Question

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

• A. $\frac{R}{2}$
• B. $\left(1-\frac{\sqrt{3}}{2}\right)R$
• C. $\frac{\sqrt{3}}{2}R$
• D. $\frac{3R}{8}$

Answer 1

Answer: (B)

Given, radius of hollow hemisphere $=R$
Coefficient of friction between the particle and the hemisphere $(\mu )=\frac{1}{\sqrt{3}}$.
We know that maximum height,
$h_{\max }=\left(1-\frac{1}{\sqrt{{\mu }^2+1}}\right)R$
$=\left(1-\frac{1}{\sqrt{{\left(\frac{1}{\sqrt{3}}\right)}^2+1}}\right)\times R$
$=\left(1-\frac{1}{\sqrt{\frac{1}{3}+1}}\right)\times R$
$=\left(1-\frac{1}{\sqrt{\frac{1+3}{3}}}\right)\times R$
or, $h_{\max }=\left(1-\frac{\sqrt{3}}{2}\right)R$