Finding the Moment of Inertia from a Point to a Ring to a Disk to a Sphere.

Photo: Rhett Allain

If you want to find the moment of inertia for a rigid object (like a disk or a sphere), you should first ask yourself two questions:

• What is the moment of inertia?
• Why am I doing this?

Here are the answers to those questions. For the moment of inertia, here is my short explanation (longer video here — but still fairly short). Imagine that I have two masses on connected together with a massless rod. Like this.

The kinetic energy for this system would be:

Although mass 1 is moving faster than mass 2 (since it has a larger circular radius), the two masses have the same angular velocity (ω). I can write the linear velocity in terms of the angular velocity.

With that substitution, I get the following expression for the total kinetic energy.

Both terms have a 1/2 and an ω², factoring that out, I get:

Now imagine that I have more than two masses — that are all rotating with the same angular velocity. This would be a rigid object. Those mr² terms can be grouped together and this is…