Slipping and Rolling on an Inclined Plane

TL;DR

This paper analyzes the motion of particles and spheres on inclined planes, deriving equations for trajectories, conditions for rolling versus slipping, and exact velocities, enhancing understanding of frictional effects in inclined plane dynamics.

Contribution

It provides a detailed analytical study of particle and sphere motion on inclined planes, including trajectory equations and conditions for rolling or slipping based on friction coefficients.

Findings

Derived parametric equations for particle trajectories.

Identified the critical friction coefficient for rolling versus slipping.

Calculated exact velocities and angular velocities of rolling spheres.

Abstract

In the first part of the article using a direct calculation two-dimensional motion of a particle sliding on an inclined plane is investigated for general values of friction coefficient ($\mu$). A parametric equation for the trajectory of the particle is also obtained. In the second part of the article the motion of a sphere on the inclined plane is studied. It is shown that the evolution equation for the contact point of a sliding sphere is similar to that of a point particle sliding on an inclined plane whose friction coefficient is $2/7}\ \mu$. If $\mu> 2/7 \tan\theta$, for any arbitrary initial velocity and angular velocity the sphere will roll on the inclined plane after some finite time. In other cases, it will slip on the inclined plane. In the case of rolling center of the sphere moves on a parabola. Finally the velocity and angular velocity of the sphere are exactly computed.