Moment of inertia

We can think of a solid rigid body as made up of many particles of masses m₁, m₂, m₃... at distances r₁, r₂, r₃... from the centre of rotation; the total rotational kinetic energy of the body will be the sum of the energies of all the particles. Then

Total kinetic energy = ½ m₁w²r₁² + ½ m₂w²r₂² + ½ m₃w²r₃² + …..

Notice that since the body is rigid the angular velocity (w) is the same for all particles although the linear velocity will be greater for particles further from the axis of rotation. We can write this as:


kinetic energy = ½ w²Smr²

where Smr² represents the sum of all terms like m₁r₁².

If we now compare the expression with that for linear kinetic energy we see that they are very similar.

The term Smr² takes the place of mass in the linear equation and it is known as the moment of inertia of the body (I).
The units for moment of inertia are kg m².


Since power is the rate at which work is done, or at which energy is transformed from one type to another, we can write:



Unlike mass, the moment of inertia of a body may be variable; it depends not only on the mass of the rotating object but also on how the mass is distributed about the axis of rotation. Therefore a wheel with a heavy rim will have a bigger moment of inertia than a uniform disc of the same mass and radius.