Rotational Kinetic Energy

First we can look at Kinetic Energy in Polar Coordinates (See Rotational Motion)

If , then

Notice that is the Moment of Inertia . So

In the case where radius is constant, we have just . We can actually find this from regular Kinetic Energy.

Kinetic Energy is additive, meaning that

So in a situation such as a ball rolling down a hill, all we have to do is add the kinetic energies of each part of the problem. Since the ball has constant radius, the rotational portion is simply . The other portion is translational using the center of mass (see Systems of Particles).