# Moments of Inertia

## Demonstration goals:

• To show the relationship between an object`s moment of inertia and the distribution of mass within it

Much of what we know about the distribution of mass within planets comes from their moments of inertia. The moment of inertia controls the rate at which a planet spins and the location of the spin axis depend on the distribution of mass in each body. We illustrate this concept by “racing” a metal ring and disk, which have the same mass but different moments of inertia, down an incline.

The moment of inertia, and can be found from where mi is the mass and pi is the distance from the axis of rotation of the ith particle. If the particles are sufficiently small, we can replace the first equation with the integral .

What do these equations mean? First, objects with more mass at the center have lower moments of inertia. Because the relationship of the moment of inertia to rotation is exactly analogous to that for inertia and motion, a body with a lower moment of inertia will rotate faster than a body with a higher moment of inertia, even if they have the same masses and sizes. Second, if the mass is distributed differently with respect to different axes, the body will have a different moment of inertia. This is important not just for planetary studies, but for things such as satellites; more than one satellite has been lost because of an incorrectly calculated moment of inertia!

We illustrate these ideas by considering disks, which we take as two-dimensional. For this experiment, you will need:

• One ring and disk set (Fischer CHS529721 \$30.00)
• Board (1 m {3 ft} long)
• Book (.25 m {10 in} thick)

1. Place the board on top of the book, so that it makes an inclined plane.

2. Pass the ring and disk around the room; ask “Which will win the race?” Record the answers.

3. Put the ring and disk at the top of the board and let them go at the same time. The disk starts off faster and rapidly pulls away from the ring, because the disk, with a smaller moment of inertia, rotates faster.

4. Set the board up on a carpet. Before you release the disk and ring, ask “Which will travel further?” Because the kinetic energy of a rotating object is given by , most students predict that the object with the larger moment of inertia will travel further. However, because the total energy of the system is constant, both objects should travel approximately the same distance.

For Discussion:
Which has the lower moment of inertia – a hollow ball or a solid one? How do they compare with a hollow ball filled with a fluid? Prove your answers!

Related pages:

Moments of inertia for various shapes