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A circular ring of mass M and radius R is rotating about its axis at an angular frequency $ω$ . Two blocks, each of mass m, are gently placed on the opposite ends of a diameter of the ring. The angular frequency of the ring becomes $ω^{r}$ . The ratio $ω^{r} / ω$ is

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$\frac{M}{(M + 2 m)}$

$\frac{2 m}{M}$

$\frac{2 M}{(M + 2 m)}$

$\frac{M}{2 m}$

answer is A.

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Detailed Solution

The angular momentum of the system will stay constant according law of conservation of angular momentum.

$\Rightarrow I ω = I^{1} ω^{r}$

$\Rightarrow M r^{2} ω = (M r^{2} + 2 m r^{2}) ω^{r}$

$\Rightarrow ω^{r} / ω = \frac{M}{M + 2 m}$

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