Paper 4, Section I, C

Dynamics and Relativity | Part IA, 2021

A rigid body composed of NN particles with positions xi\mathbf{x}_{i}, and masses mi(i=m_{i}(i= 1,2,,N)1,2, \ldots, N), rotates about the zz-axis with constant angular speed ω\omega. Show that the body's kinetic energy is T=12Iω2T=\frac{1}{2} I \omega^{2}, where you should give an expression for the moment of inertia II in terms of the particle masses and positions.

Consider a solid cuboid of uniform density, mass MM, and dimensions 2a×2b×2c2 a \times 2 b \times 2 c. Choose coordinate axes so that the cuboid is described by the points (x,y,z)(x, y, z) with axa,byb-a \leqslant x \leqslant a,-b \leqslant y \leqslant b, and czc-c \leqslant z \leqslant c. In terms of M,aM, a, bb, and cc, find the cuboid's moment of inertia II for rotations about the zz-axis.

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