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Question

Calculate the moment of Inertia of uniform solid sphere of mass M and Radius R, about its diameter



Solution

The correct option is B


Draw two spheres of radii x and (x+dx), concentric with the given solid sphere. The thin spherical shell trapped between these spheres may be treated as a hollow sphere of radius x.

Mass per volume of the solid sphere = 3M4πR3

The thin hollow sphere considered above has a surface area 4πx2 and thickness dx. Its volume is 4πx2 dx and hence its mass is

 = (3M4πR3).(4πx2 dx)

= 3Mx2 dxR3

Its moment of Inertia about diameter O is therefore,

dI = 23[3Mx2 dxR3].x2 = 2Mx4 dxR3

As x increases from 0 to R, the shell covers the whole solid sphere, therefore,

I = R02MR3x4dx = 25MR2

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