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Question: A circular ring of mass M and radius R is rotating about its axis with constant angular velocity ω. ...

A circular ring of mass M and radius R is rotating about its axis with constant angular velocity ω. Two objects each of mass m get attached to the rotating ring. The ring now rotates with an angular velocity

A

ωMM+2m\frac{\omega M}{M + 2m}

B

ωmM+2m\frac{\omega m}{M + 2m}

C

ω(M−2m))M+2m\frac{\omega\left( M - 2m) \right)}{M + 2m}

D

ω(M+2m)m\frac{\omega(M + 2m)}{m}

Answer

ωMM+2m\frac{\omega M}{M + 2m}

Explanation

Solution

In the process of attaching the objects with the ring no external torque is applied on the system. Therefore the angular momentum of the system remains constant.

⇒ Li = Lf

⇒ Ii ωi = If ωf

⇒ Iring ω. = {(Iring) + (IA) + (IB)} ω′

Putting Iring = MR2, IA = IB = mR2 we obtain

ω′ = MωM⥂+2m\frac{M\omega}{M ⥂ + 2m}