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. $\frac{7}{8}\pi r^2{n}^{2}S$
• D. $\frac{5{\left(\pi rn\right)}^2}{14S}$

Answer 1

Answer: (A)

$I=\frac{2}{5}m{r}^2,\omega =2\pi n$
$k=\frac{1}{2}I{\omega }^2=\frac{1}{2}\times \frac{2}{5}m{r}^2\times 4{\pi }^2{n}^2$
$=\frac{4}{5}{\pi }^{2}m{r}^2{n}^2$
$\Delta \theta =\frac{\Delta Q}{mS}=\frac{\frac{k}{2}}{mS}$
$=\frac{2}{5}\frac{{\pi }^{2}m{r}^2{n}^2}{mS}$
$=\frac{2{\pi }^2{r}^2{n}^2}{5S}$