Illustrated Physics
physics, drawn
Concept

Amplitude dependence — where it appears

The variation of an oscillator's period with how far it swings, which is zero for a true harmonic oscillator and small but never zero for anything else. It is second order in the amplitude for a pendulum, so halving the swing quarters the error, and removing it entirely requires a cycloid rather than a circle.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The pendulum's period against its amplitude. The exact period of a simple pendulum divided by the small-angle period, plotted against amplitude. The small-angle formula is the horizontal line at one; the exact curve leaves it slowly and then climbs without limit as the amplitude approaches a half turn.

The period that depends on the swing, computed exactly

A pendulum's period is not independent of amplitude. The exact answer is an elliptic integral, it has no elementary form, and plotting it shows precisely where the famous approximation earns its keep and where it collapses.

mechanics · Pendulum Four beads, four heights, one arrival time. One arch of a cycloid of radius 1, drawn with four beads on it at heights 0.061, 0.235, 0.592, 2.000 — a range of 33.0 to one. Each bead's time to slide, from rest and without friction, to the bottom of the arch is computed as a quadrature of ds/v along the curve as drawn, and the four answers are 1.003205 s, 1.003205 s, 1.003205 s, 1.003205 s: identical to 6.9e-13 of themselves. The closed form for this curve is π√(a/g) = 1.003205 s, which the quadrature reproduces without being told it. A bead let go at the cusp travels 5.7 times as far as the lowest one and arrives with it, because the extra distance is exactly paid for by the extra speed the extra height buys.

The curve that does not ask where it started

A pendulum's period depends on how far it swings, and the dependence is small but never zero. There is exactly one curve for which it is zero, and the reason has nothing to do with pendulums: on that curve the height above the bottom is proportional to the square of the distance travelled along it, which makes the motion harmonic by construction rather than by approximation.

mechanics · Pendulum A drive at one frequency, and what comes back at three times it. The Fourier components of the steady motion of an oscillator driven at a single frequency ω = 1.35, for drive strengths of 0.02, 0.05, 0.1. The equation is a harmonic oscillator with a cubic term added, and the components are projected out of the integrated motion rather than assumed. A linear oscillator answers only in the first column. This one answers in the third as well, because x³ of a cosine contains a cosine of three times the angle. The third harmonic grows as the drive to the power 3.00 where the fundamental grows as the power 1.00, so it is negligible at small drive and not at large — which is why nonlinearity in an instrument is a specification rather than a yes or no.

The oscillator that answers at three times the question

Push a spring hard enough that the parabola stops being the whole story, and three things happen that a linear oscillator cannot do: it emits frequencies nobody supplied, its resonance leans over, and its amplitude at one drive frequency depends on where the drive has been.

mechanics · Harmonic approximation A pendulum swung 30° with a sideways push, seen from above. The path of the bob of a 1 m pendulum free to swing in any direction, seen from directly above, released 30° from the vertical with a sideways push that would, for a small swing, make an ellipse 0.3 as wide as it is long. The path is an ellipse that turns: its long axis advances 5.30° every half swing, in the same sense as the bob goes round, where Airy's small-amplitude rate (3/8) ωab/L² gives 5.06°. Nothing outside the pendulum turns it — the pivot is fixed and there is no Earth rotation in this calculation. A pendulum on a turning Earth that is meant to show the Earth's rotation has this turning to contend with too.

The ellipse a pendulum turns by itself

Let a pendulum swing in any direction and give it a small sideways push, and its bob traces an ellipse. For a small swing the ellipse closes. For a larger one it turns, steadily, in the sense the bob is going round — with nothing outside the pendulum turning it. Airy worked out the rate in 1851, the same year Foucault hung his pendulum in the Panthéon, and the two effects are the same size for a pendulum a metre long whose ellipse is a twentieth of a millimetre wide.

mechanics · Pendulum

Named alongside it

The objects these essays reach for when they reach for this one.

Simple harmonic motionIsochronismAnharmonicityPendulumArc lengthThe arithmetic–geometric meanBrachistochroneConstraintCycloidThe elliptic integralEscapementEvolute

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