Question

A small spherical ball of mass $m$ slides without friction from the top of a hemisphere of radius $R$. At what height will the ball lose contact with surface of the sphere?

Answer 1

Let the spherical ball is at point $P$ when it looses contact with the surface, i.e $N=0$

Also let the velocity of ball be $v$.

From geometry, we get $OB=Rcos\theta$

Also $AB=AO-BO=R-Rcos\theta =R(1-cos\theta )$

Work-energy theorem:

$\Rightarrow mg\left(AB\right)=\frac{1}{2}m{v}^2-0$

$\Rightarrow$$mgR(1-cos\theta )=\frac{1}{2}m{v}^2$............... (1)

Using circular motion equation :

$\Rightarrow \frac{m{v}^2}{R}=mgcos\theta -N$

At point P:

$N=0$$⟹\frac{m{v}^2}{R}=mgR(1-cos\theta )$

From (1), we get:

$\Rightarrow mgR(1-cos\theta )=\frac{mgRcos\theta }{2}$

$\Rightarrow 1-cos\theta =\frac{cos\theta }{2}$

$\Rightarrow cos\theta =\frac{2}{3}$

Thus, $h=OB=Rcos\theta =\frac{2R}{3}$