The disk that spins faster as it stops

Spin a coin on its edge on a hard table and watch it settle. At first it spins upright, then it begins to lean, and as it leans it starts to rock round on its rim with a whirring rattle. The rattle rises in pitch, faster and faster, the coin lying flatter and flatter — and then, abruptly, it stops, flat on the table, silent. The end is not gradual. It does not trail off the way a pendulum’s swing dies away; it accelerates into the finish.

In the 1990s an engineer named Joseph Bendik turned this into a desk toy: a heavy, polished steel disk about 75 millimetres across, set spinning on a slightly concave mirror. It runs for over a minute, the whirr rising throughout, and ends with a sudden shudder. He called it Euler’s Disk, after the mathematician who had first worked out how rigid bodies spin. In 2000 the fluid dynamicist Keith Moffatt published a short paper in Nature arguing that the toy’s ending is a finite-time singularity: a point at which, according to its equations of motion, a quantity becomes infinite at a definite moment. The argument sparked a small controversy about which mechanism drives the disk to the singularity, and it has a satisfying shape.

Rolling on its own edge #

A disk spinning upright and slowly leaning over is not in the motion that produces the whirr. The whirr comes once the disk is tilted at a small angle and rolling on its rim in a circle, with its centre almost still: the point of contact runs round the rim of the disk, and at the same time round a circle on the table.

This is rolling without slipping, the contact that the grip that needs a little slipping found holding a tyre to a road: the disk’s rim at the contact point is momentarily at rest on the table, and friction supplies whatever sideways force that requires. Picture a coin that is almost flat. Its rim touches the table at one point; that point is the lowest point of the rim. Rock the coin so the lowest point moves round, and the coin rolls on its edge round its own centre. The motion the contact point traces on the table is a circle of radius $a\cos\alpha$, nearly the coin’s own radius when the tilt is small, and the rate at which it goes round is set by gravity and the tilt.

That rate comes from requiring the motion to be steady: the torque of gravity about the contact point, trying to tip the disk flat, must be exactly the rate at which the disk’s angular momentum is being turned round by the precession. The push that comes out sideways found the same balance holding up a spinning top. For a thin disk at a small tilt the result is

$$\Omega \approx 2\sqrt{\frac{g}{a\,\alpha}},$$

and the striking thing is the tilt in the denominator. A flatter disk must roll round faster to stay up.

Two rates going opposite ways #

The contact point running round the rim is not the same as the disk turning. If a mark is painted on the face of the disk, the mark turns round once each time the contact point has gone round the rim relative to the disk once more than it has gone round the table — and for a disk that is nearly flat, those two circles are almost the same size.

The face turns at $\Omega(1 - \cos\alpha)$, roughly $\Omega\alpha^2/2$, so as the tilt goes down the face slows even while the contact point speeds up. At ten degrees the contact point goes round twelve times a second and the face once every five seconds; at one degree, thirty-nine times a second and once every three minutes. This is why the head on a spinning coin seems to stand almost still near the end, rocking back and forth, while the rattle climbs: what is spinning fast is not the coin but the place where it touches.

The motion is a pure rolling motion, so its kinetic energy is small, and most of the disk’s energy is the potential energy of its raised centre. For a small tilt the total comes to

$$E \approx \tfrac{3}{2}\,M g a\,\alpha,$$

proportional to the tilt. As the disk loses energy to friction of one kind or another, the tilt has to fall in proportion. And because the contact rate goes as $\alpha^{-1/2}$, losing energy makes the contact point go faster.

A tilt that runs out #

Where the energy goes decides how fast the tilt falls. Moffatt’s proposal was the air. As the disk rolls, the thin wedge of air between its underside and the table is squeezed by the descending part of the disk and sucked in under the rising part, and the air’s viscosity resists that pumping. The narrower the wedge — the smaller the tilt — the stronger the shearing, and the rate of energy loss rises as one over the tilt squared. Write the energy as proportional to $\alpha$ and its rate of loss as proportional to $1/\alpha^2$, and the equation is

$$\alpha^2\,\frac{d\alpha}{dt} = -\text{constant} \quad\Rightarrow\quad \alpha^3 = \alpha_0^3\left(1 - \frac{t}{t_0}\right).$$

The cube of the tilt falls in a straight line, and reaches zero at a definite time $t_0$.

The second candidate is the contact itself. Rolling is not lossless: the table and the disk deform slightly where they touch, the deformation is not perfectly elastic, and the contact also slips slightly as it runs round. The simplest model of this is a rolling resistance — a small resisting torque of constant size — whose rate of energy loss is that torque times the contact rate, so proportional to $\Omega$, which goes as $\alpha^{-1/2}$. Now the equation gives $\alpha^{1/2}\,d\alpha/dt$ constant, and $\alpha^{3/2}$ falls in a straight line. Again the tilt reaches zero at a definite time.

Both models say the same qualitative thing: not an approach to flatness that goes on for ever, like a pendulum whose swing halves in each interval and never quite reaches zero, but an arrival. The energy runs out in a finite time because the rate at which it is lost grows as the energy shrinks. This is the reverse of the familiar exponential decay, in which the rate of loss is proportional to what remains and so becomes ever slower.

The same arithmetic appears in the bounces that add up to a stop, where a ball losing a fixed fraction of its speed at each bounce bounces infinitely many times in a finite total time, the intervals shrinking geometrically — and the bounces, like the disk’s rattle, crowd together into a buzz just before the ball comes to rest. A finite-time end reached by a sequence of events that speed up without limit is a common pattern wherever a loss mechanism is more effective the closer the system is to rest.

A singularity on a kitchen table #

The contact rate, which goes as $\alpha^{-1/2}$, does not merely grow as the tilt goes to zero; it goes to infinity, and it does so at the definite time $t_0$.

Plotted against the time remaining, on logarithmic axes, the rate is a straight line: a power law. Under the air model the exponent is $-1/6$ — from $\Omega \propto \alpha^{-1/2}$ and $\alpha \propto (t_0 - t)^{1/3}$ — and under rolling resistance it is $-1/3$. A millisecond before the end, the air model has the contact point going round 84 times a second and the rolling model 572. Both say infinitely fast at the end. That is what a finite-time singularity is: not a quantity growing without limit over infinite time, which is common, but one becoming infinite at a definite, predictable instant.

Such singularities are known elsewhere in physics, usually as warnings that a model is about to fail. The front that steepens until it cannot found a sound wave whose slope becomes infinite at a definite distance, where the smooth solution stops existing and a shock takes over. Fluids breaking into drops, stars collapsing and the curvature of a space-time inside a black hole all have the same shape. The coin on the table has the attraction of being cheap, safe and reproducible.

Which mechanism wins #

The exponent is what can be measured, and it is what decides between the mechanisms. Moffatt’s original estimate, from the viscosity of air and the size and weight of the toy, gave a running time of the right order — about a hundred seconds — which made the air model attractive. But within months two groups tested it directly by spinning the disk in a vacuum chamber. If air were the main loss, removing it should have made the disk run for very much longer. It did not: the running time barely changed. Measurements of the rate in the final seconds, by Easwar, Rouyer and Menon in 2002 and others since, have found exponents closer to the rolling-resistance value, and detailed models in which energy is lost through slight slipping and deformation at the contact reproduce the observations better than the air.

The disk also turns out not to reach its singularity. When the tilt is small enough, the gap between the rim and the table becomes comparable with the roughness of the two surfaces, and the contact is no longer a single point; and close to the end the rate is high enough that the disk can lose contact with the table altogether, bouncing for a few milliseconds — the final shudder or clatter. The model breaks down shortly before its singularity, as every model with a singularity must, and something else finishes the job.

None of that undoes the main point. Whatever the mechanism, the loss rate grows as the disk flattens, and so the end is reached in finite time with the contact rate soaring. Which loss is responsible changes the exponent and the pitch at which the disk falls silent, not the shape of the finish.

The pitch of the finish #

The whirr that makes the toy satisfying is the contact point striking the mirror, and its pitch is the contact rate.

In the last ten seconds the pitch rises through the bottom of the musical range, from below the piano’s lowest note through the next octave, and then stops. The ear is very good at hearing a pitch glide upward, and the effect is of a performance building to a climax. The real disk’s whirr has a rich texture — the contact point does not touch a perfect plane, and each bump adds harmonics — but its fundamental tracks the curve, and recordings of the sound are one of the ways the rate has been measured.

What the disk keeps and what it loses #

The motion has one more feature that makes it worth watching. The quantity that survives a change of shape found angular momentum conserved when a skater pulls in their arms; the disk is not isolated, since the table pushes and rubs on it, and so its angular momentum is not conserved. But it is small and stays small. The disk’s angular momentum about its own axis falls as the face’s rotation slows, and the large, rising quantity, the contact rate, is not a rotation of any material thing — it is the speed of a pattern, the point of contact, moving round the rim. The disk does not spin faster as it stops; the place where it touches does, and that place has no mass.

The top that nods before it settles found a spinning top’s axis wobbling about its slow precession, the nutation, and found friction at the tip gradually drawing it upright. The disk is the opposite case: friction draws it down, not up, and because the disk is flat rather than tall, the motion it is drawn into is rolling on its edge rather than spinning on its point. The same equations of rigid-body motion, with the same sorts of friction, produce both.

A coin, a bottle cap and a hubcap #

Nothing in the rates depends on what the disk is made of or how heavy it is, only on its radius and on gravity. At a given tilt the contact rate scales as one over the square root of the radius, so a coin 24 millimetres across whirrs about 1.8 times higher than the 75-millimetre toy at the same tilt, and a hubcap rolling to rest in a car park, several times larger, growls far lower — but every one of them shows the same rising glide and the same abrupt end. The friction at the contact must be enough to stop the rim slipping sideways, the force the force that takes what it needs found static friction supplying up to its limit; on a very smooth surface the rim can skid, and the whirr becomes a squeal.

The loss mechanisms do care about the materials. A disk on a soft surface — a tablecloth, a wooden desk — loses energy to the surface’s deformation much faster than one on glass or a mirror, which is why a coin spun on a cloth falls flat within a few seconds and the toy, a hard steel disk on a hard mirror, runs for over a minute. The heavy, highly polished design is not decoration. It reduces the rolling losses at the contact, and for the same reason it makes the role of the air relatively larger, which is part of why the air model looked plausible for the toy.

There is also a lesson here about how a dissipative system chooses its motion. The axis a leak of energy chooses found a spinning body that loses energy internally drifting towards rotation about its axis of greatest moment of inertia, because that axis holds a given angular momentum with the least energy. A spinning coin does something similar on the table: as it loses energy it is drawn from spinning upright, through rocking, into rolling on its rim, each stage a motion of lower energy, and the last stage is the one that runs out in finite time.

Small tilts, two idealised losses and a common finishing time #

The figures use the small-tilt approximation for the contact rate and the energy, which is good below about fifteen degrees and becomes poor for an upright spinning coin; the upright phase, in which the coin spins about its diameter like a top, is a different motion with its own instability, and a real coin passes through it before the rolling phase begins. The two loss models are idealisations: the air model treats the wedge of air as thin and slow, and the rolling-resistance model takes the resisting torque as constant, when in reality it depends on the contact’s speed and on the disk’s weight distribution. Real disks lose energy by several mechanisms at once, and the measured exponent reflects their combination.

Both models have been set to finish at the same time, a hundred seconds, so that their shapes can be compared; in reality each mechanism has its own constant, and the running times would differ. The disk is assumed thin and uniform, the table flat and rigid. The commercial toy’s concave mirror base helps keep the disk centred, which the figures ignore.

Still open: how a disk really stops #

Twenty years after the controversy began, the final milliseconds of a spinning disk’s motion are still not fully described. Measurements with high-speed cameras and acoustic sensors show the disk losing contact intermittently in the last moments, and simulations suggest the contact may switch between rolling and slipping, but there is no agreed model of exactly how the motion ends, or of how the ending depends on the materials. The question has practical relatives — how bearings, wheels and rotating machinery lose energy at their contacts at high speed — and a mathematical one: how real systems escape the singularities their simplest models predict.

What the spinning coin shows is a pattern worth recognising. When a loss mechanism takes energy faster the less energy there is, a system does not fade away exponentially; it reaches its end at a definite time, and quantities that depend inversely on what is left run to infinity on the way. A disk on a table loses energy throughout its spin. The whirr gets faster to the very end because the thing that speeds up is not the disk but the point where it touches.