Question

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

• A. $\frac{2}{5}\frac{{\pi }^2{r}^2{n}^2}{s}$
• B. $\frac{1}{10}\frac{{\pi }^2{n}^2}{r^{2}s}$
• C. $\frac{7}{8}\frac{\pi r^2{n}^{2}s}{1}$
• D. $\frac{5}{14}\frac{{\pi }^2{r}^2{n}^2}{s}$

Answer 1

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

Kinetic energy of rotation, E=$\frac{1}{2}I{\omega }^2$
where, I=moment of inertia of solid sphere=$\frac{2}{5}m{r}^2$
$\omega$=angular velocity=2n$\pi$
Thus, heat =Mass $\times$ specific heat$\times$temperature change =$\frac{E}{2}$
$\Rightarrow ms\Delta T=\frac{1}{10}m{r}^2{\omega }^2$
$\Rightarrow \Delta T=\frac{\frac{2}{5}{n}^2{\pi }^2{r}^2}{s}$