What minimum horizontal speed should be given to the bob of a simple pendulum of length l so tht it describes a complete circle?

Explanation

Suppose the bob is given a horizontal speed `v_0` at the bottom and it describes a complete vetical circle. Let its speed at the highest point be v. Taking the gravitatioN/Al potenial energy to be zero at the bottom the conservastion of energy gives,
`1/2 mv_0^21/2mv^2+2mgl`
or, `mv^2=mv_0^2-4mgl`.......i

The forces acting on the bob at teh highest poit are mg due to the gravity and T due to the tension in teh string. The resultant force towards the cente is therefore, `mg+T`. As the bob is moving in a circ,le, its acceleration towards the centre is `v^2/l`. Applying Newton's second law and using i,
`mg+T=mv^2/l=1/l(mv_0^2-4mgl)`
or, `mv_0^2=5mgl+Tl`
Now, for `v_0` to be minimum T should be minimum. As the minimum value of T can be zero, for minimum speed,
`mv_0^2=5mgl or, v_0=sqrt(5gl)`