A solid sphere is in purely rotational motion about its diameter. The ratio of its angular momentum and kinetic energy is $\frac{\pi}{22}$. Find angular velocity of the sphere. ( $\pi=22 / 7$ )

Angular momentum and rotational kinetic energy relate through the moment of inertia $I = \frac{2}{5}MR^2$ and angular velocity $\omega$ as $L = I\omega$ and $K = \frac{1}{2}I\omega^2$ respectively.

The ratio simplifies to $\frac{L}{K} = \frac{I\omega}{\frac{1}{2}I\omega^2} = \frac{2}{\omega}$.
Given $\frac{L}{K} = \frac{\pi}{22}$, we equate $\frac{2}{\omega} = \frac{\pi}{22}$.

Solving for angular velocity: $\omega = \frac{44}{\pi}$.

Substituting $\pi = \frac{22}{7}$ yields $\omega = 44 \times \frac{7}{22} = 14$ rad/s.

The angular velocity is therefore $14$ rad/s.