5.2: Rotational Inertia
Rotational Kinetic Energy and Rotational Inertia
While our first approach to studying dynamics for linear motion was Newton's laws (forces cause accelerations), we will find it easier to examine rotational dynamics from a standpoint of energy first. Consider an object that is rotating around a stationary center of mass. Does such an object possess kinetic energy? We might be inclined to say that it does, but with the center of mass not moving, its momentum is zero, which would make the quantity \(\frac{p^2}{2m}\) also equal to zero.
Rigid objects are collections of multiple particles, and when they are rotating, all those particles (except those right at the pivot point) are moving, which means they all have kinetic energy. At any given moment, there are particles moving in opposite directions, and if the center of mass of the object is stationary, these opposite momenta (which are vectors) cancel, Their kinetic energies, on the other hand, are not vectors, and are all positive numbers, so they can never cancel out.
In some sense, the particles comprising a rotating object can be thought of as contributing to the "internal" energy of the object as we discussed back in Section 3.2 . But doing this runs contrary to the main reason for the introduction of the mechanical/internal energy idea, which was to separate the kinetic energy of the system that we can clearly see from the kinetic energy that is concealed from us inside the confines of the system. We can clearly see rotational motion of an object, so we choose to include rotational kinetic energy in the category of "mechanical energy."
Okay, so a rotating object does possess kinetic energy. Our task now is to express that kinetic energy in terms of the rotation variables we have already defined, but all we know about kinetic energy is the linear version. In the figure below we consider the motion of a single particle within a rigid rotating object.