Question

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

• A. $[1+\frac{\sqrt{3}}{2}]R$
• B. $[1-\frac{\sqrt{3}}{2}]R$
• C. $\frac{R}{2}$
• D. $\frac{\sqrt{3}R}{2}$

Answer 1

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

Answer (2)


For equilibrium along horizontal direction
$N\sin \theta =f\cos \theta \Rightarrow f=N\tan \theta$
Now $\text{N}\tan \theta \le {\mu }_{s}N\Rightarrow \tan \theta \le \frac{1}{\sqrt{3}}$
$\tan \theta =\frac{x}{h}\Rightarrow \frac{x}{h}\le \frac{1}{\sqrt{3}}$ also, $x=\sqrt{R^2-h^2}$
$\frac{\sqrt{R^2-h^2}}{h}\le \frac{1}{\sqrt{3}}\Rightarrow \frac{R^2-h^2}{h^2}\le \frac{1}{3}$
$\Rightarrow 3R^2-3h^2\le h^2\Rightarrow 3R^2\le 4h^{\circ }\Rightarrow h^2\ge \frac{3R^2}{4}$
$h_{\text{min}}=\frac{\sqrt{3}R}{2}$
$\text{Maximum height}=R-h_{\text{min}}=R-\frac{\sqrt{3}R}{2}$
$=R[1-\frac{\sqrt{3}}{2}]$