Moment of Inertia. Moment of Inertia Defined. The moment of inertia measures the resistance to a change in rotation. Change in rotation from torque Moment of inertia I = mr 2 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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I = mr2 + mr2 = 2mr2
If it spins around one end, only the far mass counts.
I = m(2r)2 = 4mr2Two Spheres
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 integralMass at a Radius
distance r to r+Dr
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.Point and Ring
What is the moment of inertia, including the child?
Assume the merry-go-round is a disk.
Id = (1/2)Mr2 = 320 kg m2
Treat 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 m2Playground Ride
The moment of inertia depends on the distance between axes.
The moment of inertia for a rod about its center of mass:Parallel Axis Theorem
h = R/2
Iy= (1/12) Ma2
Ix= (1/12) Mb2
Iz= (1/12) M(a2 +b2)
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) MR2
Ie= (1/4) MR2Spinning Coin