Four holes of radius R cut from a thin square plate of side 4R and mass M. The moment of inertia of the remaining portion about z-axis is ( 3 8 − n π ) MR 2 . Find n :-

Question

Four holes of radius R cut from a thin square plate of side 4R and mass M. The moment of inertia of the remaining portion about z-axis is ( 3 8 ​ − n π ​ ) MR 2 . Find n :-

TestHub · Accepted Answer

The original square plate has moment of inertia about its centre

$I_{square} = \frac{1}{6} M (4 R)^{2} = \frac{8}{3} M R^{2}$

Each removed disc has mass

$m_{d} = \frac{π R ^{2}}{16 R ^{2}} M = \frac{π M}{16}$

Its centre is at distance

$d = 2 R$

Subtract the four discs using the parallel-axis theorem.

$I_{remaining} = \frac{8}{3} M R^{2} - 4 (\frac{1}{2} m_{d} R^{2} + m_{d} d^{2})$

$I_{remaining} = (\frac{8}{3} - \frac{5 π}{8}) M R^{2}$

$\frac{1}{n} = \frac{5}{8} \Rightarrow n = \frac{8}{5} = 1.6$

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