A rod of length $l$ has a velocity $v$ at $P.$ Find the velocity at $Q$ and the angular velocity of the rod.

A flat rigid body is moving in $x-y$ plane on a table. The body lies in the $x-y$ plane. At an instant it was found that some of the velocity components of its three particles $A,$ $B$ and $C$ were $v_{Ax}=4 \mathrm{m}/\mathrm{s}, v_{Bx}=3 \mathrm{m}/\mathrm{s}$ and $v_{cy}=-2 \mathrm{m}/\mathrm{s}$ respectively. At the instant the three particles $A, B$ and $C$ were located at $(0, 0), (3, 4)$ and $(4, 3)$ (all in metre) respectively in a co-ordinate system attached to the table.

(a) Find the velocity of $A, B$ and $C.$
(b) Find the angular velocity of the body.

The rod of length $l=1 \mathrm{m}$ rotates with an angular velocity $\omega =2 \mathrm{rad} {\mathrm{s}}^{-1}$ and the point $P$ moves with velocity $v=1 \mathrm{m}{ \mathrm{s}}^{-1}$ and acceleration $a=1 \mathrm{m}{ \mathrm{s}}^{-2}.$ Find the velocity and acceleration of $Q.$

The angular velocity and angular acceleration of the pivoted rod are given as $\omega$ and $\alpha$ respectively. Find the $x$ and $y$ components of acceleration of $B.$

A cylinder of radius $5 \mathrm{m}$ rolls on a horizontal surface. If the velocity of its center is $25 \mathrm{m}/\mathrm{s}.$ Find its angular velocity and velocity of the point $A.$

A disc of radius $R=\frac{1}{2} \mathrm{m}$ is rolling on a plank. If the disc is moving on the plank with angular velocity $\omega =2 \mathrm{rad}/\mathrm{s}$ and speed $v_c=2 \mathrm{m}/\mathrm{s}.$ Find the speed of the plank for pure rolling.

A uniform disc of radius $R$ rolls perfectly over two horizontal plank $A$ and $B$ moving with velocities $v$ and $2 v,$ respectively. Find the
(a) velocity of $\mathrm{CM}$ of the disc.
(b) angular velocity of the disc

A cotton reel rolls without sliding such that the point $P$ of the string has velocity $v .$ Find the:
(i) velocity of its centre $C$
(ii) angular velocity of the cotton reel.