Two beads (each of mass $m$) can move freely in a frictionless wire whose rotational inertia with respect to the vertical axis is $I$. The system is rotated with an angular velocity ${\mathrm{\omega }}_0$ when the beads are at a distance $\frac{r}{2}$ from the axis. What is the angular velocity of the system when the beads are at a distance$\mathit{ }r$ from the axis ?

  

A point object of mass m moving horizontally hits the lower end of the uniform thin rod of length $l$ and mass m and sticks to it. The rod is resting on a horizontal, frictionless surface and pivoted at the other end as shown in figure.

Find out angular velocity of the system just after collision.

The centre of mass of a uniform rod of length $10 \mathrm{meter}$ is moving with a translational velocity of $50 \mathrm{m} {\mathrm{s}}^{-1}$ on a frictionless horizontal surface as shown in the figure and the rod rotates about its centre of mass with an angular velocity of $5 \mathrm{rad} {\mathrm{s}}^{-1}$. Find out velocities $v_{\mathrm{A}} \mathrm{and} v_{\mathrm{B}}$.

A ring of radius $1 \mathrm{m}$ performs combined translational and rotational motion on a frictionless horizontal surface with an angular velocity of $3 \mathrm{rad} {\mathrm{s}}^{-1}$ as shown in the figure. Find out velocity of its centre and point $A$, if the velocity of the lowest point $v_{\mathrm{P}}$ is $1 \mathrm{m} {\mathrm{s}}^{-1}$.

A plank is moving with a velocity of $4 \mathrm{m} {\mathrm{s}}^{-1}$. A disc of radius $1 \mathrm{m}$ rolls without slipping on it with an angular velocity of $3 \mathrm{rad} {\mathrm{s}}^{-1}$ as shown in the figure. Find out the velocity of centre of the disc.

The end $B$ of uniform rod $\mathrm{AB}$which makes angle $\mathrm{\theta }$ with the floor is being pulled with a velocity $v_0$ as shown. Taking the length of the rod as $l$, calculate the following at the instant when $\mathrm{\theta }= 37º$. 

 $\left(\mathrm{a}\right)$ The velocity of end $A$

$\left(\mathrm{b}\right)$ The angular velocity of rod

$\left(\mathrm{c}\right)$ Velocity of $\mathrm{CM}$ of the rod.

A ball of radius $R=10.0 \mathrm{cm}$ rolls without slipping on a horizontal plane so that its centre moves with constant acceleration $a=2.50 \mathrm{cm} {\mathrm{s}}^{-2}$; $t=2.00 \mathrm{s}$after the beginning of motion, its position corresponds to that shown in the figure. Find: 

$\left(\mathrm{a}\right)$ the velocities of the points $A, B \mathrm{and} O$

$\text{(b)}$ the accelerations of these points.