Integration of Constraint Equations in Problems of a Disc and a Ball Rolling on a Horizontal Plane

TL;DR

This paper presents a method for integrating the differential constraint equations of a rolling disc and ball on a horizontal plane by using the natural equation of the contact point's trajectory, explicitly relating curvature to distance.

Contribution

It introduces a novel approach to solving rolling constraint equations through the natural equation of the contact point's trajectory, simplifying integration.

Findings

Successful integration of rolling constraint equations

Explicit expression of curvature in terms of arc length

Enhanced understanding of rolling motion dynamics

Abstract

The problem of a disc and a ball rolling on a horizontal plane without slipping is considered. Differential constrained equations are shown to be integrated when the trajectory of the point of contact is taken in a form of the natural equation, i.e. when the dependence of the curvature of the trajectory is explicitly expressed in terms of the distance passed by the point.