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CHAPTER5ROTATIONROTATION5.1：Descriptionofrotation5.2:ThelawofRotation5.3CalculatingMomentofInertia5.4ConservationofAngularMomentuminRotation5.5WorkandEnergyinRotation5.1DescriptionofrotationRotationisdescribedintermsofangulardisplacement,time,angularvelocity,andangularacceleration.Angularvelocityistherateofchangeofangulardisplacementandangularaccelerationistherateofchangeofangularvelocity.Theaveragesofvelocityandaccelerationaredefinedbytherelationships:Averageangularvelocity:Averageangularacceleration:wheretheGreekletterdeltaindicatesthechangeinthequantityfollowingit.: Abaraboveanyquantityindicatestheaveragevalueofthatquantity.Ifαisconstant,equations1,2,and3representacompletedescriptionoftherotation.Equation4isobtainedbyacombinationoftheothers.Example5.1:Anelevatorispulledbyacablerunningoverafixedpulleyofradiusr=0.02m.Iftheelevatorrisesfromrestwithconstantaccelerationa=0.4m/s2,Find:(a)Theangularaccelerationofthepulley;(b)Theangularspeedofthepulleyatt=2.0s;(c)thenumberofturnsofrotationofthepulleyinthis2.0s.Solution:(a)(b)(c)5.2:Thelawofrotation:Thetorqueonarigdbodyaboutafixedaxisisequaltobeproductofthemomentofinertiaoftherigidbodyaboutthesameaxisanditsangularaccelerationresulted.Example5.2Aflywheelofmassm=60kgandradiusR=0.25m,isrotatingat.Inordertobreakittostopint=5.0swithconstantdeceleration,whatforceNshouldbeappliedbythebreakontheedgeofthewheel?Assume:thefrictioncoefficientisu=0.8.Solution:Theangularacceleration:Thetorque:Example5.3AsshowninFig5.5,acordrunsoverafixedpulleyofmassMandradiusR,withoneendfixedontheedgeofthepulleyandtheotherendattachedtoablockofM.Neglectthefrictionattheaxis,Findtheaccelerationoftheblockduringfalling.Solution:5.3CalculatingMomentofInertiaMomentofinertiaisthenamegiventorotationalinertia,therotationalanalogof
mass
forlinearmotion.Itappearsintherelationshipsforthedynamicsofrotationalmotion.Themomentofinertiamustbespecifiedwithrespecttoachosenaxisofrotation.Fora
pointmass
themomentofinertiaisjustthemasstimesthesquareofperpendiculardistancetotherotationaxis,I=mr2.Thatpointmassrelationshipbecomesthebasisforallothermomentsofinertiasinceanyobjectcanbebuiltupfromacollectionofpointmasses.Becareful:JisalsowrittentobeI,theyallrefertothemomentofinertia!!!!CalculatingMomentofInertia:Whereristhedistancefromthemasselementtotheaxisofrotation.Foracontinuousmass:Example5.4Findthemomentofinertiaofauniformthincircularstripaboutanaxisthroughitscenterandperpendiculartotheplaneofthecircle.ThemassofthewholestripismandtheradiusisR.Solution:(strip)Example5.5Findthemomentofinertiaofauniformdiskaboutanaxisthroughitscenterandperpendiculartotheplaneofthedisk.ThemassofthediskismandtheradiusisRandthethicknessisLSolution:Example5.6Findthemomentofinertiaofathinuniformrodoflengthandmassm:(a)aboutaperpendicularaxisthroughoneendoftherode.(b)aboutaperpendicularaxisthroughthecenteroftherode.Solution:(a):(b):5.4ConservationofAngularMomentuminrotation
Whichtellsusthat:thetorqueonarigidbodyisequaltothetimerateofchangeofitsangularmomentum.Angularmomentum
andlinearmomentumareexamplesoftheparallels
betweenlinearandrotationalmotion.Theyhavethesameformandaresubjecttothefundamentalconstraintsofconservationlaws,the
conservationofmomentum
andtheconservationofangularmomentum
.
AngularandLinearMomentumExample5.7:AuniformrodoflengthLandmassMishangingverticallywithitsupperendpivotedonafrictionlesshorizontalaxis.AbulletofmassmisshotintothelowerendoftherodwithhorizontalverticallyV0.Findthecommonangularspeedoftherodandthebulletwhenstaringtomovetogether.Solution:Example5.8:AhorizontalturntableofmassMandradiusRrotating.Attheedgeofthetabletherestandsamanofmassmandbothareatrestrelativetotheground.Findtheangleturnedbythetablerelativetothegroundwhenthemanwalksforoneturnalongtheedgethetable.Solution:SincetheinitialangularmomentumofthesystemisZero:Then:andExample5.9Asshown,aspaceshipandisspringat.Theradiusisr=1.5m.Thetotaljetflowrateq=2.0kg/sisconstant.Thejetspeedoftheexhaustisu=50m/sandconstant.Howmanyisitneedtostopthespringofthespaceship??Solution:Whenstop:5.5WorkandenergyinRotationThekineticenergyofarotatingobjectisanalogoustolinearkineticenergyandcanbeexpressedintermsofthemomentofinertiaandangularvelocity.Thetotalkineticenergyofanextendedobjectcanbeexpressedasthesumofthetranslationalkineticenergyofthecenterofmassandtherotationalkineticenergyaboutthecenterofmass.Theexpressionsforrotationalandlinearkineticenergycanbedevelopedinaparallelmannerfromthework-energyprinciple.Example5.10:Asshown,acordrunsoverafixedpulleyofmassMandradiusR,withoneendfixedontheedgeofthepulleyandtheotherendattachedtoablockofm.Neglectthe
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