A plane of mass $M$ is looping in vertical curve path with constant speed of $v$. A man of mass "$m$" is sitting in the plane. At a certain instant $t$, when plane passes through a point $P$ of the curved path, the radius of curvature of curved path is $R$ and normal force "$N$" acting on the man exerted by seat of plane is making angle $\theta$ with vertical. [mass of man is $60\mathrm{kg}$ and $v=\frac{375}{2\sqrt{5}} {\mathrm{ms}}^{-1},R=1250\left(2+\cos \theta \right)$ metre].
Now choose the CORRECT option.

 

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 $\left(v\right)$?