On a mechanical lens

• A ball, moving upon a rough horizontal plane, is considered.
• The plane is to be divided into two regions with different coefficients of friction.
• We look for such boundary line that a parallel beam of balls will focus at a single point.
• It is shown that the boundary is defined uniquely, and its equation is derived.

In this paper, we consider the dynamics of a heavy homogeneous ball moving under the influence of dry friction on a fixed horizontal plane. We assume the ball to slide without rolling. We demonstrate that the plane may be divided into two regions, each characterized by a distinct coefficient of friction, so that balls with equal initial linear and angular velocity will converge upon the same point from different initial locations along a certain segment. We construct the boundary between the two regions explicitly and discuss possible applications to real physical systems.