A particle is placed at rest inside a hollow hemisphere of radius $R$. The coefficient of friction between…
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
$\frac{R}{2}$
$\left(1-\frac{\sqrt{3}}{2}\right) R$
$\frac{\sqrt{3}}{2} R$
$\frac{3R}{8}$
Solution
Using $h_{\max }=\left(1-\frac{1}{\sqrt{\mu^2+1}}\right) R$
$\begin{aligned} & =\left(1-\frac{1}{\sqrt{\left(\frac{1}{\sqrt{3}}\right)^2+1}}\right) \times R \quad\left(\because \mu=\frac{1}{\sqrt{3}}\right) \\ & =\left(1-\frac{1}{\sqrt{\frac{1}{3}+1}}\right) \times R=\left(1-\frac{1}{\sqrt{\frac{1+3}{3}}}\right) \times R \\ & \text { or, } h_{\max }=\left(1-\frac{\sqrt{3}}{2}\right) R\end{aligned}$