Question

A particle is placed at rest inside a hollow hemisphere of radius R. The coeffieient of friction between the particle and the hemisphere is $\mu =\frac{1}{\sqrt{3}}.$ The maximum height up to which the particle can remain stationary is:

• A. $\frac{R}{2}$
• B. $(1-\frac{\sqrt{3}}{2})R$
• C. $\frac{\sqrt{3}}{2}R$
• D. $\frac{3R}{2}$

Answer 1

The correct option is B $(1-\frac{\sqrt{3}}{2})R$

Let the body is at $\theta$ from the vertical.
Therefore force in radial direction would be $mgcos\theta$
Force along tangential direction would be a part of force due to gravity (downwards) and friction force (upwards) , $mgsin\theta ,\mu N=\mu mgcos\theta$ respectively.
So body will remain stationary when $mgsin\theta \le \mu mgcos\theta$
For maximum height: $mgsin\theta =\mu mgcos\theta \therefore tan\theta =1/\sqrt{3}\&\theta =30$
So maximum height: $R-\left(Rcos\theta \right)=R(1-cos\theta )=R(1-\frac{\sqrt{3}}{2})$