A rod of mass '$M$' and length $6 R$ is pivoted at its centre (point $P$) to rotate in vertical plane as shown. Two discs of same radius $\text{'}R\text{'}$ are joined at the ends $O,$ and $O_2$ of the rod. Mass of the discs $A$ and $B$ are $\text{'}M$ and $\text{'}2M\text{'}$ respectively. Initial angular velocities of discs are shown in figure. The system is released from rest from horizontal position. Find the angular velocity of the $\mathit{rod}$ (in $\mathrm{rad} {\sec }^{-1}$)when it is vertical. (Given $R=0.5 \mathrm{m}$ and $g=10 {\mathrm{ms}}^{-2}$)

A uniform disc of mass $m=1 \mathrm{kg}$ and radius $R=20 \mathrm{cm}$ lies flat on a smooth horizontal table. A mass less string runs halfway around it as shown in figure. One end of the string is attached to a small block of mass $m$ and the other end is being pulled with a force $F=20 \mathrm{N}.$ The circumference of the disc is sufficiently rough so that the string does not slip over it. Find acceleration of the small block (in $\mathrm{m}/{\mathrm{s}}^2$).

A metal plate which is square in shape of side length $l=2 \mathrm{m}$ has a groove made in the shape of two quarter circles joining at the centre of the plate. The plate is free to rotate about vertical axis passing through its centre. The moment of inertia of the plate about this axis is $I_0=4 \mathrm{kg}-{\mathrm{m}}^2.$ A particle of mass $m=1 \mathrm{kg}$ enters the groove at end $A$ travelling with a velocity of $v_A=2 \mathrm{m}/\mathrm{s}$ parallel to the side of the square plate. The particle move along the frictionless groove of the horizontal plate and comes out at the other end $B$ with speed $v.$ Find the magnitude of $v$ (in $\mathrm{m}/\mathrm{s}$) assuming that width of the groove is negligible.

Auniform thin stick of mass $M=24 \mathrm{kg}$ and length $L$ rotates on a frictionless horizontal plane, with its centre of mass stationary. A particle of mass $m$ is placed on the plane at a distance $x=I/3$ from the centre of the stick. This stick hits the particle elastically. Find the value of $m/M$ so that after the collision, there is no rotational motion of the stick.