Illustrated Physics
physics, drawn
Concept

Pendulum — where it appears

A mass swinging under gravity about a pivot, whose period is independent of amplitude only in the small-angle limit. It is the standard exhibit for a harmonic approximation and for its failure, and the tension in its string — a constraint force absent from the equation of motion — reaches three times the weight at the bottom of a quarter-circle swing.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Two pivots with one period, and the length between them. The period of a uniform bar 1 m long swung about a pivot, against the distance of that pivot from its centre of mass. Close to the centre the period runs away, because there is almost no restoring torque; far from it the bar behaves more and more like a simple pendulum. In between is a minimum at h = k = 0.2887 m, the radius of gyration, and because there is a minimum every period above it belongs to two pivots at once — here 0.4 m and 0.2083 m, whose product is k² and whose periods agree to 2.2e-16 seconds. Suspend the bar from those two knife edges in turn, adjust until the periods match, and the equivalent simple pendulum is the distance between them: 0.6083 m. Then g = 4π²(h+h′)/T² gives 9.8100 m/s², recovered from the drawing to 1.8e-16. The mass of the bar, the distribution of that mass, and the position of the centre of mass appear nowhere in the answer.

The length nobody has to measure

A pendulum's period depends on its length, so a pendulum will measure gravity — except that a real swinging bar has no single length, and finding where its mass effectively sits is far harder than timing it. Kater's answer was to build the instrument so that the awkward quantity cancels, leaving a distance between two knife edges that a rule can read to five figures.

mechanics · Pendulum The potential a shaken pivot creates. The effective potential of a pendulum whose pivot is shaken vertically, against the angle from hanging, for shaking rates of 8, 14, 20, 30 times the pendulum's own frequency at an amplitude of 0.12 of its length. The shaking averages to no force at all — it is up as often as down — and yet it adds a term to the potential, because the pendulum's position is correlated with the phase of the shake rather than independent of it. The added term is proportional to sin²θ, so it is largest sideways and zero at both the hanging and the inverted positions, and it turns the maximum at 180° into a minimum once 14× and 20× and 30× the natural frequency is reached. The criterion is (aΩ)² > 2gL: the shake speed must beat the speed a fall through the pendulum's own length would give. Upside down then becomes a stable equilibrium, with a restoring force and a period of its own.

Held up by a force that averages to nothing

Shake a pendulum's pivot up and down fast enough and the pendulum will stand upside down, balanced, and push back if it is nudged. The shaking supplies no average force at all — it is up as often as it is down — and the reason it nevertheless holds is that the pendulum's position and the phase of the shake are not independent.

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 Three centuries of finding nothing. The upper limit on η against the year it was set, on a logarithmic scale. Newton (1687) reached 1e-3; Bessel (1832) reached 2e-5; Eötvös (1922) reached 5e-9; Dicke (1964) reached 1e-11; Braginsky (1972) reached 1e-12; Eöt-Wash (2008) reached 2e-13; MICROSCOPE (2022) reached 1e-15. That is 12 orders of magnitude in 335 years, and every one of those measurements returned zero. A sequence of null results is not a sequence of failures. Each one is a statement that a principle assumed by every theory of gravity holds to a new level, and each new level excludes a class of theories that would have shown a departure there — a long-range force coupling to something other than mass-energy, a scalar partner to the graviton, a violation arising at some energy scale. The measurement is worth making again precisely because it has always come out the same way, which is what makes any departure decisive.

The fall that does not depend on what is falling

Everything falls at the same rate, and the statement has been tested for three hundred years by people looking for the exception. Twelve orders of magnitude have been added to the limit and every measurement has returned zero. The last three orders came not from a better instrument but from finding something bigger to fall towards.

astrophysics · Equivalence principle

Named alongside it

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

MeasurementSystematic errorAmplitude dependenceArc lengthAveragingBrachistochroneCentre of massCentrifugal forceConstraintCycloidEffective potentialEquilibrium

All concepts