The Euler disk and its dynamic finite time singularity: Investigating a fascinating acoustical and mechanical phenomenon with simple means

Abstract

Many have spun a coin or other disk on a table, observing — and listening! — that — while its motion comes to rest, the angular velocity of the contact point and the frequency of the emitted sound raise in a striking way. This dynamical system is known as the "Euler disk," and its physics is theoretically understood as a "finite time singularity" in form of an inverse power law. Various dissipation mechanisms lead to different exponents in the power law. The present investigation is about different configurations of the Euler disk using acoustical and mechanical data. We present results for the well-known Euler disk toy, well in accord with a model of viscous air dissipation. This is the first measurement using the acoustical signal for the dynamics of the Euler disk. Moreover, we present new cases where other dissipation mechanism must be dominant. Experiments are carried out with simple means and using a smartphone for recording and analysing data. This and interesting physics of the Euler disc make it a good candidate e.g. for undergraduate lab-work courses or research projects, which can spark students interest in acoustics, and more generally in applying physics to phenomena of everyday life.