Pendulum Lab

A weight on a rod is about the simplest machine there is, and for three centuries it was the best clock anyone could build. That success hides a small lie. The pendulum keeps time only as long as nobody asks too much of it.

The story usually starts with Galileo. His biographer and student, Vincenzo Viviani, claimed that his interest was sparked around 1582 by a chandelier swinging in the cathedral of Pisa. Wikipedia’s history of the pendulum dates his first recorded notes on the subject to about 1588 and his research to around 1602. What he found was isochronism: a pendulum seemed to take the same time over a wide swing as over a narrow one.

It was a useful discovery at once. A physician, Santorio Santorio, was soon measuring patients’ pulses by the length of a pendulum. In 1641 Galileo dictated to his son Vincenzo a design for a mechanism to keep a pendulum swinging. It was never finished. Christiaan Huygens did, and NIST gives the date as 1656. His first pendulum clock was wrong by less than a minute a day, and his later ones by less than ten seconds.

The small-angle trick

Why should a swing take the same time whatever its size? The force that pulls the bob back toward the bottom depends on the sine of the angle. For a small angle, measured in radians, the sine and the angle are nearly the same number. Worked out directly, the two stay within one percent of each other up to about 14 degrees.

Replace the sine with the angle and the equation of motion becomes the same one that governs a weight on a spring. Its solution is a pure sine wave, and its period is 2 pi times the square root of length over gravity. The mass is not in that formula. Neither is the size of the swing. A pendulum a metre long, under the standard gravity of 9.80665 metres per second squared, comes out at 2.0064 s.

Gravity is the only thing outside the pendulum that the formula knows about, which is why the same rod ticks differently elsewhere. NASA’s fact sheets give 1.62 metres per second squared for the Moon and 3.73 for Mars. The metre pendulum would take 4.94 seconds per swing on the Moon and 3.25 on Mars. The standard value itself is a convention, declared by the General Conference on Weights and Measures in 1901, not a measurement of any one place.

How wrong the formula gets

The formula is a limit, not a law. Galileo’s isochronism is not exactly true. Marin Mersenne and Rene Descartes found around 1636 that the period grows somewhat with the size of the swing, and Huygens analyzed the problem in his book of 1673. His cure was a pair of curved guides at the pivot that made the bob follow a cycloid instead of a circle, a path on which the timing is exact.

Clockmakers took a plainer route. They kept the swing small and as constant as they could. That is the real reason a long-case clock has a pendulum that barely moves: the less it swings, the less the swing matters.

The true period of a circular pendulum has been known for a long time, and it is not a tidy expression. Energy conservation gives the bob’s speed at every angle, and adding up the time spent along the arc leads to a complete elliptic integral of the first kind. Lima and Arun, in a paper on teaching this, write the result as four times the square root of length over gravity, times that integral evaluated at the sine of half the release angle. The small-angle formula always underestimates it.

The integral looks forbidding and computes easily. The NIST Digital Library of Mathematical Functions lists an identity due to Gauss: it equals pi divided by twice the arithmetic-geometric mean of 1 and a second number, which here is the cosine of half the release angle. That mean is found by replacing two numbers with their average and with the square root of their product, over and over. A handful of rounds gives every digit a computer can hold.

Take the metre pendulum again and let it go from 90 degrees, with the rod horizontal. The integral is then a known constant, 1.8540746773, listed in the On-Line Encyclopedia of Integer Sequences. Twice that, divided by pi, is 1.18034. The real period is 2.3682 seconds, not 2.0064, so the formula is short by 18.03 percent. Release the same pendulum from 170 degrees and the ratio is 2.4394: nearly five seconds a swing, most of it spent crawling over the top.

Small swings are not exempt, only less guilty. At 10 degrees the true period is 2.0102 seconds, an error of about 0.19 percent. Lima and Arun note that the amplitude must stay under 7 degrees for the formula to be good to 0.1 percent, the error of a hand-held stopwatch. A clock that wanders by 0.1 percent loses almost a minute and a half a day. Every one of these figures can be reproduced on the bench on this page by setting the length to one metre and changing only the release angle.

Friction, pushing and resonance

Real pendulums also run down. With a drag that grows in proportion to speed, the standard result for a lightly damped oscillator is a swing whose size shrinks along an exponential curve while it goes on oscillating. Set a damping rate of 0.2 per second on the metre pendulum, release it from 10 degrees, and after ten swings, about twenty seconds, it reaches only 1.34 degrees. The period barely notices.

That decay is what a clock’s works exist to undo. The escapement gives the pendulum a small push each swing, and the art lies in pushing without disturbing the timing. NIST’s history is a list of better ways to interfere less: the anchor escapement around 1671, George Graham’s temperature compensation in 1721, and finally the Shortt clock of 1921, in which a second pendulum did the pushing so the first could swing almost freely.

A push that arrives on a schedule of its own is a different matter. The familiar example is a playground swing: pushes timed to the swing’s own rhythm send it higher and higher, and the response is greatest when the driving frequency matches the natural one. Less damping makes that peak taller and narrower. For small swings the height of the peak is inversely proportional to the damping rate. On the bench, with the same damping of 0.2 per second, a drive of 0.02 of the pendulum’s own weight torque produces a swing under 2 degrees at 0.3 hertz and nearly 18 degrees at 0.498 hertz, the natural frequency of the metre pendulum.

Push harder and the picture stops being a tidy peak. A pendulum that is both damped and strongly driven can settle into a single repeating loop, then a double loop, then a quadruple one, as the drive is turned up. Erik Neumann’s myPhysicsLab traces that period doubling in its own simulation and finds chaos beyond it. The sine in the equation, the term the small-angle trick threw away, is what makes this possible.

One more joint

Hang a second pendulum from the bob of the first and no friction or driving is needed. David Tong’s Cambridge lecture notes write down the energy of this double pendulum in a few lines, and the equations of motion follow from it. For small angles it is still tame: two arms of equal length and mass have exactly two ways of swinging in step, a slow one with the arms together and a fast one with them opposed. For metre arms those take 2.62 and 1.09 seconds.

Tong then adds a single sentence that changes the subject: above a certain energy, the motion is chaotic. Release both arms from 120 degrees and the lower bob loops, stalls and whips over the top with no pattern that ever repeats.

The bench shows what that means in numbers. It runs two copies of the same double pendulum, the second started one millionth of a radian away from the first. For five seconds the gap stays below a ten-thousandth of a radian. By about eleven seconds the two differ by a full radian and are doing visibly different things. Start them a billionth of a radian apart instead and the split comes at about fifteen seconds. A thousandfold improvement in the starting measurement bought less than five more seconds of forecast.

What chaotic does and does not mean

Nothing here is random. The equations contain no dice, and the same starting numbers give the same motion every time. Edward Lorenz, whose famous 1963 paper on a small model of convection sharpened the study of such behavior, described deterministic equations whose slightly differing initial states evolve into considerably different ones. Wikipedia’s article on chaos theory gathers the marks of chaos: deterministic laws, nonlinearity, and a high sensitivity to initial conditions.

The same article guards against the lazy reading. Chaotic systems are predictable for a while and only then appear random: weather, it notes, can generally be forecast about a week ahead. What chaos limits is how far ahead a forecast can reach, because small uncertainties grow roughly exponentially instead of steadily.

There is order in it too. Plot speed against angle, the phase plot, and a small-angle pendulum draws a closed oval, a damped one spirals inward, and a double pendulum fills a region without retracing itself. Map which releases of a double pendulum flip over, and how soon, and the picture has a clean lens in the middle where there is too little energy to flip at all, surrounded by fine, tangled structure.

The bench has limits that should be said plainly. Its pendulums are point masses on rigid, weightless rods, with no air drag that grows with the square of speed, no flexing and no pivot friction. Its integrator takes fixed steps of two milliseconds or less and reports its own energy drift, which is small but not zero. Past the first several seconds, a chaotic run is a faithful example of what a double pendulum can do. It is not a prediction of what any particular one will do.