Pendulum — where it appears
Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
The compass that finds the axis the Earth turns about
A spinning wheel held so that its axis stays level will, on a turning Earth, swing its axis round to point at true north and stay there — not magnetic north, the geographic pole. Nothing in the instrument knows where north is; it finds the one direction about which the Earth's turning stops tilting it. Built to be used at sea, such a compass has to ignore the ship's lurches, and that forces its swing to have one particular period: 84.4 minutes, the period of a pendulum as long as the Earth's radius, of a satellite skimming the ground, and of a stone dropped through a tunnel to the far side of the planet.
The spring that swings when it was only bounced
Hang a weight on a spring and pull it straight down: it bounces straight up and down. Unless its bounce happens to be twice as fast as its swing would be as a pendulum. Then the bounce dies away and the weight starts to swing, wider and wider, until the bounce has gone completely — and then the swing dies and the bounce comes back. Two motions that look independent are trading their energy through a resonance, regularly and for ever at small amplitude. Give the spring more energy and the trading becomes chaotic: the same simple rule, with nothing random in it, produces motion that never repeats and that a hundred-millionth of a radian can redirect.
The chain that swings in Bessel's notes
Hang a chain from a hook and set it swinging. It can swing as a whole, like a pendulum, or in shapes with still points along it, as a guitar string does — but its higher notes are not two, three and four times its lowest. They are 2.295, 3.598 and 4.903 times it, the zeros of a function that Daniel Bernoulli met for the first time in exactly this problem in 1732, a century before Bessel's name was attached to it. The reason is that a hanging chain is a string whose tension grows from nothing at its free end to its whole weight at the top, and a wave crossing it changes speed all the way.
The kick that does not change the beat
A pendulum clock has to push its pendulum, or friction would stop it within minutes, and every push risks disturbing the very period the clock exists to keep. In 1826 George Airy worked out exactly when a push does no harm: at the bottom of the swing, where the pendulum is moving fastest, an impulse changes how far it swings and leaves the timing untouched. A push before the bottom makes the clock gain; after it, lose. Three centuries of escapement design are a search for a mechanism that pushes where Airy said — or that pushes in the wrong place in a way that cancels some other error.
The pendulum that must not grow when it warms
A pendulum clock keeps time by the length of its pendulum, and every material grows when it is warmed. A steel rod a metre long lengthens by a hundredth of a millimetre for each degree, and the clock loses half a second a day — a minute a fortnight between a cold winter and a warm summer, far worse than anything the clockmaker's escapement could cause. The cure, found in the 1720s, was to let the pendulum grow in two directions at once: one part downward, another upward, in exactly the proportion that leaves the bob where it was. The same trick, with different metals, kept the best clocks to a hundredth of a second a day for two centuries, and its limits are set by heat that does not spread evenly or quickly.
Named alongside it
The objects these essays reach for when they reach for this one.
IsochronismAmplitude dependenceClockEscapementMeasurementParametric resonancePrecessionRotating frameSimple harmonic motionSystematic errorAngular momentumAnharmonicity