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.
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
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
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
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
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
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
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)
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
