A thin circular ring of mass M and radius R is rotating with a constant angular velocity $2 rad s^{- 1}$ in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass m be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in ${rads}^{- 1}$ )

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

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

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

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

answer is C.

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Detailed Solution

Applying conservation of angular momentum,

L_i= L_f

Iw + 0 = Iw′ + 2 × Iw′

Iw + 0 = Iw′ + 2 × mR²w′

${MR}^{2} ω = ({MR}^{2} + 2 {mR}^{2}) ω'$

$ω' = \frac{2 M}{M + 2 m}$

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