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Q.

Find the moment of inertia of a uniform hollow cube of mass M and side length a about an axis passing through the centre of two opposite faces :

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a

$\frac{2 M a^{2}}{3}$

b

$\frac{7 M a^{2}}{3}$

c

$\frac{5 M a^{2}}{3}$

d

$\frac{5 M a^{2}}{18}$

answer is C.

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Detailed Solution

Let mass of each face be equal to m $m = M / 6$

$I = 2 \times \frac{m a^{2}}{6} + 4 \times {\frac{m a^{2}}{12} + m {(\frac{a}{2})}^{2}}$

$= \frac{m a^{2}}{3} + \frac{m a^{2}}{3} + \frac{m a^{2}}{1}$

$\frac{m a^{2} + m a^{2} + 3 m a^{2}}{3} = \frac{5 m a^{2}}{3}$ $I = \frac{5 \times \frac{M}{6} a^{2}}{3} = \frac{5 M a^{2}}{18}$

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Find the moment of inertia of a uniform hollow cube of mass M and side length a about an axis passing through the centre of two opposite faces :