Chapter 5.3, Problem 1P

The following notation is used in the problems: $M\text{ = mass}$, $\bar{x},\bar{y},\bar{z}=$ coordinates of center of mass (or centroid if the density is constant), $I=$ moment of inertia (about axis stated), $I_x,I_y,I_z=$ moments of inertia about x, y, z axes, $I_m=$ moment of inertia (about axis stated) through the center of mass.

Note: It is customary to give answers for $I,I_m,I_x$, etc., as multiples of M (for example, $I=\frac{1}{3}M{l}^2$).

Prove the “parallel axis theorem”: The moment of inertia I of a body about a given axis is $I=I_m+M{d}^2$, where $M$ is the mass of the body, $I_m$ is the moment of inertia of the body about an axis through the center of mass and parallel to the given axis, and d is the distance between the two axes.