A spinning coin's last rattle

Why does a spinning coin rattle faster and faster just before it abruptly falls silent?

Spin a coin on a table. It wobbles, settles, and then in its final second produces a rising whirr that climbs in pitch until it stops — not fading out, but cutting off.

Everything else that runs down does the opposite. A bell fades, a flywheel slows, a plucked string decays. Here something is speeding up while losing energy, and then ending abruptly rather than trailing away. Both halves of that are strange. What exactly is getting faster?

That last question is the whole puzzle, so let us not assume the answer. Put a felt-tip dot on the coin's face and film it. You will find the dot is turning slower near the end, and in the last moments barely turns at all. So the rising sound is not the coin spinning faster.

What is moving fast, then? Watch the point where the coin touches the table. Late in the life of the motion the coin is nearly flat, tipped over by only a small angle, and it is rolling on its edge — so the contact point runs around the rim. That circulation is what you hear: each circuit tips the coin down and lifts it again, drumming the table. It is the contact point that accelerates, not the coin.

Now ask what governs that circulation rate. Two facts are enough, and both are ordinary.

The first is energy. Tip a coin of radius a over until its plane makes a small angle α with the table, and its centre sits at a height of roughly *a*α above where it lies when flat. Its energy above the resting state is therefore about *Mga*α — proportional to the tilt, and going to zero as the coin flattens.

The second is the rolling motion itself. For a disk rolling steadily on its rim at a small tilt α, the standard solution has the contact point circulating at a rate that grows as one over the square root of α. Flatter means faster, steeply.

Put those together and the paradox dissolves. Energy is proportional to α; the rattle rate is proportional to α to the power minus a half. So the rattle rate goes as one over the square root of the remaining energy. Every joule the coin loses makes it louder in pitch. The acceleration is not despite the dissipation — it is caused by it.

Does it reach infinity, or merely get large? Note that the energy left is finite and shrinking, while the rate at which energy is being drained is not shrinking — if anything it grows, since the coin is thrashing the air and its contact faster and faster. A finite reservoir emptied at a non-vanishing rate empties in a finite time. So the model predicts the tilt reaching exactly zero at a definite instant, with the rattle rate diverging as it arrives. That is what the abrupt silence is: not a fade, but the model's clock running out.

Moffatt's 2000 treatment, which put the dissipation into viscous drag in the thin layer of air squeezed between disk and table, made this quantitative — the tilt falling as the cube root of the time remaining, and the rattle rate rising as the remaining time to the power minus one sixth. Those exponents are recalled, and they belong to that particular assumption about where the energy goes.

Three refinements, and one of them is a genuine open question.

First, the divergence is a property of the idealised disk, not of your coin. A real coin has a finite thickness and a rounded or bevelled edge, so once the tilt is comparable to the ratio of thickness to radius it is no longer rolling on a knife-edge, and the contact mechanics change. The singularity is regularised — cut off — before it is reached. What survives is the sharp finish, which is why the coin does not trail off.

Second, this is not the same effect as a spinning top's precession. A top precesses because gravity's torque on a body with large spin angular momentum turns that momentum sideways. The coin's late motion is dominated by the rolling constraint and by its rapidly falling energy; its own spin is nearly gone. Related family, different governing fact.

Third, and honestly: where the energy actually goes is not settled. Moffatt attributed it to air viscosity, and his exponents follow from that. A correspondence in reply reported that spinning a disk at much reduced air pressure did not extend its life the way pure air-drag would require, which points instead at dissipation at the contact — rolling friction, or slipping in a small contact patch. Later analyses have argued for contact mechanisms too. The structure of the argument survives whichever wins, because the finite-time ending follows from the energy being proportional to α and the rate to α to the minus a half; only the exponents depend on the dissipation law.

What would refute this: film a spun coin with a marked face and measure both rates. The account here requires the coin's own rotation to slow toward zero while the contact circulation diverges. If instead the marked dot were seen to spin faster and faster in step with the rising sound, the rolling-contact explanation would be simply wrong, and we would be looking at something spinning up rather than falling flat.

• Why a real coin's finite thickness cuts off the singularity, and what sets the last audible frequency.
• How the same rolling constraint explains a wobbling bottle cap, a rocking bowl, and the toy known as Euler's disk.

Every term the collection defines is gathered in the glossary.