Moment of inertia and rotational kinetic energy
Consider an object of mass m moving in a circle about an axis as in the picture below.
This can be written in terms of the angular speed of the object, , and the distance of the object from the axis of rotation, r
The quantity mr2 is defined to be the moment of inertia of the mass m. The kinetic energy can then be rewritten in terms on the moment of inertia
For a large object, we can consider it made up of many smaller pieces of mass.
Each piece of mass, mi, is at a different distance, ri, from the axis of rotation. The kinetic energy of the object is found by summing up the kinetic energies of each the small pieces of mass:
This can be written as
where is the moment of inertia of the object. This is the rotational kinetic energy of an object.
A rolling object has both rotational and translational kinetic energy, the total kinetic energy for a rolling object is
where I is the moment of inertia of the object, is the angular speed of the object, M is the mass of the object and is the velocity of the center of mass of the object.