(a) Find all elements of the principal moment of inertia matrix for a thin uniform rod of mass \(\Delta m\) and length \(D\) if the rod is oriented along the \(x\) axis and the origin of coordinates is at the center of the rod.

(b) Use the parallel-axis theorem, the perpendicular axis theorem for thin lamina, and the result of part

(a) to find the principal moment of inertia matrix for a thin square of side \(D\) and total mass \(\Delta M\) that is perpendicular to the \(z\) axis, with the origin of coordinates at the center of the square.

(c) Find the principal moment of inertia matrix for a cube of edge length \(D\) and mass \(M\).

(d) Find the moment of inertia for the cube about an axis parallel to one of the axes in part (c) and which is oriented along the middle of one face of the cube.

(e) Find the moment of inertia for the cube about an axis parallel to one of the axes in part (c) and which is oriented along the length of one corner of the cube.