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 $\mathrm{R} / 2$ is $x \omega_1$. The value of $x$ is _________.
Answer (integer)
32
Solution
<p>When sphere is of radius $R$, its mass is $M$, when radius is reduced to $\frac{R}{2}$, mass will reduced to $\frac{M}{8}$</p>
<p>Now by conservation of angular momentum</p>
<p>$$\begin{aligned}
& \left(\tau_{\mathrm{ext}}=0\right) \\
& \mathrm{L}_1=\mathrm{L}_2 \\
& \mathrm{I}_1 \omega_1=\mathrm{I}_2 \omega_2 \\
& \left(\frac{2}{5} \mathrm{MR}^2\right) \omega_1=\left(\frac{2}{5}\left(\frac{\mathrm{M}}{8}\right)\left(\frac{\mathrm{R}}{2}\right)^2\right) \omega_2
\end{aligned}$$</p>
<p>$\omega_2=32 \omega_1 \quad$ value of x is 32</p>
<p>Answer is 32</p>
About this question
Subject: Physics · Chapter: Rotational Motion · Topic: Centre of Mass
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