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

Question

Aicharya · Accepted Answer

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)

$ \begin{aligned} & \quad f \sin \phi=N \cos \phi \\ & \text { where, } f=\text { friction, } \\ & \\ & \Rightarrow \quad N=\text { normal reaction and } f=\mu N \\ & \Rightarrow \quad \mu N \sin \phi=N \cos \phi \\ & \Rightarrow \quad \tan \phi=\frac{1}{\mu}=\frac{R}{h} \text { (from figure) } \end{aligned} $ 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} $

Sand is to be piled up on a horizontal ground in the form of a regular cone of a fixed base of radius $R$.…

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