Question

A uniform solid sphere of radius $R$ has a cavity of radius $\frac{R}{2}$ as shown. Mass of this system is $M$. Find moment of inertia of this sphere about an axis passing through centre of sphere.

• A. $\frac{57}{160}M{R}^2$
• B. $\frac{16}{35}M{R}^2$
• C. $\frac{M{R}^2}{28}$
• D. $\frac{1}{20}M{R}^2$

Answer 1

The correct option is A $\frac{57}{160}M{R}^2$

Density$=\frac{M}{V}=\frac{M}{\frac{4}{3}\pi R^3}$

Mass of small sphere$=d\times V$

$=\frac{M}{\frac{4}{3}\pi R^3}\times \frac{4}{3}\pi \frac{R^3}{8}=\frac{M}{8}$

$MI$ of spehere about an axis passing through centre of sphere

$=\frac{2}{5}M{R}^2-\frac{7}{5}\frac{M}{8}\frac{R^2}{4}=\frac{2}{5}M{R}^2-\frac{7}{160}M{R}^2=\frac{57}{160}M{R}^2$