A body of mass *m* moving with velocity *v* has a kinetic energy of *mv*^{2}. A body of mass *m* rotating about a fixed axis with angular velocity *w *will have a “rotational” kinetic energy of *Iw*^{2} where *I = *Moment of Inertia of the body

A rigid body is rotating with angular velocity *w* about an axis through a point *O* which is perpendicular to the plane of the body (See Fig. 1). Derive an expression for the kinetic energy of the body in terms of *w *and the Moment of Inertia*, I *

Fig. 1 illustrates the physical model in the problem:

Consider a particle of the rotating body of mass *m* at *A* (assume |*OA*| = perpendicular distance from *A* to the axis through *O* = *r*) travelling at speed *v*. (See Fig. 1) Its kinetic energy = *mv*^{2}.

The next step is to identify the energy in the system. Then a mathematical model can be constructed:

See the “Circular Motion” example: this shows that the velocity of this particle is *v* = *rw* so that its kinetic energy can be expressed as *m*(*rw*)^{2} = *mr*^{2}*w*^{2}

To find the kinetic energy of the whole body we sum the kinetic energies of all the individual particles which make up the body. Therefore: total kinetic = S *mr*^{2}*w*^{2}. As *w* is the same for each individual particle, the total kinetic energy can be expressed as* * *w*^{2S } *mr*^{2}. The quantity S *mr*^{2} is called the Moment of Inertia and is usually denoted by the letter *I.*