Derive expressions for linear velocity at lowest point, midway and top position for a particle revolving in a vertical circle if it has to just complete circular motion without string slackening at top.

A particle of mass m performs vertical motion in a circle of radius r. Its potential energy at the highest point is ____.

(g is the acceleration due to gravity)

A small block slides down from the top of hemisphere of radius $R=3 \mathrm{m}$ as shown in the figure. The height $h$ at which the block will lose contact with the surface of the sphere is $m$. (Assume there is no friction between the block and the hemisphere)

A ball is released from rest from point $P$ of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point $Q$ is $A$ while angular position of point $Q$ is $\alpha$ with respect to point $P$. Which of the following graphs represent the correct relation between $A$ and $\alpha$ when ball goes from $Q$ to $R$?

A bullet of $10 \mathrm{g}$, moving with velocity $v$, collides head-on with the stationary bob of a pendulum and recoils with velocity $100 \mathrm{m} {\mathrm{s}}^{-1}$. The length of the pendulum is $0.5 \mathrm{m}$ and mass of the bob is $1 \mathrm{kg}$. The minimum value of $v \text{in} \mathrm{m} {\mathrm{s}}^{-1}$, so that the pendulum describes a circle. (Assume the string to be inextensible and $\mathrm{g}=10 \mathrm{m} {\mathrm{s}}^{-2}$ )

A body initially at rest and sliding along a frictionless track from a height $h$ (as shown in the figure) just completes a vertical circle of diameter $AB=D$. The height $h$ is equal to,