A particle is released from the top of a smooth hemisphere.find the angle rotated by the radius through the particle, when it loses contract with the hemisphere. Also sketch variation of normal contact force with `cos theta`, where `theta` is an angle rotated by the radius through the particle.

Explanation

`h=R-Rcostheta`
`v^(2)=u^(2)+2gh=0+2gR(1-cos theta)`
`mg cos theta-N=(mv^(2))/(R )=(m)/(R )2gR(1-cos theta)`
`N=3mg cos theta-2mg`
when the particle loses contact, normal contact force vanishes.
`0=3mg cos theta-2mg`
`cos theta=(2)/(3)impliestheta=cos^(-1)((2)/(3))`
`N v//s cos theta`
`N=3mg cos theta-2mg`
`theta=0impliescostheta=1impliesN=mg`
`cos theta=(2)/(3)impliesN=0`
shape: `y=3mgx-2mg` (straight line)
`y=Mx+C(M=3mg`, slope is positive, `C=-2mg`, intercept is negative)