Stochastic Model for the Motion of a Particle on an Inclined Rough Plane and the Onset of Viscous Friction

TL;DR

This paper introduces a stochastic model explaining the viscous friction and velocity scaling observed in particles moving on inclined rough planes, aligning with experimental results and providing insights into the underlying mechanisms.

Contribution

A novel one-dimensional stochastic model that reproduces experimental behaviors of particles on rough inclines, explaining viscous friction and velocity-radius scaling.

Findings

Frictional force is proportional to velocity, not velocity squared.

Velocity scales as a power of the particle radius.

Model reproduces phase diagram features of particle motion.

Abstract

Experiments on the motion of a particle on an inclined rough plane have yielded some surprising results. For example, it was found that the frictional force acting on the ball is viscous, {\it i.e.} proportional to the velocity rather than the expected square of the velocity. It was also found that, for a given inclination of the plane, the velocity of the ball scales as a power of its radius. We present here a one dimensional stochastic model based on the microscopic equations of motion of the ball, which exhibits the same behaviour as the experiments. This model yields a mechanism for the origins of the viscous friction force and the scaling of the velocity with the radius. It also reproduces other aspects of the phase diagram of the motion which we will discuss.