Consider a sphere of mass M and radius R centered at origin. The density of material of the sphere is =A r^ , where r is the radial distance, and A are constants. If the moment of inertia of the sphere about the axis passing through centre is 6 7 M R^2 , then the value of is

Step-by-step solution

Given, density of sphere, p=A r^ (where, r= radial distance and A and are constants) Consider an elemental spherical shell of radius r and thickness d r Mass of elemental spherical shell, aligned & d m= Volume Density & d m= (4 r^2 ) d r A r^ =4 A r^ 2+ d r aligned Mass of entire solid sphere, aligned & M=4 A ₀^R r^ 2+ d r & M=4 A [ r^ 3+ 3+ ]₀^R= 4 A 3+ R^ 3+ aligned Now, moment of inertia of elemental spherical shell is d I= 2 3 (d m) r^2= 2 3 (4 A r^ 2+ d r ) r^2 Moment of inertia of entire solid sphere, I= ₀^R