Oscillations of a suspended slinky

TL;DR

This paper analyzes the oscillations of a suspended slinky, revealing distinct motion patterns in different parts and deriving a simple formula for the oscillation period based on its length and gravity.

Contribution

It introduces a discrete model of the slinky and compares it to a continuous approach, providing new insights into its oscillatory behavior and period.

Findings

Upper part performs triangular oscillation

Bottom part performs near harmonic oscillation

Oscillation period is T=√(32L/g)

Abstract

This paper discusses the oscillations of a spring (slinky) under its own weight. A discrete model, describing the slinky by $N$ springs and $N$ masses, is introduced and compared to a continuous treatment. One interesting result is that the upper part of the slinky performs a triangular oscillation whereas the bottom part performs an almost harmonic oscillation if the slinky starts with "natural" initial conditions, where the spring is just pulled further down from its rest position under gravity and then released. It is also shown that the period of the oscillation is simply given by $T=\sqrt{32 L/g}$, where $L$ is the length of the slinky under its own weight and $g$ the acceleration of gravity independent of the other properties of the spring.