Question
Find the moment of inertia of disc of mass M and radius R about an axis passing through centre and perpendicular to its plane.
Solution
Let σ is the surface mass density of the disc.
Therefore,
Therefore,
Let AB be the axis passing through the center and perpendicular to the plane of disc.
Consider an arbitrary ring of radius x and thickness dx concentric with disc.
The mass of the ring of radius x and thickness dx is,
dm = σ2πxdx
The moment of inertia of this elementary ring about AB axis is,
dl = dmx2 = σ2πx3 dx
To find the moment of inertia of the disc about AB axis, integrate the above equation from x - 0 to x = R. 
