p253_07c
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about techniquesTRANSCRIPT
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Moment of Inertia
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Moment of Inertia DefinedThe moment of inertia measures the resistance to a change in rotation.Change in rotation from torqueMoment of inertia I = mr2 for a single mass
The total moment of inertia is due to the sum of masses at a distance from the axis of rotation.
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Two SpheresA spun baton has a moment of inertia due to each separate mass.
I = mr2 + mr2 = 2mr2
If it spins around one end, only the far mass counts.
I = m(2r)2 = 4mr2mrm
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Mass at a RadiusExtended objects can be treated as a sum of small masses.
A straight rod (M) is a set of identical masses Dm.The total moment of inertia is
Each mass element contributes
The sum becomes an integralaxislength Ldistance r to r+Dr
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Rigid Body RotationThe moments of inertia for many shapes can found by integration.
Ring or hollow cylinder: I = MR2Solid cylinder: I = (1/2) MR2
Hollow sphere: I = (2/3) MR2Solid sphere: I = (2/5) MR2
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Point and RingThe point mass, ring and hollow cylinder all have the same moment of inertia.
I = MR2
All the mass is equally far away from the axis.The rod and rectangular plate also have the same moment of inertia.
I = (1/3) MR2
The distribution of mass from the axis is the same.RMMRaxislength Rlength RMM
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Playground RideA child of 180 N sits at the edge of a merry-go-round with radius 2.0 m and mass 160 kg.What is the moment of inertia, including the child?Assume the merry-go-round is a disk.Id = (1/2)Mr2 = 320 kg m2Treat the child as a point mass.W = mg, m = W/g = 18 kg.Ic = mr2 = 72 kg m2
The total moment of inertia is the sum.I = Id + Ic = 390 kg m2mMr
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Parallel Axis TheoremSome objects dont rotate about the axis at the center of mass.
The moment of inertia depends on the distance between axes. The moment of inertia for a rod about its center of mass:axisMh = R/2
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Perpendicular Axis TheoremFor flat objects the rotational moment of inertia of the axes in the plane is related to the moment of inertia perpendicular to the plane.MIx = (1/12) Mb2Iy = (1/12) Ma2abIz = (1/12) M(a2 + b2)
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Spinning CoinWhat is the moment of inertia of a coin of mass M and radius R spinning on one edge?The moment of inertia of a spinning disk perpendicular to the plane is known.Id = (1/2) MR2
The disk has two equal axes in the plane.The perpendicular axis theorem links these.Id = Ie + Ie = (1/2) MR2Ie = (1/4) MR2MRMRnextIdIe