Find the moment of inertia of uniform ring of mass M and radius R about a diameter.

Explanation

Let AB and CD be two mutually perpendicular diameters of the ring. Take them ax X and Y-axes and the line perpendicular to the plane of the ring through the centre as the Z-axis. The moment of inertia of the ring about the Z-axis is `I = "MR"_2`. As the ring is uniform, all of its diameter equivalent and so `I_x = I+y`, From perpendicular axes theorem,
`I_z = I_x + I_(y')` Hence `I_x = I_z/2 = ("MR"^2)/2`
Similarly, the moment of inertia of a uniform disc about a diameter is `MR^2//4 `