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 rin

Question

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 rad s - 1 ).

TestHub · Accepted Answer

As the masses are gently placed we can apply conservation of angular momentum.

Initial moment of Inertia $I_{1} = M R^{2}$

Final moment of inertia $I_{2} = M R^{2} + (2 m) R^{2}$

By conservation of angular momentum

$I_{1} ω_{1} = I_{2} ω_{2}$ $\Rightarrow M R^{2} \times 2 = (M + 2 m) R^{2} \times ω_{2}$

$\Rightarrow ω_{2} = \frac{2 M}{M + 2 m} rad s^{- 1}$

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