Moment of InertiaMoment of Inertia DefinedTwo SpheresMass at a RadiusRigid Body RotationPoint and RingPlayground RideParallel Axis TheoremPerpendicular Axis TheoremSpinning CoinMoment of InertiaMoment of InertiaMoment of Inertia DefinedMoment of Inertia DefinedThe moment of inertia measures the resistance to a The moment of inertia measures the resistance to a change in rotation.change in rotation.•Change in rotation from torqueChange in rotation from torque•Moment of inertia Moment of inertia II = = mrmr22 for a single mass for a single massThe total moment of inertia is due to the sum of The total moment of inertia is due to the sum of masses at a distance from the axis of rotation.masses at a distance from the axis of rotation.NiiirmI12Two SpheresTwo SpheresA spun baton has a moment A spun baton has a moment of inertia due to each of inertia due to each separate mass.separate mass.•II = = mrmr22 + + mrmr22 = 2 = 2mrmr22If it spins around one end, If it spins around one end, only the far mass counts.only the far mass counts.•II = = mm(2(2rr))22 = 4 = 4mrmr22mrmMass at a RadiusMass at a RadiusExtended objects can be Extended objects can be treated as a sum of small treated as a sum of small masses.masses.A straight rod (A straight rod (MM) is a set of ) is a set of identical masses identical masses mm..The total moment of inertia isThe total moment of inertia isEach mass element Each mass element contributescontributesThe sum becomes an The sum becomes an integralintegralaxislength Ldistance r to r+r2)( rmIrrLMIrLMm2)/()/(2302)3/1()3/)(/()/(MLLLMIdrrLMILRigid Body RotationRigid Body RotationThe moments of inertia for many shapes can found The moments of inertia for many shapes can found by integration.by integration.•Ring or hollow cylinder: Ring or hollow cylinder: II = = MRMR22•Solid cylinder: Solid cylinder: II = (1/2)= (1/2) MRMR22•Hollow sphere: Hollow sphere: II = (2/3)= (2/3) MRMR22•Solid sphere: Solid sphere: II = (2/5)= (2/5) MRMR22Point and RingPoint and RingThe point mass, ring and The point mass, ring and hollow cylinder all have the hollow cylinder all have the same moment of inertia.same moment of inertia.•II = = MRMR22All the mass is equally far All the mass is equally far away from the axis.away from the axis.The rod and rectangular The rod and rectangular plate also have the same plate also have the same moment of inertia.moment of inertia.•II = (1/3) = (1/3) MRMR22The distribution of mass from The distribution of mass from the axis is the same.the axis is the same.RMMRaxislength R length RMMPlayground RidePlayground RideA child of 180 N sits at the A child of 180 N sits at the edge of a merry-go-round edge of a merry-go-round with radius 2.0 m and mass with radius 2.0 m and mass 160 kg.160 kg.What is the moment of What is the moment of inertia, including the child?inertia, including the child?Assume the merry-go-round Assume the merry-go-round is a disk.is a disk.•IIdd = (1/2) = (1/2)MrMr22 = 320 kg m = 320 kg m22Treat the child as a point Treat the child as a point mass.mass.•WW = = mgmg, , mm = = WW//gg = 18 kg. = 18 kg.•IIcc = = mrmr22 = 72 kg m = 72 kg m22The total moment of inertia is The total moment of inertia is the sum.the sum.•II = = IIdd + + IIcc = = 390 kg m 390 kg m22mMrParallel Axis TheoremParallel Axis TheoremSome objects don’t rotate Some objects don’t rotate about the axis at the center about the axis at the center of mass.of mass.The moment of inertia The moment of inertia depends on the distance depends on the distance between axes. between axes. The moment of inertia for a The moment of inertia for a rod about its center of mass:rod about its center of mass:2MhIICMaxisMh = R/222222)12/1()4/1()3/1()2/()3/1(MRIMRMRIRMIMRCMCMCMPerpendicular Axis Perpendicular Axis TheoremTheoremFor flat objects the rotational For flat objects the rotational moment of inertia of the axes moment of inertia of the axes in the plane is related to the in the plane is related to the moment of inertia moment of inertia perpendicular to the plane.perpendicular to the plane.MIx = (1/12) Mb2Iy = (1/12) Ma2abIz = (1/12) M(a2 + b2)yxzIII Spinning CoinSpinning CoinWhat is the moment of What is the moment of inertia of a coin of mass inertia of a coin of mass MM and radius and radius RR spinning on spinning on one edge?one edge?The moment of inertia of a The moment of inertia of a spinning disk perpendicular spinning disk perpendicular to the plane is known.to the plane is known.•IIdd = (1/2) = (1/2) MRMR22The disk has two equal axes The disk has two equal axes in the plane.in the plane.The perpendicular axis The perpendicular axis theorem links these.theorem links these.•IIdd = = IIee + + IIee = (1/2) = (1/2) MRMR22•IIee = (1/4) = (1/4)