question_answer

A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity co. If two objects each of mass m be attached gently to opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity

A)  \[\frac{\omega M}{M+m}\]                   

B)  \[\frac{\omega (M-2m)}{M+2\,m}\]

C)  \[\frac{\omega M}{M+2\,m}\]               

D)  \[\frac{\omega (M+2m)}{M}\]

Correct Answer: C

Solution :

As masses are added to the ring gently, there will be no torque and angular momentum will be conserved   \[l\omega =l'\omega '\] \[\Rightarrow \,\,\,\,\,\,\,\,\,M{{R}^{2}}\omega \,\,=\,\,\left( M{{R}^{2}}+2m{{R}^{2}} \right)\omega '\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\omega '=\frac{M\omega }{M+2m}\]