Initial angular velocity of a circular disc of mass $M$ is $\omega_1$. Then two small spheres of mass $m$ are attached gently to diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?
(1) $\left( \frac{M+m}{M} \right) \omega_1$
(2) $\left( \frac{M+m}{m} \right) \omega_1$
(3) $\left( \frac{M}{M+4m} \right) \omega_1$
(4) $\left( \frac{M}{M+2m} \right) \omega_1$

A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with an angular speed $\omega$. Two particles having mass $m$ each are now attached at diametrically opposite points. The angular speed of the ring will become:
(1) $\omega \frac { M } { M + m }$
(2) $\omega \frac { M + 2 m } { M }$
(3) $\omega \frac { M } { M + 2 m }$
(4) $\omega \frac { M - 2 m } { M + 2 m }$

Q7. Two planets $A$ and $B$ having masses $m _ { 1 }$ and $m _ { 2 }$ move around the sun in circular orbits of $r _ { 1 }$ and $r _ { 2 }$ radii respectively. If angular momentum of $A$ is $L$ and that of $B$ is 3 L , the ratio of time period $\left( \frac { T _ { A } } { T _ { B } } \right)$ is:
(1) $\left( \frac { r _ { 2 } } { r _ { 1 } } \right) ^ { \frac { 3 } { 2 } }$
(2) $\frac { 1 } { 27 } \left( \frac { m _ { 2 } } { m _ { 1 } } \right) ^ { 3 }$
(3) $27 \left( \frac { m _ { 1 } } { m _ { 2 } } \right) ^ { 3 }$
(4) $\left( \frac { r _ { 1 } } { r _ { 2 } } \right) ^ { 3 }$

Two disc having same moment of inertia about their axis. Thickness is $t _ { 1 }$ and $t _ { 2 }$ and they have same density. If $R _ { 1 } / R _ { 2 } = 1 / 2$, then find $t _ { 1 } / t _ { 2 }$.
(A) 4
(B) $1 / 4$
(C) $1 / 16$
(D) 16