How a Pendulum Keeps Time

Pull a weight on a rod or string to one side and let it go. It swings back and forth, and every swing takes the same time as the one before. That steady beat ran the most accurate clocks in the world for close to three centuries, from the first pendulum clock, built by Christiaan Huygens in 1656 , until quartz clocks overtook them in the 1930s.

The steadiness is stranger than it looks. A pendulum pulled back a long way has much farther to travel than one barely nudged, so you might expect it to take longer.

Here are two identical pendulums, each a little under a meter long. The left one always starts from 5°; the right one starts from . Try a matching 5° or a wide 40° swing, and watch the two bobs cross the dashed middle line:

The wider swing has much farther to travel, yet both bobs reach the middle at almost the same moment. Each takes about 2 seconds to swing out and back, a time called the period, and the two readouts differ by under a percent until the right swing passes about 20°. Leave a 40° swing running for half a minute and it does fall behind; that small mismatch comes back later, because it cost clockmakers dearly.

A beat that barely depends on how hard the pendulum was started is exactly what a clock needs. Nobody can start a pendulum the same way twice, and friction shrinks every swing a little. If the timing followed the size of the swing, the clock would wander every time either changed. To see why it doesn't, we have to look at what drives the swing.

The pull toward the middle

Two forces act on the bob: its weight, pulling straight down, and the rod, pulling along its own length. The rod can't speed the bob up or slow it down along its path; it only keeps the bob on a circle. So split the weight in two. One part points along the rod, and the rod cancels it. The other points along the path, and it's the pull that drives the bob back toward the middle.

Drag the bob to change the angle and watch how the pull changes in the plot:

At the bottom there's no pull at all, and it grows as the bob moves out. For small angles it grows in a straight line: twice the angle, twice the pull. The orange curve sits right on the dashed line there, within half a percent at 10°. Farther out it bends below the line, because the pull is the weight times the sine of the angle, and the sine levels off toward 1 at 90°.

That straight-line stretch is what keeps the beat steady. Start the bob twice as far out and at every matching point on its path it feels twice the pull, so it picks up twice the speed. Twice the distance at twice the speed takes the same time. Any force that grows in step with the distance from the middle gives a swing whose period doesn't depend on its sizePhysicists call this simple harmonic motion. A weight bouncing on a spring, a guitar string, and the air in an organ pipe all move this way for small motions..

Trading height for speed

The pull explains the timing; energy explains the motion as a whole. A bob lifted to one side holds energy in its height, the same way a ball held above the floor does. As it swings down, that height turns into speed, and on the way up the speed turns back into height. At the ends of the swing the bob stops for an instant and all its energy is in its height. At the bottom it's all in its speed.

Set the swing to and follow the two kinds of energy through one period:

The two bars trade places twice every period, once on each side, and the height bar is full exactly when the speed bar is empty. The dashed total never moves: energy isn't made or lost in the swing, only moved from one form to the other. (Real pendulums do lose a little to the air and the pivot, which we'll come to.)

This bookkeeping makes one prediction we can check directly. Since all of the height turns into speed at the bottom, the bob should arrive there exactly as fast as anything that dropped through the same height, whatever path it took. Pull the bob back to and let it race a ball dropped from the same height:

The ball arrives first, because it falls straight down while the bob starts out moving almost sideways. But when each reaches the bottom it is moving at the same speed, which energy fixes at v = \sqrt{2\red{g}h}: about 3.1 m/s from half a meter up, whatever the path. A pendulum is a way of falling slowly and then climbing back up, over and over, without ever losing track of how high it started.

What sets the beat

If the size of the swing doesn't set the period, what does? A pendulum is only a rod, a bob, and the gravity pulling on it. Change the length of the rod, the mass of the bob, and the world it hangs on, and watch the period on the curve. Start with the seconds pendulum of a grandfather clock, then take it to the Moon:

The mass makes no difference. A heavier bob is pulled down harder, but it's also harder to get moving, by exactly the same factorThat the two factors match exactly is not obvious. Newton tested it with pendulums of gold, silver, lead, glass, sand, salt, wood, water, and wheat, and found no difference to about one part in a thousand .. The length does matter, though not in proportion: a pendulum four times longer takes only twice as long, and the green dot rides a square-root curve. Weaker gravity slows everything down, which is why the same pendulum takes 4.9 s on the Moon. For small swings the period is

where \green{L} is the length from the pivot to the center of the bob and \red{g} the strength of gravity, 9.81 m/s² at Earth's surface. A pendulum 0.994 m long has a period of 2.00 s, so each swing in one direction takes one second. Clockmakers call it a seconds pendulum, and it's the one swinging inside a classic grandfather clock.

Every swing at once

So far we've followed one swing at a time. There's a picture that shows every possible swing together. At any instant a pendulum is described by two numbers: its angle, and how fast that angle is changing, its angular speed. Plot one against the other and the pendulum's whole history becomes a single curve, called its phase portrait.

Click or drag anywhere in the plane on the right to start the pendulum with that angle and that speed. Try a small swing, a big one, and one that goes over the top:

A small swing traces a loop around the middle, nearly an ellipse: the angle and the speed take turns being largest, as the energy bars did. Bigger swings make bigger loops that grow pointed at the ends. Start the pendulum fast enough and it no longer swings at all; it goes over the top and keeps turning, and its curve becomes a wave that never closes. The dashed curve between the two families belongs to the one motion that creeps up to the very top and balances there forever, never quite arriving.

Turn on friction and every curve becomes a spiral that winds into the point at the center, the pendulum hanging still. Even a pendulum that starts out turning over the top gets captured: it loses a little energy each turn until it can no longer clear the top, and from then on it swings.

Big swings run slow

The formula assumes the straight-line part of the orange curve. Past it, the pull grows more slowly than the distance, so a wide swing takes longer to come back. This seconds pendulum hangs from a stiff rod, so it can swing past horizontal without going slack. Widen its swing to and follow its period on the curve:

Up to about 10° the curve is almost flat: the period is only 0.2% longer than for a tiny swing. At 90° it's 18% longer, and near 180° it grows without limit, because a bob balanced almost upside down barely moves at allFor a swing of amplitude θ₀ the period is close to T_0\,(1 + \theta_0^2/16) up to about 40°. The exact answer needs an elliptic integral, which this page computes with the arithmetic–geometric mean..

The flat start is why pendulums work as clocks, and the slow rise is why clockmakers worried anyway. A clock doesn't care about the period itself, which it can be adjusted for; it cares about changes in the period. If the swing of a clock grows or shrinks, from a stronger mainspring, thicker oil, or a draft, the period changes with it. This is called circular error. Set a clock's usual swing and how much it grows:

A clock swinging 4° each way that grows to 5° loses about 15 seconds a day, which is a lot for a clock that hopes to be right to a second a week. The curve gets steeper as the swing grows, so the same 1° change costs a turret clock swinging 8° about 30 seconds a day, and a precision regulator swinging 1.5° almost nothing. That's the first lesson clockmakers learned: keep the swing small, and keep it the same.

Huygens's curve

Huygens took a cleverer route. If the circle is the problem, change the path. He proved in 1659 that there is one curve on which a sliding bead takes exactly the same time to reach the bottom from any starting point: the cycloid, the curve traced by a point on the rim of a rolling wheel . On a circle the pull falls behind the straight line at large angles; on a cycloid it rises steeply enough that the pull stays exactly in proportion to the distance along the path, however far out the bead starts.

Here two beads slide without friction on a circle (left) and two on a cycloid (right), both tracks the same small-swing period. The first bead always starts at the top; release the second from of that height:

On the circle the higher bead always arrives a little after the lower one, and the gap grows the higher it starts. On the cycloid the two arrive together every time, and keep swinging in step as long as you watch. Huygens hung his pendulums on a flexible strip between two curved metal cheeks, shaped so the strip wrapped around them and steered the bob along a cycloid instead of a circle.

It was a beautiful piece of mathematics that didn't work very well in practice. The strip bent and wore, the cheeks added friction, and getting their shape exactly right was harder than keeping the swing small. Later clockmakers dropped the cheeks and attacked circular error directly: small, steady swings, and a gentle way of keeping them going.

Losing energy

A real pendulum doesn't swing forever. Each swing it pushes air aside, flexes its suspension a little, and rubs at the pivot, and each of those turns a sliver of its energy into heat. Left alone, the swing shrinks and dies. How quickly is measured by a single number, the quality factor Q: the bigger it is, the longer the pendulum keeps ringing.

Set Q to and watch the swing fade:

The envelope, the dashed curve, falls by the same fraction every swing, which makes it an exponential decay: a pendulum with Q = 12 loses half its swing every 2.6 swings, one with Q = 50 every 11. The rule of thumb is that the swing halves every 0.22\,Q swings.

For a clock, a high Q matters for more than economy. The less energy a pendulum loses per swing, the smaller the push needed to replace it, and the less any irregularity in that push can disturb the beat. A good regulator clock's pendulum has a Q in the thousands or tens of thousands , so it would keep swinging for hours on its own. That's why precision pendulums have heavy, streamlined bobs, thin spring suspensions instead of knife edges, and in the best clocks a case pumped down to a partial vacuum.

Pushing in time

To keep the pendulum going, something has to put back the energy friction takes. The obvious way is to push it, and anyone who has pushed a child on a swing knows the push has to come at the right moment. Push in time with the swing and it grows; push at the wrong rhythm and the pushes fight the swing as often as they help.

Here a seconds pendulum is pushed back and forth by a small force with a rhythm you set, from slower to faster than its natural beat. Try pushing exactly in time, or a little too fast:

The swing peaks when the push matches the pendulum's own beat, a resonance. The higher the Q, the taller and narrower the peak: at Q = 80 a push that alone would budge the bob by 0.15° builds a 12° swing, but only if its rhythm is within a percent or two. Watch the push at resonance: it is strongest exactly as the bob passes through the middle, a quarter of a swing ahead of the swing itself. Pushing at the middle puts energy in most efficiently, because that's where the bob is moving fastest.

A clock can't push with its own independent rhythm; the pendulum has to set the rhythm and the push has to follow. But the lesson about timing carries over, and the astronomer George Airy worked out the rule exactly in 1826 : a push given as the pendulum passes through the middle changes the size of the swing but not its timing. A push anywhere else changes the timing too.

Here a pendulum swinging 3° gets one small push in the direction it's moving, worth 5% of its top speed. Move where the push happens, from well before the middle to the middle to well after, and see how much longer or shorter that swing takes:

A push before the middle hurries the bob toward a point it was going to reach anyway, and the swing comes back early. A push after the middle sends it farther out, and it comes back late. Exactly at the middle the two effects balance, and only the size of the swing changes. A real clock gives a push like this every single swing, so any push off the middle becomes a steady error, and any change in the push becomes a change in the rate. Good clocks are designed so that the push lands as close to the middle as possible, and stays the same from swing to swing.

Counting the swings

The part of a clock that gives these pushes, and counts the swings while it does, is the escapement. A toothed wheel, turned by a falling weight or a spring, is held still by the pendulum and released one tooth at a time. Each release moves the hands one step, and as the tooth slips past it gives the pendulum its small push.

First, the escapement reduced to what it does. Each time the bob passes the middle, the seconds hand moves on and the pendulum gets a push. Set the push to none and watch the swing fade, then raise it again:

With no push the hand stops at once, because nothing turns the wheel, and the swing shrinks. Raise the push and the swing grows back until each push exactly replaces what friction drains in one swing. If the swing has already shrunk too far to release the wheel, the button starts it again.

Now make the push stronger. The swing settles wider, but the time per tick barely moves: doubling the push doubles the swing and lengthens each tick by less than 1%. That's the steady beat from the first figure doing its job, helped by Airy's rule, since this escapement pushes as the bob passes the middle. The pendulum keeps the time. The escapement keeps it going and counts.

The real mechanism that does this, the anchor escapement, appeared around 1670 and is usually credited to Robert Hooke or William Clement. An anchor-shaped arm rocks with the pendulum; its two hooked ends, the pallets, take turns catching the teeth of the escape wheel. Turn the model to see how they meet:

As the pendulum swings one way, one pallet lifts out of the wheel's path and a tooth slides along its slanted face, pushing it, and so the pendulum, a little. Then the tooth drops free and the wheel jumps until the other pallet catches the next tooth. On the return swing the roles reverse. Each swing lets the wheel advance exactly half a tooth, so the wheel's position counts the swings, and gears pass that count on to the hands.

The early anchor escapement pushed the wheel backward a little at each catch, a recoil you can see in the second hand of an old clock. George Graham's deadbeat version of the 1710s shaped the pallets so the wheel stops dead, and gave the push over a short stretch near the middle of the swing, close to Airy's ideal a century before Airy wrote it downThe model here gives the push in the middle of each swing, like a deadbeat escapement, and leaves out the recoil, the drop, and the exact pallet shapes..

Heat and cold

With the swing small and the push in the right place, the length of the rod is the last thing to control, and it won't stay still. Almost everything grows when it warms. A steel rod grows by about 12 parts in a million for each degree Celsius, which sounds like nothing until it's multiplied by the 86,400 seconds in a day. Because the period goes as the square root of the length, the clock's rate changes by half the change in length.

This clock was set to keep perfect time at 20 °C. Warm or cool the room to and try a rod of brass, pine, or Invar:

A steel rod costs about half a second a day for every degree, so a room that warms by 10 °C between winter and summer makes a good clock lose five seconds a day. Brass is worse; straight-grained pine, oddly, is much better, and clockmakers used varnished wooden rods for exactly that reason. Invar, a nickel–iron alloy found in 1896 that barely expands at all, made the problem nearly vanish, and its discoverer, Charles-Édouard Guillaume, won a Nobel Prize for it.

Before Invar, the trick was to fight one expansion with another. John Harrison's gridiron pendulum of the 1720s hangs the bob from a frame of rods: steel rods hang down, and their growth lowers the bob, while brass rods stand up and their growth raises it. Brass grows about 1.65 times as much as steel per degree, so if the brass rods add up to about 1/1.65 of the steel, the two cancel. Warm it by :

The plain rod's bob sinks steadily as it warms. The gridiron's bob, hung from the same amount of steel and then lifted again by the brass, stays where it was. Turn the brass rods off, replacing them with steel, and it sinks too. Harrison claimed his regulator clocks kept time to about a second a month, and the same idea of cancelling one error with another runs through his later sea clocksThe other classic fix was George Graham's mercury pendulum, whose bob is a jar of mercury: as the steel rod lengthens, the mercury expands upward in the jar and lifts the center of the bob back..

Gravity isn't the same everywhere

There's one more thing a pendulum's length depends on, and it isn't in the clock at all. Equation (1) has gravity in it, and gravity varies across the Earth. The planet spins, which flings things outward a little at the equator, and it bulges there for the same reason, putting the equator about 21 km farther from the center than the poles. Both effects weaken gravity toward the equator.

In 1672 the astronomer Jean Richer took a pendulum clock regulated in Paris to Cayenne, in French Guiana, and found it lost about two and a half minutes a day. He had to shorten the pendulum to make it keep time. Newton used it as evidence that the Earth spins and bulges . Drag the point to a latitude, or go to Paris, Cayenne, or the North Pole:

Gravity at sea level is about 9.780 m/s² at the equator and 9.832 m/s² at the poles, a difference of half a percent. A seconds pendulum has to be 991 mm long in Cayenne and 994 mm in Paris. Carried south without adjustment, a Paris clock runs slow by about two minutes a day by the formula; Richer's measured two and a half included other errors of the voyage, but the sign and the size told the story. Within a few decades pendulums were being carried around the world on purpose, not to tell time, but to measure the shape of the Earth.

How good a pendulum can get

Every section of this page is one of the errors clockmakers spent three centuries chasing. Keep the swing small and steady, so circular error stays constant. Give the push near the middle, gently, so it hardly touches the timing. Make the pendulum heavy and streamlined, so its Q is high and the push can be small. Cancel the growth of the rod with heat. Set it for the gravity where it hangs.

The last great pendulum clocks did all of these at once. The Shortt clock of the 1920s kept its master pendulum in a sealed tank at low pressure, swinging freely and touched only by a tiny impulse every 30 seconds, while a second, slave pendulum did all the counting. The best of them kept time to a few thousandths of a second a day , good enough that observatories began to see irregularities in the Earth's own rotation in their records. Within a few years, quartz crystals, which vibrate many thousands of times a second with a Q in the hundreds of thousands, took over, and then atoms took over from quartz.

Every one of those clocks, quartz and atomic alike, still works the way Huygens's did: something that oscillates at a steady rate, a way to keep it going without disturbing it, and a way to count. The pendulum was simply the first oscillator steady enough to be worth counting.

References

1. Christiaan Huygens, Horologium Oscillatorium (Paris, 1673). English translation by Richard J. Blackwell, The Pendulum Clock (Iowa State University Press, 1986).
2. Isaac Newton, Philosophiæ Naturalis Principia Mathematica (London, 1687), Book III: the pendulum experiments on different materials and the discussion of Richer's observations at Cayenne.
3. George Biddell Airy, “On the disturbances of pendulums and balances, and on the theory of escapements”, Transactions of the Cambridge Philosophical Society 3 (1830), 105–128.
4. Robert J. Matthys, Accurate Clock Pendulums (Oxford University Press, 2004).
5. Gregory L. Baker and James A. Blackburn, The Pendulum: A Case Study in Physics (Oxford University Press, 2005).