Question

A body of mass m is suspended from a string of length R. What is the minimum horizontal velocity that should be given to the body in its lowest position so that it may complete the vertical circle?

Answer 1

The tendency of the string to become slack is maximum when the particle is at the topmost point of the circle.
At the top, tension is given by $T=\frac{m{v}_t^2}{R}-mg,where{v}_t=$ speed of the particle at the top.
$\Rightarrow \frac{m{v}_t^2}{R}=T+mg$
For $v_t$ to be minimum, $T=0\Rightarrow v_t=\sqrt{gR}$
If $v_b$ be the critical velocity of the particle at the bottom, then from conservation of energy
$mg\left(2R\right)+\frac{1}{2}m{v}_t^2=0+\frac{1}{2}m{v}_b^2$
Thus, $v_b=\sqrt{5gR}$