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 ________

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