Inertia

Abstract

When we make things rotate we are concerned with couples and moments of inertia. Just as a force is a vector, so a couple is represented by a vector in the direction of the axis about which it rotates. The integral of such a couple will be angular momentum. But while a moving mass has linear momentum in the direction in which it is moving, a revolving body can have momentum that is not aligned with the axis of its angular velocity. That is where the inertia tensor comes in. An object will have three mutually perpendicular principal axes of inertia about which it can spin undisturbed. For a ball these are any three perpendicular directions, having the same moment of inertia about each. But when a more general object spins about some other axis than a major axis, strange things can happen. But first in this chapter we will see the principles behind the inertia tensor, when we regard the body as an assembly of point masses. We later see that when solid objects are combined to make a bigger object, we first calculate the inertia tensor resulting from point masses at their centres of gravity, then add on the sum of their individual inertia tensors. Of course these must all be aligned about the same set of axis directions. So we can work out the inertia tensor of a boomerang by regarding it as a pair of sticks joined at right-angles.