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

Answer 1

Answer: (A)

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