Question

The moment of inertia of a disc about an axis perpendicular to it and passing through its centre is MR'/2. What is the moment of inertia 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 MR²/2, 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,

I₁ = Ix + Iy

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

Ix = ly

and I = 21,

But I₂ = MR²/2

So finally, I, I,/2 = MR²/4

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