Motion of a rolling sphere on an azimuthally symmetric surface

Abstract

This paper analyzes the translational motion that a sphere rolling over an azimuthally symmetric surface, under the presence of a constant gravitational field, and with the rolling-without-slipping condition, exhibits in two different situations: with and without friction with air, where the latter is expressed as a power-series function of the sphere’s translational speed. In order to achieve this, the equations of motion for each case are obtained through the use of Lagrangian Mechanics and are subsequently solved by numerical computation in Wolfram Mathematica. For the frictionless case, periodic behavior and a conservation law for the angular coordinate have been found, along with the condition under which an effective potential energy can be approximated as well as the relationships between initial conditions that produce gravitational-like trajectories for the motion of the sphere. The equations of motion derived for the case with friction are found to predict the energy loss and general decay of the sphere’s motion. Likewise, the normal force over the sphere as a function of time is obtained through the method of Lagrange's Undetermined Multipliers, and thus, the general conditions that the motion must satisfy in order to be described by the obtained models. Overall, this research provides insight into the type and characteristics of the motion performed by the system in these two cases, both through equations and their numerical solutions for different surfaces and initial conditions.