Approximate solution to the motion of a rolling ellipse

Abstract

In this paper, we theoretically analyze the motion of an ellipse rotating on a horizontal floor without slipping. The contact point between the ellipse and the floor is shifted back and forth against the center of gravity of the ellipse depending on its inclination, and the static frictional force is also changed its direction and prevents the ellipse from slipping. The torque due to the normal force exerted on the contact point causes the rotation to be accelerated and decelerated. While the exact form of the equation of motion of the rolling ellipse is nonlinear, we take the linear approximation of small deviations from the vertically standing and horizontally lying states, and its approximate solutions are given by a hyperbolic and a trigonometric function. We will show that the linear approximate solutions are suitable for university students in physics courses to understand the rotational motion of the ellipse, especially the nature of the forces acting on the ellipse.