The rotational kinetic energy of an object with moment of inertia is

where is the angular velocity of the object.

Details

The rotational kinetic energy is just the sum of the kinetic energies for each little point mass in the object. It’s easy to see this, if we write the sum of each little mass , moving with a velocity a distance away from the center of the rotating object:

In this expression I’ve related the speed of each point with the speed of the whole body through . Then the last thing I did was use the definition of the moment of inertia, .

References

Moore (2e): Unit C, pg 160