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  4. A thin circular ring of mass M and radius r is rotating about its axis with a constant angular velocity ω. Four objects each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be

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Q. A thin circular ring of mass $M $and radius $r $ is rotating about its axis with a constant angular velocity $\omega$. Four objects each of mass $m$, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be

AIPMTAIPMT 2003System of Particles and Rotational Motion

Solution:

According to conservation of angular momentum,
$L=I \omega=$ constant.
Therefore, $I_{2} \omega_{2}=I_{1} \omega_{1}$
or $\omega_{2}=\frac{I_{1} \omega_{1}}{I_{2}}=\frac{M k^{2} \omega}{(M+4 m) k^{2}}=\frac{M \omega}{M+4 m}$.