First slide

Angular momentum

Question

A solid sphere is rotating about a diameter at an angular velocity $ω$ If it cools so that its radius reduces to $\frac{1}{n}$ of its original value, its angular velocity becomes

Moderate

A

$\frac{ω}{n}$
B

$\frac{ω}{n^{2}}$
C

$nω$
D

$n^{2} ω$

Solution

We know that $I_{1} ω_{1} = I_{2} ω_{2}$
For solid sphere, the moment of inertia about diameter is given by
$\begin{array}{l} I = \frac{2}{5} {mr}^{2} \\ ∴ \frac{2}{5} {mr}_{1}^{2} ω_{1} = \frac{2}{5} {mr}_{2}^{2} ω_{2} \\ or {r_{1}}^{2} ω = {(\frac{r_{1}}{n})}^{2} ω_{2} \\ ω_{2} = n^{2} ω \end{array}$

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