Question

(a) Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be 2MR^2/4, where M is the mass of the sphere and R is the radius of the sphere. 

(b) Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be 2MR²/4, find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

Answer 1

(a) Moment of inertia of sphere about its diameter = \(\frac {2}{5}\)MR² 

Using Parallel axes theorem, Moment of inertia of this sphere about any tangent to the sphere is = 1_CM + Md²

= \(\frac {2}{5}\)MR² + MR²

= \(\frac {7}{5}\)MR²

(b) Moment of inertia of the of disc about its diameter = \(\frac {MR^2}{4}\)

|_AB= MR²/4

1_AB = MR²/4

Using the perpendicular axes theorem for planar objects,

1_PQ = 21_AB = MR²/2

using parallel axes theorem,

|1_PQ| = 1_{PQ +} MR²

= MR²/2 + MR² = \(\frac{3}{2}\)MR².