Suppose a particle of mass $m$ is attached to an inextensible string of length $R$ and is undergoing vertical - circular motion.
The centripetal force is given by the tension & its weight:
$$T - mg\cos\theta = m\dfrac{v^2}{R}$$. Now, the velocity is decreasing since it is being retarded by $mg\sin\theta$. So, $T - mg\cos\theta$ will change.
Now, my book says
The particle will complete the circle if the string does not slack even at the highest point ie. at $\theta = \pi$. Thus tension in the string should be greater than or equal to zero at $\theta = \pi$.
Now, how can tension be zero upward? Even if it is zero, why doesn't it slack then?
Then the book jot down an equation:
So, $$T \geq 0 \quad \text{at} \quad \theta = \pi.$$ In critical case, substituting $\ T = 0\ $ and $\ \theta = \pi\ $ in the above mentioned equation, we get $$mg = m\dfrac{{v_{min}}^2}{R} \implies v_{min} = \sqrt{gR}.$$ After some kinematic calculation, we get the intitial velocity to be $\sqrt{5gR}$. Thus, to complete the circle, the particle must have velocity $\geq \sqrt{5gR}$.
Now, here the author always wrote $v_{min}$. What is $\text{min}$? Meant to say, why it should be minimum?? I'm not getting the sense.