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  4. Moment of inertia of an annular disc of mass M having inner and outer radius r and R respectively about an axis passing through centre of mass and perpendicular to the plane of the disc is

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Q. Moment of inertia of an annular disc of mass $M$ having inner and outer radius $r$ and $R$ respectively about an axis passing through centre of mass and perpendicular to the plane of the disc is

Solution:

$I = \int^{R}_{r}\frac{M}{\pi\left(R^{2}-r^{2}\right)}\left(2\pi xdx\right)x^{2}$
$=\frac{M}{2\left(R^{2}-r^{2}\right)}\left(R^{4}-r^{4}\right)$
$=\frac {M}{2}(R^2+r^2)$