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