Question

Moment of inertia of a thin hallow sphere of mass $M$ and radius $R$ about an axis passing through its centre is:

• A. ${MR}^2$
• B. $M{R}^2/2$
• C. $2/3{MR}^2$
• D. $2/5{MR}^2$

Answer 1

The correct option is C $2/3{MR}^2$

Consider a hollow sphere of radius $R$.

Consider a strip(ring shape) on this surface of angular width $d\theta$.

The strip being at position $\theta$ from horizon.

Mass of elemental strip $dm=\frac{M}{4\pi R^2}\cdot 2\pi R\cos \theta \cdot R\cdot d\theta$

$\Rightarrow dm=\frac{M}{2}\cos \theta \cdot \theta$

Elemental Moment of Inertia $dI=\frac{M}{2}\cos \theta \cdot d\theta \times R^2{\cos }^2\theta$

$dI=\frac{M}{2}{R}^2{\cos }^3\theta \cdot d\theta$

$\Rightarrow I={\int }_{\frac{-\pi }{2}}^{\frac{\pi }{2}}dI=\frac{M{R}^2}{2}{\int }_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{\cos }^3\theta \cdot d\theta$

$\Rightarrow I=\frac{2}{3}M{R}^2$