Sand is to be piled up on a horizontal ground | vedclass

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(D) Let $h$ be the height of the sand cone with base radius $R$. The angle of repose $\alpha$ is the maximum angle at which the sand can be piled with

Vedclass · Accepted Answer

$\frac{\mu \pi R^3}{3}$

Vedclass · Answer

(D) Let $h$ be the height of the sand cone with base radius $R$. The angle of repose $\alpha$ is the maximum angle at which the sand can be piled without slipping. This angle is related to the coefficient of static friction $\mu$ by $	an \alpha = \mu$.In the cone,the angle $\alpha$ is formed by the slant height with the horizontal ground. From the geometry of the cone,$	an \alpha = \frac{h}{R}$.Equating the two expressions for $	an \alpha$,we get $\frac{h}{R} = \mu$,which implies $h = \mu R$.The volume $V$ of a cone is given by $V = \frac{1}{3} \pi R^2 h$.Substituting $h = \mu R$ into the volume formula,we get $V_{\max} = \frac{1}{3} \pi R^2 (\mu R) = \frac{\mu \pi R^3}{3}$.

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

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