Question

A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $\left(E_{\text{sphere}}/E_{\text{cylinder}}\right)$ will be

• A. 3 : 1
• B. 2 : 3
• C. 1 : 5
• D. 1 : 4

Answer 1

The correct option is C 1 : 5

Kinetic energy of sphere is given by
$E_{\text{sphere}}=\frac{1}{2}{I}_{\text{sphere}}{\omega }_{\text{sphere}}^2$
$E_{\text{sphere}}=\frac{1}{2}\times \frac{2}{5}m{R}^2\times {\omega }_{\text{sphere}}^2$

Kinetic energy of cylinder is given by
$E_{\text{cylinder}}=\frac{1}{2}{I}_{\text{cylinder}}{\omega }_{\text{cylinder}}^2$
$E_{\text{cylinder}}=\frac{1}{2}\times \frac{1}{2}m{R}^2\times {\omega }_{\text{cylinder}}^2$

Given, ${\omega }_{\text{cylinder}}=2{\omega }_{\text{sphere}}$

$\therefore \frac{E_{\text{sphere}}}{E_{\text{cylinder}}}=\frac{1}{5}$