# Sand is 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 α.the maximum volume of sand that can be piled up,without the sand slipping on the surface is?

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by ritheesha

2014-11-25T04:29:07+05:30

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See diagram.
ABC is a cone of sand of base radius R and height h.  The base AB is circular.

Let us consider a point P on the slanting edge of the cone. On a small sand particle at P, there are three forces acting.  Force of friction, Normal reaction and weight.  To have the sand particle at P not to slide, we must have the forces balance each other.

consider the force components in the horizontal direction.

F Sin Ф = N Cos Ф,               but,    F = μ N
μ N Sin Φ = N Cos Φ
=>    Tan Φ = 1/μ
But tan Φ = R / h
hence      h = μ R

Maximum volume of cone = 1/3 * π R² h = π R³ μ /3