The Motion Of A Spring Released From Uniform Circular Motion

TL;DR

This study investigates the dynamic behavior of a spring released from uniform circular motion, combining numerical modeling and video analysis to reveal how the spring's extension, angles, and collapse time depend on rotor angular velocity.

Contribution

It provides a detailed analysis of spring motion during release from circular motion, highlighting effects of angular velocity on spring extension and angles, supported by simulations and experimental data.

Findings

Spring extension and drag angle increase with rotor angular velocity.

Droop angle decreases as rotor angular velocity increases.

Collapse time and bend angle are independent of angular velocity.

Abstract

A weak spring is connected at one end to a rotor turning at constant angular velocity. The spring extends to a stretched length as determined by the spring mass, rest length, spring constant, rotor radius and rotor angular velocity. When released from the rotor, the inner end of the spring pulls away as expected, causing a wave to travel down the spring as it collapses. During this time interval, the outer end of the spring continues to move along its original circular path in uniform circular motion, as if the spring were still connected to the rotor. This is analogous to the effect of a hanging Slinky released from rest whose bottom end remains at a fixed position above the ground until a wave from the top of the Slinky reaches the bottom of the Slinky. Values from a numerical model and measurements from video analysis show that upon release the inner end travels along a circle of similar radius as the outer end. The effect appears as a series of alternating semi-circles. In addition, the simulation and data agree that (1) the spring extension and drag angle increase with the angular velocity of the rotor; (2) the droop angle decreases with angular velocity of the rotor; (3) the collapse time and bend angle of the collapsing spring are independent of the angular velocity.