Calculate the moment of Inertia of uniform solid sphere of mass M and Radius R, about its diameter
25MR2
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 = R∫02MR3x4dx = 25MR2