Question

A particle slides on. the surface of a fixed smooth spherce starting from the topmost point. Find the angle rotated, by the radius through the particle, when it leaves contact with the sphere.

Answer 1

Let the velocity be v when the body leaves the surface,

From the free body diagram,

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

[because normal reaction]

$v^2=Rgcos\theta$ ...(i)

Again from work energy principle

Change in K.E.=work done

$\Rightarrow \frac{1}{2}m{v}^2-0=mg(R-Rcos\theta$

$\Rightarrow v^2=2gR(1-cos\theta$ ...(ii)

From (i) and (ii),

$Rgcos\theta =2gR(1-cos\theta )$

$3gRcos\theta =2gR$

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

$\theta =co{s}^{-}1\left(\frac{2}{3}\right)$