The moment of inertia of a solid sphere of mass M and radius R is (2/5)MR2
Moment of inertia
Moment of inertia is defined as the quantity expressed by the body resisting angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation.
Let us consider a sphere of radius R and mass M. A thin spherical shell of radius x, mass dm and thickness dx is taken as a mass element. Volume density (M/V) remains constant as the solid sphere is uniform.
M/V = dm/dV
M/[4/3 × πR3] = dm/[4πx2.dx]
dm = [M/(4/3 × πR3) ]× 4πx2 dx = [3M/R3] x2 dx
I = ∫ dI = (2/3) × ∫ dm . x2
= (2/3) × ∫ [3M/R3 dx] x4
=( 2M/R3)× 0∫R x4 dx
Limits: As x increases from 0 to R, the elemental shell covers the whole spherical surface.
I = (2M/R3)[x5/5]R0
= (2M/R3)× R5/5
Therefore, the moment of inertia of a uniform solid sphere (I) = 2MR2/5.