Question

Sand is to be piled up on a horizontal ground in the form of a regular cone of a fixed base of radius $R$. Coefficient of static friction between the sand layers is $\mu .$ Maximum volume of the sand can be piled up in the form of cone without slipping on the ground is

• A. $\frac{\mu R^3}{3\pi }$
• B. $\frac{\mu R^3}{3}$
• C. $\frac{\pi R^3}{3\mu }$
• D. $\frac{\mu \pi R^3}{3}$

Answer 1

Answer: (D)

Let $h=$ critical height of sand cone of radius $R$
Then, for a sand particle to be in equilibrium (it must no slips to the ground)

$f\sin \varphi =N\cos \varphi$
where, $f=$ friction, $N=$ normal reaction and $f=\mu N$
$\Rightarrow \mu N\sin \varphi =N\cos \varphi$
$\Rightarrow \tan \varphi =\frac{1}{\mu }=\frac{R}{h}$ (from figure)
So, maximum volume of sand cone that can be formed over level ground is
$V_{\max }=\frac{1}{3}\pi R^{2}h=\frac{1}{3}\pi R^2(\mu R)=\frac{\mu \pi R^3}{3}$