A worker wishes to pile a cone of sand onto a circular area...

A worker wishes to pile a cone of sand onto a circular area in his yard. The radius of the circle is $R$, and no sand is to spill onto the surrounding area (Fig. 6-46). If $\mu^{\text {stat }}$ is the static coefficient of friction between each layer of sand along the slope and the sand beneath it (along which is might slip), show that the greatest volume of sand that can be stored in this manner is $\pi \mu^{\text {stat }} R^{3} / 3$. (The volume of a cone is $A h / 3$, where $A$ is the base area and $h$ is the cone's height.)

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Key Concepts

Geometric Properties of Cones

A cone is a three?dimensional geometric shape with a circular base tapering smoothly to a point called the apex. The key formula here is that the volume of a cone is one third of the product of its base area and its height. This concept underpins the translation of physical dimensions into a quantity (volume) that can be optimized under certain constraints.

Static Friction and Slope Stability

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. In this context, the static coefficient of friction between sand layers determines the maximum angle (often called the angle of repose) at which the sand pile remains stable without collapsing or slipping. This principle is fundamental in analyzing when a granular material remains in equilibrium on a slope.

Optimization Under Stability Constraints

The problem involves maximizing the stored volume of sand while ensuring that no sand spills over the boundary. This introduces an optimization constraint where the geometric configuration (the cone's slope or height) must not exceed the maximum stable slope determined by the static friction. This combination of geometric optimization and static equilibrium analysis is key in many problems dealing with material stability under limiting physical forces.

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