In our study of rotational dynamics we skipped over exactly how to calculate the rotational inertia of a solid body. The process for calculating this quantity is quite complicated, and requires quite a bit of calculus. Thus we devote a section to calculating this quantity.
Consider a small section of a rod, a radius r from the axis of rotation, and with a mass δm, as shown below: Because the volume of the section of the rod is sufficiently small, we can calculate the moment of inertia of this single piece: I = δmr2. To find the moment of inertia of the entire rod, we sum over all pieces of a similar size that compose the rod:
This integral equation is the basic equation for the moment of inertia of a solid body.
Even with this equation, it is quite difficult to calculate the moment of inertia of a solid body. We will go through an example to show how it is done. Let us simply return to the example of the solid rod of length L, and mass M, rotated about its center, as shown below. Let us denote the cross sectional area of the rod by A. Thus the volume of the small element of mass, dV = Adx, where dx is the length of the small element of mass. Thus, if we denote the density of the rod by ρ, then we can describe dm in terms of dx:
Thus we now have an integral we can evaluate. We simply need to determine the limits. If we denote the axis of rotation to be at x = 0, then we simply integrate from -L/2 to L/2:
This is the equation for the moment of inertia of a thin rod, and it agrees with measured values.
In general, the moment of inertia of a solid body varies with MR2, where R is the measure of the radius, or length of a given object. To find the exact value of the moment of inertia, however, the complicated calculus is required.