Suppose the body is rotating about an axis through O with a constant angular velocity ω. A particle A of mass m, at a distance r 1 from O describes its own circular path. If v 1 is its linear velocity along the tangent to the path, at the instant shown then v 1= r 1ω and the kinetic energy of A =(1/2) m 1 v 12=(1/2) m 1 v 12ω2
It follows that the kinetic energy of the whole body is the sum of the kinetic energy of its component particles. If these have masses m 1, m 2, m 3, etc. and are distributed at distances r 1, r 2, r 3, etc. from O then since all the particles have the same angular velocity (the body is rigid).
The quantity ∑ m i r i represents the sum of the mr 2 values depends on the mass and distribution of and is taken as a measure of the moment of inertia of the body about the axis in question. It is denoted by the symbol I.
Therefore, the rotational K.E. of the body is
