Paper 4, Section I, C

A rigid body composed of $N$ particles with positions $\mathbf{x}_{i}$, and masses $m_{i}(i=$ $1,2, \ldots, N)$, rotates about the $z$-axis with constant angular speed $\omega$. Show that the body's kinetic energy is $T=\frac{1}{2} I \omega^{2}$, where you should give an expression for the moment of inertia $I$ in terms of the particle masses and positions.

Consider a solid cuboid of uniform density, mass $M$, and dimensions $2 a \times 2 b \times 2 c$. Choose coordinate axes so that the cuboid is described by the points $(x, y, z)$ with $-a \leqslant x \leqslant a,-b \leqslant y \leqslant b$, and $-c \leqslant z \leqslant c$. In terms of $M, a$, $b$, and $c$, find the cuboid's moment of inertia $I$ for rotations about the $z$-axis.

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