A spherical ball of mass m is kept at the highest point in the space between two fixed, concentric spheres A and B (see fig.). The smaller sphere A has a radius R and the space between the two spheres has a width d. The ball has a diameter very slightly less than d. All surfaces are frictionless. The ball is given a gentle push (towards the right in the figure). The angle made by the radius vector of the ball with the upward vertical is denoted by θ.  What is the total normal reaction force exerted by the spheres on the ball in terms of angle θ ?

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 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 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 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,

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 $\mathrm{g}=10 \mathrm{m} {\mathrm{s}}^{-2}$ )