The surprising dynamics of a chain on a pulley: Lift-off and snapping

TL;DR

This paper investigates the complex dynamics of a chain on a pulley when one end is pulled with constant acceleration, revealing lift-off, acceleration reversal, and whip-like snapping through experiments, simulations, and theory.

Contribution

It introduces a novel analysis of chain motion with one end pulled, highlighting dramatic lift-off and snap phenomena not previously characterized.

Findings

Chain lifts off from pulley at finite time

Free end accelerates faster than pulled end

Chain exhibits whip-like snapping behavior

Abstract

The motion of weights attached to a chain or string moving on a frictionless pulley is a classic problem of introductory physics used to understand the relationship between force and acceleration. Here, we consider the dynamics of the chain when one of the weights is removed and, thus, one end is pulled with constant acceleration. This simple change has dramatic consequences for the ensuing motion: at a finite time, the chain `lifts off' from the pulley and the free end subsequently accelerates faster than the end that is pulled. Eventually, the chain undergoes a dramatic reversal of curvature reminiscent of the crack, or snap, of a whip. We combine experiments, numerical simulations, and theoretical arguments to explain key aspects of this dynamical problem.