The pendulum that must not grow when it warms

The pendulum and its small lie found that a pendulum’s period depends on its length and on the strength of gravity and, for small swings, on nothing else: $T = 2\pi\sqrt{L/g}$. That is what makes a pendulum a clock. The period that depends on the swing and the kick that does not change the beat attacked the two corrections a clockmaker can control through the mechanism — the dependence on the size of the swing, and the disturbance of the escapement that keeps it going. A third correction is not in the mechanism at all. It is in the length.

Every material expands when warmed, by a fraction per kelvin called its coefficient of expansion: about twelve parts per million for steel, nineteen for brass. Since the period goes as the square root of the length, a pendulum that lengthens by a fraction $\alpha$ slows by $\alpha/2$, and a clock counts its seconds by its pendulum’s swings. A part per million is a tenth of a second a day. A pendulum rod is therefore a thermometer, and a clock with a plain rod reads its room’s temperature as a drift in the time.

Half a second a day for a degree #

The first figure draws the problem. A seconds pendulum, whose half-swing takes one second, is 0.994 metres long to its centre of oscillation. Made with a steel rod, it loses half a second a day for every degree its rod warms, and a brass rod is worse. A house without heating, in the climate of London in the eighteenth century, might range from near freezing in January to over twenty degrees in July; an uncompensated pendulum clock in it would run fast all winter and slow all summer, by several seconds a day. For a household clock that was tolerable. For an astronomical regulator, timing the transits of stars to determine longitude or to catalogue positions, it was not: a second of time is fifteen seconds of arc of the sky’s rotation.

Two materials on the figure point at the two ways out. Seasoned pine, cut so that the rod runs along the grain, expands less than half as much as steel, and wooden pendulum rods were used for that reason in the eighteenth century and later. But wood absorbs moisture and swells with the humidity far more than it expands with the temperature, and a wooden rod exchanges one thermometer for a hygrometer. Invar, at the bottom, is the twentieth century’s answer, and it comes at the end. The eighteenth century’s answer was not to look for a material that did not expand but to use two that did.

Two expansions that cancel #

The trick is that the period depends on the distance from the suspension to the bob, not on the length of any single piece of metal. If the bob hangs from a steel rod, it moves down as the steel expands. If something then lifts it — a second rod of a different metal, fixed at its lower end and pushing up — the bob moves up as that metal expands. Make the upward expansion equal the downward one and the bob stays where it is.

John Harrison, a carpenter who became the greatest clockmaker of his century, built the arrangement around 1726, and the second figure draws it to scale for steel and brass. The condition is that the total length of steel times its coefficient equal the total length of brass times its coefficient. Since brass expands $1.65$ times as much as steel, it needs only 0.605 of steel’s length to cancel it. The difference between the two lengths is the pendulum’s working length, so the steel must be 2.53 times as long as the pendulum and the brass 1.53 times — two and a half metres of steel, folded back and forth in a frame of parallel rods so that it fits a metre-long case. That folding is what gives the gridiron its name and its look, a grid of nine rods in the classic form, every one doubled side by side so that the bob hangs straight.

The design works by the same principle as two glasses that cancel a derivative, where a lens of crown glass and one of flint, whose colours spread in opposite directions, combine into an achromat whose focus does not move with wavelength. Here the derivative is of length with temperature, and the two metals cancel it to first order. That essay noted the parallel in passing; the second-order term is where the gridiron’s limits begin, and so is the arithmetic of how quickly each rod warms.

A jar of mercury #

George Graham, Harrison’s contemporary and the leading instrument maker in London, had found a different cure a few years earlier, in 1721. His bob was a glass jar of mercury. As the rod lengthened and lowered the jar, the mercury expanded and its level rose, carrying the centre of the bob upward. Mercury’s expansion is large — 182 parts per million per kelvin by volume, fifteen times steel’s linear rate — so a modest column suffices. The third figure computes the height. A jar widens as it warms, which lowers the level by twice the glass’s linear expansion, and the centre of the mercury rises by half of what remains; balancing that against the rod gives about fourteen centimetres of mercury for a steel rod of a seconds pendulum. Real columns were a little taller, because the jar’s stirrup and the rest of the bob also expand downward with the rod.

The mercury pendulum is simpler than the gridiron, has one adjustment instead of many — the amount of mercury — and was the standard compensation in observatory regulators through the nineteenth century. Its weakness is the one the next figure shows.

Heat that does not arrive all at once #

Compensation is exact only when every part has the same temperature, and parts warm at different rates. A slender steel rod in moving air reaches the room’s temperature in a quarter of an hour or so; a jar of several kilograms of mercury, with less surface for its bulk, takes hours. The fourth figure models a sudden warming of the room by a degree, with time constants of that kind assumed for each part. The uncompensated rod lengthens within a few minutes and the clock loses steadily from then on. The gridiron’s steel and brass are rods of similar thickness and warm together, so it hardly moves. The mercury pendulum’s rod warms first and lowers the bob before the mercury has risen to meet it; for an hour or two the clock runs slow, until the mercury catches up and the rate returns to its old value. The seconds lost in the interval are never recovered.

That is a design trade the eighteenth-century makers understood. A gridiron, with its many rods of similar section, follows temperature changes faithfully but is heavy, has many joints that can stick, and needs careful matching of its metals; a mercury pendulum is easy to adjust but lags. In an observatory vault whose temperature changes by a fraction of a degree a day, the lag hardly matters. In a room heated by a fire in the morning, it does.

When the top of the case warms first #

The other enemy is that air stratifies. Warm air rises, and the top of a clock case is commonly warmer than its foot by a degree or more. If the room warms unevenly — the upper air warming more — the parts of a compensated pendulum see different changes according to where they sit. The last figure computes the consequence for a gradient that grows by one kelvin per metre. A mercury pendulum is the vulnerable design: its rod runs from top to bottom and warms by the average, but all its compensation is concentrated at the bottom, which warms least, and the clock loses a quarter of a second a day. A gridiron’s steel and brass both span the whole height and share the average, so it stays compensated, which was one of the arguments for gridirons in the finest clocks. Clockmakers learned to keep regulators in cellars, to insulate their cases and in some installations to stir the air in them with a small fan, so that the pendulum saw one temperature from end to end.

The same cure in a balance wheel #

The problem was not confined to pendulums. A watch or a marine chronometer keeps time by a balance wheel oscillating on a spiral spring, and its period depends on the wheel’s moment of inertia and the spring’s stiffness, both of which change with temperature — the wheel grows, and, worse, the steel spring loses stiffness as it warms, by more than its expansion alone would suggest. An uncompensated balance loses several seconds a day per degree, ten times worse than a pendulum. Harrison’s sea clocks, built to find longitude for the Board of Longitude from the 1730s onward, carried the gridiron’s principle into a form that fitted a moving ship: a strip of brass riveted to a strip of steel, which curls as it warms because brass expands more, arranged so that its curling shortened the active length of the balance spring and stiffened it as the steel weakened. Later chronometers put bimetallic strips into the rim of the balance itself, so that the rim’s weights moved inward as the wheel warmed and the moment of inertia fell to match the weaker spring. The bimetallic strip, now in every thermostat, began as a way of making a clock ignore the weather.

The alloy that hardly expands #

The eventual answer was a material, discovered by accident. In 1896 Charles-Édouard Guillaume, working at the International Bureau of Weights and Measures near Paris on materials for standard metre bars, found that an alloy of iron with 36 per cent nickel expanded by little more than a part per million per kelvin, a tenth as much as either of its constituents. He called it invar, for invariable, and received the Nobel Prize in Physics in 1920 for the discovery, one of very few awarded for metrology. A pendulum rod of invar loses a twentieth of a second a day per degree, and pendulums of the early twentieth century, from Sigmund Riefler’s regulators in Munich to the Shortt clocks of the 1920s, used invar rods, often with a small additional compensating collar under the bob to remove the last trace.

The reason invar does not expand is magnetic. In most solids, warming raises the vibration of atoms and the average spacing between them grows. Invar is a ferromagnet, and its magnetism also affects its volume: the magnetic ordering of its iron and nickel moments pushes the atoms apart, so the magnetised alloy is larger than it would be otherwise. Warming weakens the magnetic order, and the volume it was holding open shrinks, almost exactly cancelling the ordinary thermal expansion over a range around room temperature. The cancellation is the same idea as the gridiron’s, achieved inside the crystal by two mechanisms of opposite sign. What holds a magnet together found the exchange interaction ordering a ferromagnet’s moments; in invar the same interaction, through its dependence on the spacing of the atoms, holds the length of a pendulum constant.

Where compensation stops #

Compensation cancels the first-order effect of temperature on length and leaves the rest. The coefficients of expansion themselves change with temperature, so a pendulum compensated at one temperature is slightly over- or under-compensated at another, a second-order error of a few thousandths of a second a day per degree squared. Temperature also changes the air’s density, and so its buoyancy on the bob and the drag on it, effects that a vacuum case removes. Pressure changes alter the rate directly, by about a tenth of a second a day for ten millibars, which is why the Shortt clock’s pendulum swung in a sealed tank pumped to low pressure. And the support moves: a pendulum hung from a wall swings with the wall, and the stiffness of the suspension spring changes with temperature too.

A pendulum used not to keep time but to measure gravity has the same problem in reverse. The length nobody has to measure followed Kater’s reversible pendulum, which finds $g$ from a period and a distance between two knife-edges; the distance is a length of metal at the temperature of the room, and every survey of gravity made with such pendulums in the nineteenth century had to record the temperature and correct for the bar’s expansion, by the same part in ten thousand for every ten degrees that drives a clock.

Each correction was worked out and applied, and the residual errors of the best pendulum clocks of the 1920s were a few thousandths of a second a day. Below that, the Earth’s own tides — which change local gravity by a part in ten million twice a day — and the slow creep of the metals set the limit, and the quartz clocks of the 1930s, keeping time by an oscillator whose frequency depends on a crystal’s elasticity rather than on gravity, took over.

Every clock has a temperature coefficient of some kind, and the habit of compensating for it outlived the pendulum. A quartz crystal’s frequency drifts with temperature unless it is cut at a particular angle to its crystal axes where the drift passes through zero, and the cut is chosen exactly as the gridiron’s lengths were, to cancel a derivative. Even the atoms of an atomic clock are affected: the clock that runs slow because it is warm found that a gas of clock atoms at temperature $T$ ticks slow by $3kT/2mc^2$ from time dilation alone, a part in $10^{13}$ for caesium at room temperature, which is why atomic fountains toss laser-cooled atoms. A steel pendulum’s rate changes by six parts in a million per kelvin and a room-temperature caesium atom’s by about one part in $10^{15}$; the scale of the problem has shrunk by nine orders of magnitude, and it has never gone away. Only at absolute zero would it, since the two laws absolute zero will not allow require every coefficient of expansion to vanish there.

What a rate curve leaves out #

The figures model the rod and the compensation as one-dimensional, with uniform temperatures in each part and with time constants put in by assumption. A real pendulum has a suspension spring, a bob of finite size whose own expansion moves its centre of oscillation, cross-pieces and screws whose expansions add small terms, and parts whose thermal contact with each other and with the air varies with the air’s motion. None of that is drawn. The figures also say nothing about humidity, which affects wooden rods far more than temperature does, or about the slow ageing of metals, whose dimensions creep for years after manufacture. A real clock’s error budget is a long list, and temperature is merely the first and largest entry.

Still open: how far a pendulum can be pushed #

The question of the ultimate pendulum clock is no longer one the world’s timekeeping depends on, but it has not been closed. Pendulum gravimeters and pendulum clocks built with modern materials, in vacuum and in temperature-stabilised enclosures, could in principle reach stabilities that would make them sensitive instruments of gravity, and experiments with long-period pendulums and torsion pendulums continue to test gravity at short ranges. How far below a thousandth of a second a day a freely swinging pendulum can be made to hold its rate, once temperature, pressure and the support are controlled, has not been measured with modern means.

The habit worth carrying away is to cancel a sensitivity rather than wait for a material without it. A pendulum loses $\alpha/2$ of its rate per kelvin — half a second a day for steel — and two metals arranged to expand in opposite directions, 2.52 metres of steel against 1.52 of brass for a seconds pendulum, or a rod against fourteen centimetres of mercury, cancel it exactly for slow and uniform changes, leaving only the errors of heat that arrives unevenly or late. Wherever a quantity drifts with one variable, two parts that drift oppositely can hold it still, and the residual is set by how well the two parts share the same conditions.