisspinning.True

or

false

question:Thetrajectoryofeachpointonarigidbodyisacurve,anditsmotioncannotbetranslational.4.1.2BasicmotionofaRigidBody4-1KinematicsofRotationsaboutaFixedAxisCompositionofmotionsTranslationandrotationaroundafixedaxisaretwobasicmotionsofarigidbody.Variouscomplexmotionscanbedecomposedintoseveralbasicmotions4.1.2BasicmotionofaRigidBody4-1KinematicsofRotationsaboutaFixedAxisTherotationoccursaboutanaxisthatdoesnotmove.RotationaxisRotationplaneReferenceline74-1KinematicsofRotationsaboutaFixedAxis4.1.3RotationaboutaFixed-axis4-1TheKinematicsofRotationsaboutaFixedAxisTherotationoccursaboutanaxisthatdoesnotmove.4.1.3RotationaboutaFixed-axis4-1TheKinematicsofRotationsaboutaFixedAxisRotationaxisPBody

zO

ReferencelinexyEachparticlemovesinacircleinitsownplaneofrotation.RotationplaneZeroangularposition

Right-handrule

4.1.3RotationaboutaFixed-axis4-1TheKinematicsofRotationsaboutaFixedAxisRotationaboutaFixed-axiszRotationaxisPBody

O

ReferencelinexyRotationplaneZeroangularposition

Motionfunctionofarotatingbody:=(t)

ispositivewithcounterclockwisedirectionandnegativewithclockwisedirection.4-1TheKinematicsofRotationsaboutaFixedAxisRotationaboutaFixed-axisRotationaxisP

Body

zO

ZeroangularpositionReferencelinexy

Rotationplane

Axisz

Axisz

SpeedingupSlowingdown4-1TheKinematicsofRotationsaboutaFixedAxisRelatingwithLinearandAngularVariablesPosition

Velocity

Acceleration

AngularLinear

Displacement

Aristotlesaid:givemealever,Icanmovetheearth.4.2.1Torque4-2ThedynamicsofrigidbodyrotationaboutafixedaxisThequantitativemeasureofthetendencyofaforcetocauseorchangetherotationalmotionofabodyiscalledtorque.LineofactionMomentarmdUnits:N

mDefinitionoftorquerelativetoapointVectorformoftorqueRight-handrule4.2.1Torque4-2ThedynamicsofrigidbodyrotationaboutafixedaxisDefinitionoftorquerelativetoarigidbodyorVectorformoftorqueRight-handrule

A

4.2.1Torque4-2Thedynamicsofrigidbodyrotationaboutafixedaxis4.2.2Thelawofrotation(Newton’ssecondlawforrotation)Supposethatthereareexternalforceandinternalforceactingontherigidbody.Forithmasselement:

mi,ri,Fi,φi,fi,θiConsiderthetangentialcomponentMultiplyingthisequationbyri

Forallelements=0RotationLaw4.2.2Thelawofrotation(Newton’ssecondlawforrotation)IfarigidbodyconsistsofafewparticlesIfarigidbodyconsistsofagreatmanyofparticles(itiscontinuous)MassofrigidbodyRotationalinertiadependsonfollowingfactors:DistributionofthemassofrigidbodyPositionofrotationaxis4-3CalculatingtheRotationalInertia4-3CalculatingtheRotationalInertiaIfarigidbodyconsistsofafewparticles

Ifarigidbodyconsistsofagreatmanyofparticles(itiscontinuous)

MassofrigidbodyRotationalinertiadependsonfollowingfactors:DistributionofthemassofrigidbodyPositionofrotationaxis

Lineardistribution:Surfacedistribution:Volumedistribution:LinearmassdensitySurfacemassdensityVolumemassdensity

ExampleCalculatetherotationalinertiaofathin,uniformrodofmassMandlengthL,aboutdifferentaxesasshown.

Solution:TherotationaxisthroughtheleftendTherotationaxisthroughthecenterperpendiculartolength

ExampleCalculatetherotationalinertiaofathin,uniformringofmassMandradiusR,iftherotationaxisisthroughcenterandperpendiculartothecircularplane.RExampleCalculatetherotationalinertiaofathin,uniformdiskofmassMandradiusR,iftherotationaxisisthroughcenterandperpendiculartothecircularplane.RExampleCalculatetherotationalinertiaofathin,uniformringofmassM.ItsinnerradiusisR1,outerradiusisR2,iftherotationaxisisthroughcenterandperpendiculartothecircularplane.R1R2ExampleIftherotationaxisisthroughcenterandperpendiculartothecircularplane,calculatetherotationalinertiaof(1)athin,uniformringofmassMandradiusR.(2)athin,uniformdiskofmassMandradiusR.

Solution:（1）（2）

ExampleShowthattherotationalinertiaofauniformannularcylinder(orring)ofinnerradiusR1,outerradiusR2,andmassMis,iftherotationaxisisthroughthecenteralongtheaxisofsymmetry.

Dividethecylinderintothinconcentriccylindricalringsorhoopsofthicknessdr,eachwithmass

isthemassdensityofthebody.

Then

Solution:TableParallel-axisTheoremIfJistherotationalinertiaofabodyoftotalmassMaboutanyaxis,andJcomistherotationalinertiaaboutanaxispassingthroughthecenterofmassandparalleltothefirstaxisbutadistancehaway,thenExampleDeterminetherotationalinertiaofasolidcylinderofradiusR,andmassMaboutanaxistangenttoitsedgeandparalleltoitssymmetryaxis,asshowninthefigure.Solution:

FromTable5-1Sinceh=R,wehave

4-3CalculatingtheRotationalInertiaParallel-axisTheorem

Checkpoint24-3CalculatingtheRotationalInertiaPerpendicular-axisTheorem（onlyvalidforplaneobjects）

Thesumoftherotationalinertiaofaplanebodyaboutanytwoperpendicularaxesintheplaneofthebodyisequaltotherotationalinertiaaboutanaxisthroughtheirpointofintersectionperpendiculartotheplaneoftheobject.since

Therefore

ProofofPerpendicular-axisTheorem4-3CalculatingtheRotationalInertia

Ring：

A

ring

and

a

disk

with

thesame

radius

Rand

massM,rolldowntheslopefromthesameheight.Whoreachesthegroundfirst?Why?4-3CalculatingtheRotationalInertia4.4ApplicationofthelawofrotationForallelements=0RotationLawAsshowninFigure,auniformdisk,withmassM=3.0kgandradiusR=20cm,mountedonafixedhorizontalaxle.Ablockwithmassm=1.5kghangsfromamasslesscordthatiswrappedaroundtherimofthedisk.Findtheaccelerationofthefallingblock,theangularaccelerationofthedisk,andthetensioninthecord.Supposethatthereisnofrictionattheaxle(g=9.8N/kg).

Solvingequations,Forthepulley:Form:

Angularacceleration:Tension:

ExampleExampleAnAtwoodMachineconsistsoftwomasses,m1

andm2,whichareconnectedbyamasslessinelasticcordthatpassesoverapully.IfthepulleyhasradiusRandrotationalinertiaIaboutitsaxle,Determinetheaccelerationofthemassm1andm2,andcomparetothesituationwheretherotationalinertiaofthepulleyisignored.33Solution:Form1:

Form2:

Forthepulley:

Example

34Solution:Chooseathinringofradiusrandthicknessdr,thenitsareaisdS=2πrdranditsmassisdm=σdS,inwhichσ=m/(πR2)isthesurfacemassdensity.ThefrictionforceexertedonthisringisthetorqueofthefrictionforceontheringisThetotaltorqueofthefrictionforceontheentirecircularplateisExampleThen,theangularaccelerationoftheplatecanbeobtainedbythelawofrotationβ=M/Jz=-4μg/3RWhentheplatestops,itsangularvelocityisSupposethattheplaterotatedConsequently,

Thenumberofrotationsis

beforeitsstop,thereshouldbe35

Theworkdonefrom

i

f:

4-5KineticEnergyandWorkinRotationMotion4.5.1WorkdonebytorqueArigidbodyistreatedasacollectionofparticles.

miisthemassoftheithparticleandviisitsspeed.

AsSo

DefineKineticenergyofrotation

4-5KineticEnergyandWorkinRotationMotion4.5.2KineticEnergyofrotation384-5KineticEnergyandWorkinRotationMotion394-5KineticEnergyandWorkinRotationMotionAforceactsonaparticle(one-dimensionalmotion)Atorqueactsonarigidbod,(rotationaboutafixedaxis)Example

ConservationofMechanicalEnergyTheworkdonebythegravityisFromthework

kineticenergytheorem,

θmgA

4-6AngularMomentumofaRigidBodyandConservationofAngularMomentumDefinitionofAngularMomentumforaparticleTheparticle’sangularmomentumwithrespecttooriginO

Magnitude:

Direction:Right-handruleUnit:Kg.m2/s4.6.1AngularmomentumofarigidbodyExampleAparticlemoveswithuniformvelocityparalleltothey-axisinthexy-plane.Thepathisdescribedby

Findtheangularmomentumoftheparticleabouttheorigin.Wemustfirstfindthespeedoftheparticle,

Then

UsingtheresultforvectorproductsreducestoTheAngularMomentumofaRigidBodyRotatingAboutaFixedAxis

Note:Sometimes,wedropsthesubscriptz,butyoumustrememberthattheangularmomentumdefinedbyL=Iistheangularmomentumabouttherotationaxis.Also,Iinthatequationistherotationalinertiaaboutthataxis.4.6.2AngularmomentumtheoremofrigidbodyRotationaxisPBody

O

ReferencelinexyRotationplaneZeroangularpositionFromthedefinitionofangularmomentumofarigidbody

Differentiatingeachsidewithrespecttotime,

So

Note:(1)Mnetmeansthenetexternaltorque.(2)Torqueandangularmomentummustbemeasuredrelativetothesameaxis.Newton’sSecondLawinAngularForm4.6.2AngularmomentumtheoremofrigidbodyNewton’sSecondLawinAngularFormThe(vector)sumofallthetorquesactingonarigidbodyisequaltothetimerateofchangeoftheangularmomentumofthatbodywithrespecttothesameaxis.4.6.2Angularmomentumtheoremofrigidbody4.6.3ConservationofAngularMomentumBytheNewton’ssecondlawinangularform

Ifnonetexternaltorqueactsonthesystem:

or

Ifthenetexternaltorqueactingonasystemiszero,theangularmomentumofthesystemremainsconstant,nomatterwhatchangestakeplaceinthesystem.

Note:(1)HoldbeyondthelimitationofNewton’smechanics.(2)Applicationofthelawofconservationofangularmomentum.4.6.3ConservationofAngularMomentum4.6.4ConservationofAngularMomentuminEngineeringTechnology50ExamplelO

Solution：Thisproblemcanbedividedintothreesimpleprocesses(1)Selectthesystemcomposedoftherodandtheearthastheresearchobject

(2)

Selectthesystemcomposedoftherodandtheblockastheresearchobject,theangularmomentumandtheenergyofthesystemisconserved.

51ExamplelO

Solution：Thisproblemcanbedividedintothreesimpleprocesses(3)Selecttheobjectblockastheresearchobject,lettheobjectblockslideontheh