Question

Calculate the amount of inertia of
(a) a ring of mass $M$ and radius $R$ about an axis coinciding with the diameter of the ring
(b) a thin disc about an axis coinciding with the diameter.

Answer 1

Let $X$ and $Y$ axes be along two perpendicular diameters of the ring
By symmetry $I_x=I_y$ and by perpendicular axis theorem $I_z=I_x+I_y$
But we know that $I_z=M{R}^2$
$M{R}^2=I_x+I_y=2I_x$
$I_x=I_y=\frac{M{R}^2}{2}$
Similarly for a thin disc (i.e a circular plate), the moment of inertia about a diameter is $I=\frac{1}{2}\left(\frac{M{R}^2}{2}\right)=\frac{1}{4}M{R}^2$