The moment of inertia of the pulley system as shown in figure is $3 {\mathrm{kgm}}^2 .$ Its radius is $1\mathrm{m}.$ The system is released from rest ,find the linear velocity of the block when it has descended through $40 \mathrm{cm}.$(Assume that there is no slipping between string & pulley and string is light)$[\mathrm{Take} g=10 \mathrm{m} {\mathrm{s}}^{-2} ]$

   

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 $1 \mathrm{m} {\mathrm{s}}^{-1}$.