Explanation

(b) The ratio of rotational kinetic energy to total kinetic energy is
$\frac{KE_{R}}{KE_{T}} = \frac{\frac{1}{2}\ I\omega^{2}}{\frac{1}{2}I\omega^{2} + \frac{1}{2}\ Mv^{2}}$
The moment of inertia of a ring
= $\frac{1}{2}MR^{2}$
$\therefore$ $\frac{KE_{R}}{KE_{T}} = \frac{\frac{1}{2}\ \left( \frac{1}{2}MR^{2} \right)\left\lbrack \frac{v}{R} \right\rbrack^{2}}{\frac{1}{2} \times \frac{1}{2}MR^{2}\left\lbrack \frac{v}{R} \right\rbrack^{2} + \frac{1}{2}\ Mv^{2}}$
$= \frac{\frac{1}{2}\left( \frac{1}{2}Mv^{2} \right)}{\frac{1}{2}\left( \frac{1}{2}Mv^{2} \right) + \frac{1}{2}Mv^{2}}$
$\frac{KE_{R}}{KE_{T}} = \frac{1}{3}$