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Physics Question on System of Particles & Rotational Motion

A thin circular ring of mass M and radius r is rotating about its axis with a constant angular velocity ω\omega. Two objects, each of mass m, are attached gently to the opposite ends of a diameter of the ring. The wheel now rotates with an angular velocity

A

ωMM+m\frac{\omega M}{M+m}

B

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

C

ωMM+2m\frac{\omega M}{M+2 m}

D

ω(M+2m)M\frac{\omega( M +2 m )}{ M }

Answer

ωMM+2m\frac{\omega M}{M+2 m}

Explanation

Solution

Apply conservation of angular momentum.
Li=LfL _{ i } = L _{ f }
MR2ω=(M+2m)R2ω′MR ^{2} \omega =( M +2 m ) R ^{2} \omega'
ω′=MωM+2m\omega'=\frac{ M \omega}{ M +2 m }