Question

A solid sphere with uniform density and radius $ R $ is rotating initially with constant angular velocity ($ \omega_1 $) about its diameter. After some time during the rotation, it starts losing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius becomes $ \frac{R}{2} $ is $ \omega_2 $. The value of $ x $ is ________.

Hint:

Conservation of angular momentum is key when the object loses mass uniformly but retains its shape and rotation.

Answer 1

Correct Answer: 32

When the sphere is of radius \( R \), its mass is \( M \), and when the radius is reduced to \( \frac{R}{2} \), the mass will reduce to \( \frac{M}{8} \). 
This is due to the conservation of angular momentum. 
Using the conservation of angular momentum (\( \tau_{\text{ext}} = 0 \)): \[ I_1 \omega_1 = I_2 \omega_2 \] \[ \left( \frac{2}{5} M R^2 \right) \omega_1 = \left( \frac{2}{5} \times \frac{M}{8} \times \left( \frac{R}{2} \right)^2 \right) \omega_2 \] Simplifying this: \[ \omega_2 = 32 \omega_1 \] 
Thus, the value of \( x \) is 32.