Question

A solid sphere of mass $M$ and radius $R$ is rotating about its diameter. A hollow sphere of same mass and same external radius is also rotating with an angular speed twice that of solid sphere. The ratio of their kinetic energies of rotation $(E_S:E_H)$ will be

• A. $3:20$
• B. $1:10$
• C. $20:3$
• D. $10:1$

Answer 1

The correct option is A $3:20$

We know
Moment of inertia of solid sphere $I_S=\frac{2}{5}M{R}^2$
Moment of inertia of hollow sphere $I_H=\frac{2}{3}M{R}^2$
Also,
Kinetic energy of rotation $E=\frac{1}{2}I{\omega }^2$
For solid sphere, $E_S=\frac{1}{2}{I}_S{\omega }_S^2$
For hollow sphere, $E_H=\frac{1}{2}{I}_H{\omega }_H^2$
Given:
${\omega }_H=2{\omega }_S$
$\therefore \frac{E_S}{E_H}=\frac{\frac{1}{2}{I}_S{\omega }_S^2}{\frac{1}{2}{I}_H{\omega }_H^2}$
$=\frac{\frac{1}{2}\times \left(\frac{2}{5}M{R}^2\right){\omega }_S^2}{\frac{1}{2}\times \left(\frac{2}{3}M{R}^2\right)(2{\omega }_S{)}^2}$
$=\frac{2}{5}\times \frac{3}{2}\times \frac{1}{4}$
$\Rightarrow \frac{E_S}{E_H}=\frac{3}{20}$