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Q.
From a solid sphere of mass $M$ and radius $R$, a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is
JEE AdvancedJEE Advanced 2015System of Particles and Rotational Motion
Solution:
$I =\frac{ Mx ^{2}}{6}$
edge length : (x)
$2 R=\sqrt{3} x$
$x =\frac{2 R }{\sqrt{3}}$
Now,
mass of cube :
$m=\frac{M}{\left(\frac{4}{3} \pi R^{3}\right)}\left(\frac{2 R}{\sqrt{3}}\right)^{3}$
$\left(\frac{3 M}{4 \pi R^{3}}\right)\left(\frac{8 R^{3}}{3 \sqrt{3}}\right)$
$m =\frac{2 M }{\sqrt{3} \pi}$
$I =\frac{1}{3}\left(\frac{2 M }{\sqrt{3} \pi}\right)\left[\frac{4 R ^{2}}{3}\right]$
$=\frac{4 MR ^{2}}{9 \sqrt{3} \pi}$
