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. $\frac{2}{5}\frac{{\pi }^2{n}^2{r}^2}{S}$
• B. $\frac{1}{10}\frac{{\pi }^2{n}^2}{r^{2}S}$
• C. $\frac{7}{8}\pi r^2{n}^{2}S$
• D. $5{\left[\frac{rnx}{145}\right]}^2$

Answer 1

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

Given frequency, v = n rotation/s
$\omega =2\pi v$
=$2\pi n$
kinetic energy=$\frac{1}{2}I{\omega }^2$
=$\frac{1}{2}\times \left(\frac{2}{5}m{r}^2\right)\left(4{\pi }^2{n}^2\right)$
50% of kinetic energy = heat energy used
$\frac{1}{2}\times \frac{4}{5}m{r}^2{\pi }^n{n}^2=ms\Delta T$
$\Delta T=\frac{2}{5}\frac{r^2{\pi }^2{n}^2}{S}$