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1、-CalculationofMomentsofInertiaforRigidObjectsofDifferentGeometries-ParallelAxisTheoremAPPhysicsCMrs.CoyleRemember:AngularandLinearQuantitiesDisplacementsSpeedsAccelerationsRemember:RotationalKineticEnergyandMomentofInertiaThetotalrotationalkineticenergy

2、oftherigidobjectisthesumoftheenergiesofallitsparticlesIiscalledthemomentofinertiaRemember:MomentofInertiaMomentofInertia,I,isameasureoftheresistanceofanobjecttochangesinitsrotationalmotion.MomentofInertiaisanalogoustomassintranslationalmotion.Whentheobjec

3、tismadeupofpointmassesyoucalculatemomentofinertiausing:MomentofInertiaofaRigidObjectorintermsofdensity:Whenobjectsofdifferentgeometricshapesareinvolved,(ie.cylinders,hoops,rods,spheres)wewillusegeometricrelationshipstoevaluatetheintegral.Notethatforthefol

4、lowingexamplesthattheaxisofrotationwillcoincidewiththeaxisofsymmetryoftheobjects.Ex:MomentofInertiaofaUniformThinHoop–forAxisperpendiculartotheplaneattheCenteroftheHoopAssumerisconstantIftheObjecthasUniformDensity:wewillusetheconstantdensityexpressiontoev

5、aluatetheintegral.DensityExpressionsVolumetricMassDensity–>massperunitvolume:r=m/VLinearMassDensity–>massperunitlengthofarodofuniformcross-sectionalarea:l=m/L=rAFaceMassDensity–>massperunitthicknessofasheetofuniformthickness,t:s=rtEx:MomentofInertiaofaUni

6、formRigidRodforAxispassingthroughcenterofmassl=m/L=rAdm=ldxEx:MomentofInertiaofaUniformSolidCylinderforaxisalongzaxisForconcentricshellswithradiusr,thicknessdrandlengthLr=m/VParallel-AxisTheoremIsusedtofindI,iftheaxisofrotationdoesnotcoincidewiththeaxisof

7、symmetrybutisparalleltotheaxisthroughthecenterofmassoftheobject.ParallelAxisTheoremI=ICM+MD2-ICMisthemomentofinertiaabouttheaxisthroughthecenterofmass-DisthedistancefromthecenterofmassaxistothearbitraryaxisMomentofInertiaforaRodRotatingAroundOneEndD=½LFor

8、auniformdensityobjectmadeupofvariousshapesthetotalmomentofinertiaisthesumofthemomentsofinertiafortheindividualobjects.Example#25Auniformthinsoliddoorhasheight2.20m,width0.870mandmass23.0kg.Finditsmomentofinertiafror