Euler’s disk
A heavy disk spinning on a flat surface wobbles faster and faster, producing a rising chirp that accelerates toward an abrupt stop. The precession frequency diverges as a finite-time singularity — one of the few places in classical mechanics where a physical quantity approaches infinity.
Ω(t) ∝ (tstop − t)−1/3 • E ∝ α(t)2 • α → 0 as t → tstop
The phenomenon
Spin a coin on a table and listen. As it loses energy, it wobbles faster and faster, producing a distinctive rising buzz that accelerates until the coin suddenly drops flat. Euler’s disk is a heavy, precision-machined version of this: a thick chrome disk on a polished concave mirror, designed to spin for minutes and produce a dramatic chirp.
The finite-time singularity
The remarkable physics: the precession frequency Ω diverges as (tstop − t)−1/3. This means the frequency approaches infinity in finite time. In practice, the disk reaches thousands of Hz before air resistance and surface imperfections cause the abrupt final collapse. It is one of the clearest examples of a finite-time singularity in a tabletop experiment.
Energy dissipation
As the tilt angle α decreases, the disk’s gravitational potential energy drops as α2. The energy loss accelerates dramatically near the end — most of the energy is dissipated in the final fraction of a second. The energy graph shows this characteristic “cliff” at the end.
What causes it to stop?
The dominant dissipation mechanism is debated. Rolling friction, air viscosity (the thin air layer between disk and surface), and vibration all contribute. The exponent −1/3 comes from the assumption of viscous rolling friction. Different dissipation models give slightly different exponents, but all predict frequency divergence.