Illustrated Physics
physics, drawn
Mechanics

The pendulum, and the small lie that makes it simple

A pendulum's period does not depend on how far it swings. This is one of the most useful false statements in physics, and it is worth knowing exactly how false.
11 min read 7 figures Approximations that lieWhat stays the same

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.

A pendulum displaced 32°A pendulum bob on a rod, with gravity resolved into a component along the rod and the restoring component perpendicular to it.θmgmg sin θ
Fig. 1 A pendulum displaced from vertical. Gravity pulls straight down; only the component perpendicular to the rod does anything, and that component is mgsin⁡θmg\sin\theta.

Why it nearly works

The rod is rigid, so the bob can only move along an arc. Of the weight mgmg 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, mgsin⁡θmg\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=Lθs = L\theta, Newton’s second law gives

mLθ¨=−mgsin⁡θ,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⁡θ\sin\theta with θ\theta:

θ¨=−gLθ.\ddot{\theta} = -\frac{g}{L}\theta.

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

T=2πL/gT = 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.

A pendulum displaced 12°A pendulum bob on a rod, with gravity resolved into a component along the rod and the restoring component perpendicular to it.θmgmg sin θ
Fig. 2 A smaller displacement. The arc and the chord have become nearly the same length, and the restoring force has become nearly proportional to the displacement — which is the whole of the approximation, drawn.

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 cost of assuming sin θ = θPercentage error in replacing sine of an angle by the angle itself, against the angle in degrees. Below about ten degrees it is under half a percent; by sixty it is over fifteen.02040608005101520amplitude (degrees)0.13%0.51%2.1%4.7%11.1%20.9%“small angle” is a claim about tolerance, not about physics
Fig. 3 The percentage by which sin⁡θ\sin\theta falls short of θ\theta, against the angle. The error is quadratic near zero, which is why the approximation is so forgiving at first and stops being forgiving so suddenly.

At five degrees the error is under a tenth of a percent. At ten degrees it is half a percent. By thirty it is nearly five percent, and by sixty it is over fifteen — 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≈2πLg(1+θ0216),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.

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.

Energy trading places through one swingKinetic and potential energy at successive moments of a swing. Each column has the same total height: whatever one loses the other gains.totalone end of the swingpassing the bottomthe other endkineticpotential
Fig. 4 Kinetic and potential energy at successive moments of a swing. Every column has the same total height: whatever one loses the other gains.

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 pendulum's phase portraitAngle plotted against angular velocity. Closed loops are swinging back and forth; the open curves above and below are rotating all the way round; the dashed curve between them is the separatrix.−π−π/2π/2π-22angleangular velocityat rest, hanging downbalanced upside downclosed: swingingopen: going over the topdashed: the boundary
Fig. 5 Angle plotted against angular velocity. Each curve is a level set of the energy — a possible history of the pendulum, traced out over and over.

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 same equation, elsewhere

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

Energy trading places through one stretchKinetic and potential energy at successive moments of an oscillating spring. Each column has the same total height: whatever one loses the other gains.totalfully stretchedpassing the rest lengthfully compressedkineticpotential
Fig. 6 The same accounting for a mass on a spring. Nothing has changed but the labels: the potential is stored in the spring rather than in height, and the trade is identical.

The equation x¨=−kx\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.

A travelling wave, caught at one instantA sine wave plotted against position at a fixed moment. The wavelength is the distance between repeats; the ghosted curve is the same wave a moment later.00.511.522.5-101positionone wavelength
Fig. 7 A wave. Each point of the medium is an oscillator obeying this same equation, coupled to its neighbours — which is all that separates a pendulum from acoustics.

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

Later rungs: the exact period as an elliptic integral, and what that function looks like plotted against amplitude. 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 gg 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.

Where this goes next

27°mgNmg sin θf The slope, and the two directions that make it easy An inclined plane looks like a harder problem than a flat one. Split the weight into two components chosen to suit the slope and it becomes an easier one. 00.511.522.5-101positionone wavelength A wave is a shape that travels, and nothing else does In a wave on water, no water goes anywhere. What moves is the shape — and separating the two motions is the whole of wave physics. 00.20.40.60.8100.20.4range20°35°45°60°75°same speed, five angles45° goes furthest; 20° and 70° tie The angle that throws furthest, and why nobody notices Forty-five degrees is the answer, and the maximum is so flat that a throw ten degrees off loses almost nothing. Both halves of that are worth drawing.