(C) In the absence of any external torque,the angular momentum of the system remains conserved.$L_{initial} = L_{final}$$I \omega = I' \omega'$Initial

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(C) In the absence of any external torque,the angular momentum of the system remains conserved.$L_{initial} = L_{final}$$I \omega = I' \omega'$Initially,the moment of inertia of the ring about its axis is $I = MR^2$.When two objects of mass $m$ are placed at opposite ends of a diameter,they are at a distance $R$ from the axis of rotation. The new moment of inertia becomes $I' = MR^2 + mR^2 + mR^2 = (M + 2m)R^2$.Using the conservation of angular momentum:$MR^2 \omega = (M + 2m)R^2 \omega'$$\omega' = \frac{MR^2 \omega}{(M + 2m)R^2}$$\omega' = \frac{\omega M}{M + 2m}$

Class 11 Physics System of Particles and Rotational Motion Conservation of angular momentum (combined translation and rotational motion)

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$A$ thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects,each of mass $m$,are gently placed at opposite ends of a diameter of the ring. The new angular velocity of the ring will be

A
$\frac{\omega M}{M + m}$
B
$\frac{\omega (M - 2m)}{M + 2m}$
C
$\frac{\omega M}{M + 2m}$
D
$\frac{\omega (M + 2m)}{M}$

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System of Particles and Rotational Motion

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Conservation of angular momentum (combined translation and rotational motion)

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$A$ thin ring of mass $M$ and radius $R$ is rotating with an angular velocity $\omega$ about an axis passing through its center and perpendicular to its plane. If $4$ point masses each of mass $m$ are placed gently at the opposite ends of two mutually perpendicular diameters,what will be the new angular velocity of the system?

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$A$ thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects,each of mass $m$,are gently placed at opposite ends of a diameter of the ring. The new angular velocity of the ring will be

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Initial angular velocity of a circular disc of mass $M$ is $\omega_{1}$. Then two small spheres of mass $m$ are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?

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