Question

A metal sphere of radius r and specific heat s is rotated about an axis passing through its centre at a speed of n rotations per second. It is suddenly stopped and 50 % of its energy is used in increasing its temperature. Then the rise in temperature of the sphere is :

• A. $\frac{2{\pi }^2{n}^2{r}^2}{5s}$
• B. $\frac{{\pi }^2{n}^2}{10r^{2}s}$
• C. $\frac{7}{8}{\pi }^2{r}^2{n}^{2}s$
• D. $\frac{5{\left(\pi rn\right)}^2}{14s}$

Answer 1

The correct option is A $\frac{2{\pi }^2{n}^2{r}^2}{5s}$

moment of inertia of a sphere
$I=\frac{2}{5}m{r}^2$
$\omega =2\pi nrad/s$
$KE=\frac{1}{2}I{\omega }^2$
$=\frac{1}{2}.\frac{2}{5}m{r}^2\times {\left(2\pi n\right)}^2$
$=\frac{4}{5}m{\pi }^2{r}^2{n}^2$
Half of this energy is converted into heat
$dQ=\frac{4}{5}m{\pi }^2{n}^2{r}^2\times \frac{1}{2}$
Specific heat
$s=\frac{1}{m}\frac{dQ}{dT}$
$Dt=\frac{dQ}{ms}$
$dT=\frac{2/5m{\pi }^2{r}^2{n}^2}{ms}$
$dT=\frac{2{\pi }^2{r}^2{n}^2}{5s}$