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(C) According to the principle of conservation of angular momentum,the initial angular momentum is equal to the final angular momentum because no exte

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$\left(\frac{M}{M+4 m}\right) \omega_{1}$

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(C) According to the principle of conservation of angular momentum,the initial angular momentum is equal to the final angular momentum because no external torque acts on the system.The initial moment of inertia of the disc is $I_{1} = \frac{1}{2} M R^{2}$.The final moment of inertia of the system,after attaching two spheres of mass $m$ at the edge (distance $R$ from the center),is $I_{2} = \frac{1}{2} M R^{2} + m R^{2} + m R^{2} = \frac{1}{2} M R^{2} + 2 m R^{2} = R^{2} (\frac{M}{2} + 2m) = \frac{1}{2} R^{2} (M + 4m)$.Using $L_{initial} = L_{final}$:$I_{1} \omega_{1} = I_{2} \omega_{2}$$\frac{1}{2} M R^{2} \omega_{1} = \frac{1}{2} R^{2} (M + 4m) \omega_{2}$Solving for $\omega_{2}$:$\omega_{2} = \left(\frac{M}{M + 4m}\right) \omega_{1}$.

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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