The pendulum, and the small lie that makes it simple

Galileo is supposed to have noticed it watching a lamp swing in the cathedral at Pisa, timing it against his own pulse: the lamp took the same time to swing whether it was moving a lot or a little. Out of that observation came the pendulum clock, and out of the pendulum clock came three centuries of navigation.

The observation is false. It is very nearly true, which is a different thing, and the gap between nearly and exactly is where the physics lives.

Why it nearly works #

The rod is rigid, so the bob can only move along an arc. Of the weight $mg$ pulling straight down, the part along the rod is taken up by tension and accomplishes nothing. What remains is the component perpendicular to the rod, $mg\sin\theta$, and that is the entire restoring force.

That decomposition is not specific to pendulums. It is the same move that makes the forces on an inclined plane tractable — resolve the weight along and across whatever the constraint permits, and discard the half the constraint absorbs. A pendulum is a block on a slope whose angle changes as it moves.

Writing the arc displacement as $s = L\theta$, Newton’s second law gives

$$mL\ddot{\theta} = -mg\sin\theta,$$

which is where the trouble starts. That equation has no solution in elementary functions. The sine of the angle, not the angle, is what appears — and a differential equation with a sine of the unknown inside it is a genuinely hard object.

So the standard move is to assume the angle is small and replace $\sin\theta$ with $\theta$:

$$\ddot{\theta} = -\frac{g}{L}\theta.$$

This is the equation of simple harmonic motion, it has a clean solution, and its period

$$T = 2\pi\sqrt{L/g}$$

contains no amplitude at all. That is Galileo’s observation, and it is a consequence of the substitution rather than of the pendulum.

Two absences in that formula are as informative as what is present. The mass is missing, so a heavy bob and a light one on the same string keep the same time — the same cancellation that makes all objects fall together. And the amplitude is missing, which is the claim under examination.

How big is the lie #

The substitution is worth measuring rather than waving at, because “small angle” is a claim about a tolerance, not about physics.

The lie is an eighth of a per cent at five degrees and over twenty at sixty. That is worth having as a pair of numbers rather than as a formula, because it explains both halves of the situation: why the approximation was good enough to run clocks for three centuries, and why it becomes useless the moment the amplitude is allowed to grow. The error is third order, so it is negligible and then suddenly not.

At five degrees the error is an eighth of a percent. At ten degrees it is half a percent. By thirty it is nearly five percent, and by sixty it is over twenty — at which point calling the motion simple harmonic is no longer an approximation but a different claim.

There is a second-order correction that captures most of the discrepancy in the period:

$$T \approx 2\pi\sqrt{\frac{L}{g}}\left(1 + \frac{\theta_0^2}{16}\right),$$

and the shape of it is the useful part. The correction grows as the square of the amplitude, which is exactly why the approximation feels exact for a while and then does not. Halving the swing quarters the error.

A pendulum clock swinging through a few degrees gains or loses seconds a day from this term — a discrepancy of the same order as the relativistic correction a satellite clock needs, arrived at by a far more mundane route. That is the reason clock escapements are designed to keep the amplitude constant rather than merely to keep the pendulum going: the period depends on amplitude, so an amplitude that drifts is a clock that drifts.

The quadratic shape of the error is a general feature rather than a coincidence about sines. Any smooth function replaced by its tangent line agrees with it to first order and disagrees at second, so the error of a linearisation always starts as a square. The same statement governs the flatness of the projectile range near forty-five degrees and the paraxial approximation in lens design, which uses this identical substitution on this identical function and fails in this identical way.

What the approximation costs, in seconds #

The correction term is small enough to be dismissed and consequential enough that three centuries of clockmaking were organised around it, so it is worth turning into a number.

A seconds pendulum swinging through five degrees runs slow by $\theta_0^2/16$, which in radians is about one part in 2,100 — 41 seconds a day. That is not a subtle effect; it is the difference between a clock and an ornament. It does not, however, matter in the slightest, because it is a constant offset: the pendulum is simply cut to a length that absorbs it, and the clock is correct.

What matters is the drift. Let the amplitude decay from five degrees to four, as it will if the escapement is delivering slightly less energy than the air is taking, and the correction falls to 26 seconds a day. The clock gains about fifteen seconds a day for no reason a reader of the dial could ever discover. The pendulum has not changed length, the gravity has not changed, and the timekeeping has moved by a quarter of a minute because the swing got smaller.

That is the real cost of the amplitude dependence, and it explains a piece of clock design that looks otherwise perverse. An escapement’s job is not to keep the pendulum going — a slow leak of energy would do that. Its job is to keep the amplitude constant, which is a much harder specification, and it is why the good escapements deliver their impulse in a brief kick at the bottom of the swing, where the bob is moving fastest and the impulse perturbs the timing least.

Huygens saw the whole problem in 1673 and solved it exactly. A bob constrained to a cycloid rather than a circle has a period genuinely independent of amplitude — the cycloid is the tautochrone, and the proof is one of the loveliest results of the century. He built clocks with curved metal cheeks at the suspension to bend the string into a cycloid as it swung.

They kept worse time than the ones without. The cheeks introduced friction and made the suspension’s flexure amplitude-dependent in its own right, and the errors they added were larger than the error they removed. The exact solution was available, was implemented, and lost — which is the most useful thing this anchor has to say about approximations. The question is never whether a model is exact. It is whether its error is smaller than the errors introduced by the machinery required to avoid it.

The other cost is arithmetic. Refusing the approximation means the exact period, which is a complete elliptic integral of the first kind: a function with no expression in elementary terms, tabulated in the eighteenth century and evaluated numerically ever since. The trade is between an exact answer nobody can write down in closed form and an approximate answer that fits on one line and can be reasoned with. For three hundred years the second was worth more.

Energy, which never needed the approximation #

Nothing so far has mentioned energy, and the swing can be described completely without solving any equation of motion.

Height and speed trade against each other at every moment of the swing, and their sum does not move: whatever the bob loses in one it gains in the other, so a column drawn at any instant of the motion has the same total height as a column drawn at any other. That is a statement about the whole swing rather than about an instant of it, and it is the reason the energy argument survives where the period formula does not.

The bob is highest at the two ends of the swing and fastest at the bottom, and the sum of the two is fixed. This holds at any amplitude — the approximation was never used — which is why energy arguments give the speed at the bottom of a sixty-degree swing exactly, while the period formula gives its timing badly.

That division of labour is typical. Conservation laws answer questions about endpoints without requiring the middle to be solved, and they keep working long after the approximations that make the middle solvable have failed.

What the formula cannot show #

The period formula describes a pendulum that swings. It has nothing at all to say about a pendulum that goes over the top, and a real pendulum on a rigid rod can do that.

The phase portrait shows every possible motion at once, and it separates them into two kinds that the formula treats as one. Near the centre are closed loops: the pendulum swings back and forth, and the loop closes because the motion repeats. Those loops are almost perfect ellipses for small swings, which is simple harmonic motion, drawn.

Further out the loops distort — visibly non-elliptical, flattened at the sides — and that distortion is the amplitude dependence. The pendulum lingers near the top of its swing, where the restoring force is weakest.

Beyond a certain energy the curves stop closing. The pendulum has enough energy to pass straight over the pivot and it never turns around; it just keeps rotating. Those are the open curves running off the top and bottom of the diagram.

Between the two families is a single curve that belongs to neither: the pendulum arrives at the top with exactly zero velocity left. It never quite gets there — it takes infinite time — and it never falls back. That curve is the separatrix, and it is the boundary between swinging and spinning.

The portrait also shows what it cannot show, which is dissipation. Every curve on it is closed or unbounded, because the energy is fixed along each one. A real pendulum loses energy and spirals slowly inward across the curves toward the centre, and that spiral is a motion the diagram has no line for. The level sets map the possible histories of an idealised pendulum; the real one drifts between them.

The point that is stable and the point that is not #

The phase portrait also makes visible something the equation states without emphasis: the pendulum has two equilibria, and they could not be more different.

Hanging straight down, at the centre of the diagram, the pendulum is stable. Nudge it and it comes back — the closed loops around that point are the coming back.

Balanced exactly upside down, at the edges of the diagram, it is also in equilibrium: the force is zero, and left perfectly alone it would stay. But every curve near that point leads away from it. An arbitrarily small disturbance grows, and the sign of the disturbance decides which way it falls.

That is the difference between a minimum and a maximum of the potential energy, drawn in a way that a graph of the potential does not quite convey. Both are flat spots. Only one of them is somewhere a physical system will ever be found — and the asymmetry between them is, in miniature, a version of the reason the world has a direction in time: configurations that are unstable are not places anything is ever discovered sitting.

The quantity that is not in the formula #

The period contains a length and a gravitational field and nothing else. There is no mass in it, and hanging a lead bob and a cork bob of the same size on the same string gives the same swing.

That cancellation is not bookkeeping. The pull on the bob is proportional to its gravitational mass — how strongly gravity couples to it — and the resistance to being accelerated by that pull is its inertial mass, which is a different property with a different definition. The two appear on opposite sides of the equation and divide out only because they happen to be numerically equal.

So a pendulum is a test of that equality, and it was used as one immediately. Newton reports in the Principia swinging pairs of equal-length pendulums with bobs of gold, silver, lead, glass, sand, common salt, wood, water and wheat, and finding no difference he could detect — about one part in a thousand. Bessel repeated it in 1832 with far better technique and reached one part in sixty thousand.

Modern versions use torsion balances and satellites rather than swinging bobs, and the equality now holds to around one part in $10^{15}$. It has never been explained by anything in Newtonian mechanics, where it is a coincidence; it is instead the observation that general relativity takes as its starting point. The absence of $m$ from a formula on this page is the whole of the reason a falling laboratory feels like empty space.

Where the model stops #

The small-angle substitution is the approximation this essay is named for, and it is not the only one holding the figures up. Four more are built into the phrase simple pendulum, and each fails somewhere a real pendulum is asked to work.

The rod is massless and the bob is a point. A real pendulum is a distributed body, and what governs its period is not its length but its moment of inertia about the pivot divided by the first moment of its weight. The formula becomes $T = 2\pi\sqrt{I/mgd}$, and the two agree only when the mass is concentrated far from the pivot. A uniform rod swinging about one end has the period of a point mass at two-thirds of its length, not at its end — a 22 per cent error in period for anyone who measures the rod and trusts the simple formula.

There is no air. Air does three separate things and only one of them is the obvious one. It damps, so the amplitude decays and the period drifts with it by the mechanism above. It buoys, which reduces the effective gravity by the ratio of the densities — about one part in 7,000 for a brass bob, worth 12 seconds a day. And it is dragged along with the bob, adding to the inertia without adding to the weight. Kater’s measurements of $g$ in the 1810s were limited by the third of these, which nobody had thought to include.

The pivot is perfect. A knife-edge deforms elastically under load, so the axis of rotation shifts by a few microns during the swing, and it shifts by an amount that depends on the amplitude. This is the error that stopped precision pendulum work at around one part in $10^7$ and was never solved — it was made irrelevant by quartz.

The support is fixed and the ground is still. A pendulum hung from anything with compliance shares energy with its mount, and two clocks on the same wall will synchronise, which Huygens also discovered and called an odd kind of sympathy. And the ground turns underneath: the plane of the swing is fixed relative to the stars rather than to the room, which is a nuisance for a clock and the entire point of Foucault’s pendulum.

Every one of these is smaller than the small-angle error at ordinary amplitudes, and every one of them became the limiting error once the small-angle error was engineered away. That is the usual shape of a model’s domain of validity — not a wall, but a sequence of walls, each revealed by removing the one in front of it.

The same equation, elsewhere #

The reason the small-angle pendulum is worth this much attention is that almost nothing in the essay was about pendulums.

Draw the same accounting for a mass on a spring and nothing changes but the labels. The potential is stored in the extension rather than in the height, the kinetic term is the same term, and the columns trade in the same proportions — which is the sense in which the two systems are not analogous but identical, one equation wearing two sets of units.

The equation $\ddot{x} = -kx$ — restoring influence proportional to displacement — describes a mass on a spring, a charge oscillating in a circuit, a molecule vibrating in a bond, a ship rolling, and a bridge deck that would rather not. Each of those has its own constants and its own units, and all of them produce a sine wave in time for the same reason.

Each point of a medium carrying a wave is an oscillator obeying the same equation, moving only up and down and coupled to its neighbours. That is where this equation goes next: a single oscillator gives a period, a chain of them gives a wave speed, and the small-angle approximation that makes one tractable is the same one that makes the other linear.

Couple a line of these oscillators together and a wave appears; confine the line at both ends and only certain frequencies survive. Neither of those subjects introduces any physics not already present in a swinging weight. They add coupling and boundaries, and inherit the oscillator whole.

And each of them has its own version of the lie. The spring obeys Hooke’s law until it does not; the circuit is linear until the components saturate; the molecule is harmonic near the bottom of its potential well and anharmonic further up — which is precisely why the overtones of a real piano string come out sharp of the exact multiples. In every case the linear model is the first term of an expansion, exact nowhere and excellent nearby, and in every case the interesting behaviour — resonance shifting with amplitude, oscillations that do not quite repeat, chaos in the driven case — lives in the terms that were dropped.

The ladder from here #

The next rung is the exact period — the elliptic integral, and what that function looks like plotted against amplitude. After it: damping, and the three regimes either side of critical. The driven pendulum and resonance. The double pendulum, where two of these joined together stop being predictable at all. Foucault’s pendulum, which is the same object used to detect that the ground is turning. Kater’s reversible pendulum, which measures $g$ to five figures. And the escapement, which is the engineering answer to everything this essay has said about amplitude.

Galileo’s observation was wrong and it was worth making. Most good observations are wrong somewhere.