A particle is moving along a vertical circle of radius $R.$ At $P,$ what will be the velocity of particle (assume critical condition at $C) ?$

A particle of mass $m$ attached to the end of string of length $l$ is released from the horizontal position. The particle rotates in a circle about $O$ as shown. When it is vertically below $O,$ the string makes contact with a nail $N$ placed directly below $O$ at a distance $h$ and rotates around it. For the particle to swing completely around the nail in a circle,

A ball is released from the top of vertical circular pipe. Find angle $\mathrm{\theta }$ with vertical where the ball will lose contact with inner side wall of pipe and start moving with outer side wall. (Thickness of pipe is small as compared to radius of circle)

For a particle moving under the action of a variable force, kinetic energy-position graph is given, then