question_answer

A thin circular ring of mass M and radius r is rotating about its axis with a constant angular velocity 0). Two objects each of mass muare placed gently on the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity.

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

B) \[\frac{\omega (M+2m)}{m}\]

C) \[\frac{\omega (M-2m}{m=2m}\]                           

D) \[\frac{\omega M}{M=m}\]

Correct Answer: A

Solution :

                Angular momentum \[J=pr\] \[J=m\upsilon \times r\]          ????(1) Put \[\upsilon =r\omega \](D in equation (1)                                 \[J=m{{r}^{2}}\omega \] Now, by  conservation  of  angular momentum Initial angular momentum = final angular momentum                 \[M{{r}^{2}}\omega =M{{r}^{2}}\omega +m{{r}^{2}}\omega +m{{r}^{2}}\omega \]                                 \[\omega =\frac{M\omega }{(M+2m)}\]