Question

Calculate the moment of inertia of a uniform disc of mass M and radius R about a diameter.

Answer 1

Let AB and CD be two mutually perpendicular diameters of the disc. Take them as X and Y-axes and the line perpendicular to the plane of the disc through the centre as the Z-axis. The moment of inertia of the ring about the Z-axis is $I=\frac{1}{2}M{R}^2$. as the disc uniform, all of its distance are equivalent and so $I_x=I_y$, From the perpendicular axes theorem,
$I_z=I_x+I_y$ Hence $I_x=\frac{I_z}{2}=\frac{M{R}^2}{4}$