Rotating rod and ball

TL;DR

This paper analyzes a mechanical system with a rotating infinite rod and a ball, classifying possible motions, proving existence and uniqueness, and studying asymptotic behaviors of the system.

Contribution

It introduces a comprehensive classification of motion types and provides rigorous proofs of existence, uniqueness, and asymptotic analysis for the rotating rod and ball system.

Findings

Five types of motion are possible in the system.

Existence and uniqueness of motion are established.

Asymptotic behaviors of hit intervals and distances are characterized.

Abstract

We consider a mechanical system consisting of an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod rotates uniformly around one of its points. The ball is reflected elastically when colliding with the rod and moves freely between consecutive hits. A sliding motion along the rod is also allowed. We prove the existence and uniqueness of the motion with a given position and velocity at a certain time instant. We prove that only 5 kinds of motion are possible: a billiard motion; a sliding motion; a billiard motion followed by sliding; a sliding motion followed by a billiard one; and a constant motion when the ball is at the center of rotation. The asymptotic behaviors of time intervals between consecutive hits and of distances between the points of hits on the rod are determined.