A solid sphere with uniform density and radius R is rotating initially with constant angular velocity…
A solid sphere with uniform density and radius R is rotating initially with constant angular velocity $\left(\omega_1\right)$ about its diameter. After some time during the rotation its starts loosing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become $R / 2$ is $x \omega_1$. The value of $x$ is ________
Solution
When sphere is of radius $R$, its mass is $M$, when radius is reduced to $\frac{\mathrm{R}}{2}$, mass will reduced to $\frac{\mathrm{M}}{8}$ Now by conservation of angular momentum $\begin{aligned} & \left(\tau_{e x t}=0\right) \\
& L_1=L_2 \\
& I_1 \omega_1=I_2 \omega_2 \\
& \left(\frac{2}{5} M R^2\right) \omega_1=\left(\frac{2}{5}\left(\frac{\mathrm{M}}{8}\right)\left(\frac{\mathrm{R}}{2}\right)^2\right) \omega_2 \end{aligned}$ $\omega_2=32 \omega_1$ value of x is 32 Answer is 32