$A$ particle is moving along a horizontal circle of radius $r$ under a centripetal force $F = -\frac{c}{r^2}$,where $c$ is a constant. Then,the total energy of the particle is

$A$ small sphere of mass $m$ is suspended by a light and inextensible string of length $l$ from a point $O$ fixed on a smooth inclined plane of inclination $\theta$ with the horizontal. The sphere is moving in a circle on the incline as shown. If the sphere has a velocity $u$ at the top most position $A$, then:

$A$ ball whose size is slightly smaller than the width of a tube of radius $R = 2.5 \, m$ is projected from the bottommost point of a smooth tube fixed in a vertical plane with a velocity of $v_0 = 10 \, m/s$. Let $N_1$ be the normal reaction exerted by the inner side of the tube and $N_2$ be the normal reaction exerted by the outer side of the tube on the ball. Which of the following statements is correct?

$A$ weightless thread can bear a tension of up to $3.7 \, kg \, wt$. $A$ stone of mass $500 \, g$ is tied to it and revolved in a circular path of radius $4 \, m$ in a vertical plane. If $g = 10 \, m/s^2$,then the maximum angular velocity of the stone will be ........ $rad/s$.

$A$ small block slides down from the top of a hemisphere of radius $R=3\, m$ 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)

You may have seen in a circus a motorcyclist driving in vertical loops inside a 'death well' (a hollow spherical chamber with holes,so the spectators can watch from outside). Explain clearly why the motorcyclist does not drop down when he is at the uppermost point,with no support from below. What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is $25 \; m$?