The bobo of a pendulum at rest is given a sharp hit to impart a horizontal velocity u where l is the length of the pendulum. Find the angle rotated by the string before it becomes slack.

Explanation

Let stirng slacks at B.

Applying COM, E at A and B, `DeltaU+DeltaK=0`
`mgh+[1/2mv^2-1/2mu^2]=0`
`1/2mv^2-1/2mu^2=-mgh`
`v^2=u^2-2g(1+1costheta)`
When string slacks tension in the string becomes zero. The component of the weight in radial direction provide centripetal force at this position. From FBD of bob, we can write
`(mv^2)/(l)=mg cos theta`
`v^2=lg cos theta` (ii)
From equation (i) and (ii), we get
`3gl-2glcostheta=glcostheta`
`3cos theta=1impliestheta=cos^-1(1/3)`

Hence, the angle rotated by the string before it becomes slack is
`implies alpha=pi-theta=pi-cos^-1(1/3)`