On the support of a body by a surface with random roughness

TL;DR

This paper analyzes the probability distribution of support points for a body resting on a rough surface, providing mathematical solutions for intervals and circles, with applications in static mechanics.

Contribution

It introduces a probabilistic model for support points on rough surfaces and solves related geometric support problems for intervals and circles.

Findings

Support points are almost surely at two locations for an interval.

Derived probability distributions for support points with fine-grained roughness.

Applications demonstrated in static equilibrium problems.

Abstract

Suppose an interval is put on a horizontal line with random roughness. With probability one it is supported at two points, one from the left, and another from the right from its center. We compute probability distribution of support points provided the roughness is fine grained. We also solve an analogous problem where a circle is put on a rough plane. Some applications in static are given.