A solid sphere with uniform density and radius R is rotating init… - Vidyadip

Question

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 __________ .

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Answer

32
When sphere is of radius R , its mass is M , when radius is reduced to $\frac{R}{2}$, mass will reduced to $\frac{M}{8}$
Now by conservation of angular momentum $\left(\tau_{\text {ext }}=0\right)$
$L _1= L _2$
$I _1 \omega_1= I _2 \omega_2$
$\left(\frac{2}{5} MR ^2\right) \omega_1=\left(\frac{2}{5}\left(\frac{ M }{8}\right)\left(\frac{ R }{2}\right)^2\right) \omega_2$
$\omega_2=32 \omega_1$ value of $x$ is 32

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