Question

A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Four objects each of mass $m$ are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be

• A. $\frac{(M+4m)\omega }{M}$
• B. $\frac{(M-4m)\omega }{M+4m}$
• C. $\frac{M\omega }{4M}$
• D. $\frac{M\omega }{M+4m}$

Answer 1

The correct option is D $\frac{M\omega }{M+4m}$

$L_i=L_f$
$L_i=M{r}^2\omega$
$L_f=(I_r+I_m)=(M{r}^2+4m{r}^2){\omega }^{\prime }$
Since $L_i=L_f$,

$M{r}^2\omega =(M{r}^2+4m{r}^2){\omega }^{\prime }$

${\omega }^{\prime }=\frac{M{r}^{2}w}{(M{r}^2+4m{r}^2)}=\frac{M\omega }{M+4m}$