Terminal retrograde turn of rolling rings

TL;DR

This paper investigates the unexpected retrograde spiral turn in rolling rings, revealing how hollow geometry influences air drag and friction to produce complex motion, with implications for designing advanced surface-effect flying objects.

Contribution

It introduces a novel observation of retrograde turning in rolling rings and explains the underlying physics involving air drag and friction effects.

Findings

Rolling rings exhibit a retrograde spiral turn in their motion.

Hollow geometry affects rotational air drag, influencing the trajectory.

Frictional force changes direction at the inflection point, causing the turn.

Abstract

We report an unexpected reverse spiral turn in the final stage of the motion of rolling rings. It is well known that spinning disks rotate in the same direction of their initial spin until they stop. While a spinning ring starts its motion with a kinematics similar to disks, i.e. moving along a cycloidal path prograde with the direction of its rigid body rotation, the mean trajectory of its center of mass later develops an inflection point so that the ring makes a spiral turn and revolves in a retrograde direction around a new center. Using high speed imaging and numerical simulations of models featuring a rolling rigid body, we show that the hollow geometry of a ring tunes the rotational air drag resistance so that the frictional force at the contact point with the ground changes its direction at the inflection point and puts the ring on a retrograde spiral trajectory. Our findings have potential applications in designing topologically new surface-effect flying objects capable of performing complex reorientation and translational maneuvers.