Consider a particle of mass m attached to a string and revolved in vertical circle of radius r, At every instant of motion there are only two forces acting on the particle (a) its weight mg, vertically downwards, which is constant and
(b) the force due to the tension along the string, directed along the string and towards the centre.
Its magnitude changes periodically with time and location.The particle may not complete the circle if the string slackens before the particle reaches at the top. This requires that the particle must have some minimum speed.
At the top position (Point A) : Let v_{a }be the speed of the particle and T_{A }the tension in the string. Here both, weight mg and force due to tension T_{A} are downwards, i.e. towards the centre. In this case, net force on the particle towards the center O is T_{A} + mg is their resultant as the centripetal force.
∴ T_{A} +mg = \(mvA^2\over r\) ……..(1)
or minimum possible speed at this point (or if the motion is to be realized with minimum possible energy),
T_{A} = 0
∴ 0 + mg =\(mvA^2\over r\)
\(\therefore{mv A^2\over r}\)= mg
That is the particle’s weight alone is the necessary centripetal force at the point A.
∴ v_{A}^{2 }= rg
∴ v_{A} = \(\sqrt{rg} \ \ .......2\)