A spherical ball of mass $m$ is kept at the highest point in the space between two fixed, concentric spheres $\mathrm{A}$ and $\mathrm{B}$ (see fig.). The smaller sphere $\mathrm{A}$ has a radius $R$ and 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 $\theta$.  What is the total normal reaction force exerted by the spheres on the ball in terms of angle $\theta$?

Two spheres of equal masses are attached to a string of length $2 \mathrm{m}$ as shown in figure. The string and the sphere are then whirled in a horizontal circle about $O$ at a constant rate. What is the value of the ratio?
$\frac{\left(\mathrm{Tension} \mathrm{in} \mathrm{the} \mathrm{string} \mathrm{between} P \mathrm{and} Q\right)}{\left(\mathrm{Tension} \mathrm{in} \mathrm{the} \mathrm{string} \mathrm{between} P \mathrm{and} O\right)}=$?

A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass $m$ moves against the wall with a speed $v$. Which of the following curve represents the correct relation between the normal reaction on the block by the wall $\left(N\right)$ and speed of the block $\left(v\right)$?