A worker wishes to pile a cone of sand on to a circular area in his yard. The radius of the circle is r; no sand is to spill on to the surrounding area. If μ is the static coefficient of friction between each layer of sand, show that the greatest volume of sand that can be stored in this manner is 1/3πμr³

N =F_gy = mgcosθ

f = F_gx = mg sinθ < f_{s max} = µN=  µmgcos theta--> μ > tan theta

The surface of the cone has the greatest slope and the height of the cone is

μ=tan theta = h/r = rμ

V_cone = A*h /3 = piμ_s r³/3

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