Derive an expression for the velocity of the body moving in a vertical circle. And also find a tension at the top of the circle.

Explanation

(i) A body of mass (m) attached to one end of a massless and inextensible string executes circular motion in a vertical plane with the other end of the string fixed. The length of the string becomes the radius `vec(r )` of the circular path.
(ii) The motion of the body by taking the free body diagram (FBD) at a position where the position vector `vec(r )` makes an angle `theta` with the vertically downwards direction and the instantaneous velocity is as shown in Figure.
There are two force acting on the mass.
1. Gravitational force which acts downward
2. Tension along the string.

Applying Newton's second law on the mass, In the tangential direction,
`mg sin theta = ma_(t)`
`mg sin theta =- m((dv)/(dt))`
where, `a_(t)=-((dv)/(dt))` is tangential retardation
In the radial direction,
`T - mg cos theta = ma_(r ), T - mg cos theta = (mv^(2))/(r )`
where, `a_(r )=(v^(2))/(r )` is the centripetal acceleration.