A uniform rod of mass $m$ and length $l$ is kept vertical with the lower end clamped. It is slightly pushed to let it fall down under gravity. Find its angular speed when the rod is passing through its lowest position. Neglect any friction at the clamp. What will be the linear speed of the free end at this instant?

A uniform rod of length $l$ is released from rest such that it rotates about a smooth pivot. Find the angular speed of the rod when it becomes vertical.

A uniform disc of mass $m$ is fitted (pivoted smoothly) with a rod of mass $m/2$ . If the bottom of the rod is pulled with a velocity $v$ , it moves without changing its angle of orientation and the disc rolls without sliding. Find the kinetic energy of the system $(\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{}+\mathrm{}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c})$ .

The steel balls $A$ and $B$ have a mass of $500 \mathrm{g}$ each and are rotating about the vertical axis with an angular velocity of $4 \mathrm{r}\mathrm{a}\mathrm{d} {\mathrm{s}}^{-1}$ at a distance of $15 \mathrm{c}\mathrm{m}$ from the axis. Collar $C$ is now forced down until the balls are at a distance of $5 \mathrm{cm}$ from the axis. How much work must be done to move the collar down?

A cylinder of mass $m$ is rotated about its axis by an angular velocity $\omega$ and lowered gently on an inclined plane as shown in figure. Then:

Figure shows a smooth inclined plane fixed in a car accelerating on a horizontal road. The angle of incline $\mathrm{\theta }$ is related to the acceleration $a$ of the car as $a=g\tan \theta$. If the sphere is set in pure rolling on the incline,
 

A sphere is moving on a smooth surface with linear speed $v_0$ and angular velocity ${\omega }_0.$ It finds a rough inclined surface and it starts climbing up: