On a mechanical lens

TL;DR

This paper analyzes the dynamics of a sliding ball under dry friction, revealing how the plane's division into regions with different friction coefficients influences convergence points, with explicit boundary construction and potential applications.

Contribution

It introduces a model dividing the plane into regions with distinct friction coefficients and explicitly constructs the boundary, advancing understanding of frictional dynamics.

Findings

Balls with equal initial velocities converge to the same point from different initial positions.

The plane can be divided into two regions with different friction coefficients affecting motion.

Explicit boundary between regions is constructed and discussed for applications.

Abstract

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.