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.
Best Answer
There are two senses to $v_{\mathrm{min}}$. Firstly it means that the speed of the particle must be greater or equal to $v_{\mathrm{min}}$ at $\theta = \pi$ or else it will not complete the cycle. Instead it will follow a ballistic trajectory until such time that tension comes back into the string. Secondly, the speed of the particle is not constant: it is fastest at the very bottom and slowest at the very top. Thus $v_{\mathrm{min}}$ it is the slowest speed of the particle at any point around the cycle for a particle that just completes the cycle.
EDIT: The reason that there is no tension in the string is that the centripetal acceleration is perfectly balanced by the particle's weight when $v=v_{min}$ at the very top. If the particle were travelling faster here then the "excess" centripetal acceleration is taken up by tension in the string.