Chapter 5.3, Problem 7P

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, $I=\frac{1}{3}M{l}^2$).

A rectangular lamina has vertices (0, 0), (0, 2), (3, 0), (3, 2) and density xy. Find

(a) $M$,

(b) $\bar{x},\bar{y}$,

(c) $I_{x,}{I}_y$,

(d) $I_m$ about an axis parallel to the z axis. Hint: Use the parallel axis theorem and the perpendicular axis theorem.