Solution
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
/2, 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. As shown in Fig., we take three concurrent axes through the centre of the disc, O as the x,y,z axes ;x and y-axes lie in the plane of the disc and z is perpendicular to it. By the theorem of perpendicular axes,
I
z
=I
x
+I
y
Now, x and y 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
I
x
=I
y
and I
z
=2I
x
But I
z
=MR
2
/2
So finally, I
x
=I
z
/2=MR
2
/4
Thus the moment of inertia of a disc about any of its diameter is MR
2
/4.