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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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moment of inertia defined
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 = 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 spheres
A 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 = 4mr2

Two Spheres

m

m

r

mass at a radius
Extended 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 integral

Mass at a Radius

distance r to r+Dr

length L

axis

rigid body rotation
Rigid Body Rotation
  • The moments of inertia for many shapes can found by integration.
    • Ring or hollow cylinder: I= MR2
    • Solid cylinder: I= (1/2)MR2
    • Hollow sphere: I= (2/3)MR2
    • Solid sphere: I= (2/5)MR2
point and ring
The 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.

Point and Ring

M

R

R

M

M

M

length R

length R

axis

playground ride
A 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 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 m2

Playground Ride

m

M

r

parallel axis theorem
Some objects don’t 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:

Parallel Axis Theorem

h = R/2

M

axis

perpendicular axis theorem
Perpendicular Axis Theorem
  • For flat objects the rotational moment of inertia of the axes in the plane is related to the moment of inertia perpendicular to the plane.

Iy= (1/12) Ma2

b

Ix= (1/12) Mb2

M

a

Iz= (1/12) M(a2 +b2)

spinning coin
What 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) MR2

Ie= (1/4) MR2

Spinning Coin

R

M

M

R

Id

Ie

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