Question

give the expression for moment of inertia of a circular disc of radius R about its diameter​

Answer 1

ANSWER

We assume the moment of inertia of the disc about an axis perpendicular to it and through its centre to be known; it is MR^2/2MR

(where M is the mass of the disc and R is its radius.)

The disc can be considered to be a planar body. Hence the theorem of perpedicular axes is applicable to it. we take three concurrent axes through the centre of the disc, O as the x, y, zx,y,z axes ; x;x and yy-axes lie in the plane of the disc and zz is perpendicular to it. By the theorem of perpendicular axes,

Iz=Ix+Iy

Now, xx and yy axes are along two diameters of the disc, and by symmetry the moment of inertia of the disc is the same about any diameter. Hence

Ix=Iy

and Iz=2Ix

But Iz=MR²/2

So finally, Ix=Iz/2=MR²/4

Thus the moment of inertia of a disc about any of its diameter is MR²/4