Question

Calculate the moment of Inertia of uniform solid sphere of mass $M$ and Radius $R$, about its diameter

• A. $\frac{2}{3}M{R}^2$
• B. $\frac{2}{5}M{R}^2$
• C. $\frac{5}{3}M{R}^2$
• D. $\frac{5}{2}M{R}^2$

Answer 1

The correct option is B

$\frac{2}{5}M{R}^2$

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 $x$.

$\therefore$ Mass per volume of the solid sphere = $\frac{3M}{4\pi R^3}$

The thin hollow sphere considered above has a surface area $4\pi x^2$ and thickness $dx$. Its volume is $4\pi x^{2}dx$ and hence its mass is

= $\left(\frac{3M}{4\pi R^3}\right).\left(4\pi x^{2}dx\right)$

= $\frac{3M{x}^{2}dx}{R^3}$

Its moment of Inertia about diameter $O$ is therefore,

$dI$ = $\frac{2}{3}\left[\frac{3M{x}^{2}dx}{R^3}\right].x^2$ = $\frac{2M{x}^{4}dx}{R^3}$

As $x$ increases from $0$ to $R$, the shell covers the whole solid sphere, therefore,

$I$ = $\underset{0}{\overset{R}{\int }}\frac{2M}{R^3}{x}^{4}dx$ = $\frac{2}{5}M{R}^2$