In the given system, when the ball of mass m is released, it will swing down the dotted arc.
a. How fast will it reach the lowest point in its swing? A nail is located at a distance d below the point of suspension.

b. Show that d must at `0.6l`, if the ball is to swing completely around a circle centered along the nail.
c. If `d=0.6l`, find the change in tension in the string just after it touches the nail.

Explanation

a. Radius of the circle at nail `=l-d`. To complete the circle centered at nail, the speed at the bottom must be at least `=sqrt(5g(l-d))`
From A to B conserving mechanical energy `DeltaK+DeltaU=0`:
Loss in GPE=Gain in KE
`mgl=1/2mv^2impliesv=sqrt(2gl)`
b. To complete the circle: `sqrt(2gl)=sqrt(5g(l-d))`
`implies5l-5d=2limpliesd=3/5l=0.6l`
c. Just before touching the nail, the ball is moving in a circle of radius l.
`impliesT_(before)-mg=(mv^2)/(l)`
`=mg+(mv^2)/(l)=mg+2mg=3mg`

Just after touching the nail, the ball is moving in a circle of radius `(l-d)`.
`T_(after)-mg=(mv^2)/((l-d))`
`=mg+(mv^2)/(l-d)=mg+(m(2gl))/(0.4l)`
`implies Tension=6mg`
Hence, the tension in the string changes from `3mg` to `6mg` as it touches the nail.