Find the moment of Inertia of a uniform hollow sphere of mass M and radius R, about its diameter
Let us consider a small area element of Rdθ, at an angle θ with x axis as shown in figure.
It forms a ring of radius R Sinθ.
The width of this ring is Rdθ and its periphery is 2πR sinθ. Hence, the area of the ring = (2πRsinθ)(Rdθ)
Mass per unit Area of the given sphere = M4πR2
∴ Mass of ring element = (M4πR2)(2π RSinθ)(Rdθ)
= M2Sinθ dθ
Now, the moment of Inertia of this elemental Ring about OX is
dI = (M2Sinθ dθ)(RSinθ)2
= M2R2Sin3θ dθ
As θ increases from 0 to π, the elemental ring cover the whole spherical Hollow surface.
Therefore, I = π∫0MR2Sin3θ dθ = 23MR2