Illustrated Physics
physics, drawn
Concept

Cantilever — where it appears

A beam fixed at one end and free at the other, the simplest element that bends. Its lowest resonant frequency rises in proportion to its thickness and falls as the square of its length, which sets the size of a watch's quartz fork and the sensitivity of a force microscope's probe.

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

Where a speck lands decides how much it weighs. The fractional drop in frequency of a cantilever's first and second bending modes when a small mass lands on it, per unit of that mass as a fraction of the beam's, against where along the beam it lands, from the clamp to the free tip. The response is the square of the mode's displacement at the landing point divided by twice the mode's effective mass: a quarter of the beam for both modes (0.250 and 0.250), since each is measured against its own tip. At the tip both modes respond fully — the first by 2.0 times the landed fraction, the second by 2.0. Near the clamp neither responds, so a speck landing there is invisible. Halfway along, the first mode drops by 0.23 and the second by 1.02; at 0.783 of the length the second mode has a node and does not notice the speck at all. A single frequency cannot tell a light speck at the tip from a heavier one further in.

The weight a frequency can find

A spring's frequency is the square root of its stiffness over the mass it carries, so adding a little mass lowers the note by half the fraction added. Nothing in that rule cares about gravity, and nothing in it cares about size. It weighs astronauts in orbit by the swing of a chair, films a fraction of an atom thick on a quartz plate, and single molecules on a beam a few micrometres long — and it improves without limit as the thing that vibrates gets lighter. What it cannot do on its own is say where the added mass sat, and that turns out to matter as much as how much there was.

mechanics · Harmonic approximation The parabola every quartz watch follows. The frequency error of a watch's quartz fork, in parts per million, against its temperature: a downward parabola, −0.034 ppm per degree squared, about a turnover at 25 °C, where the fork runs exactly at its nominal rate; with the right-hand scale in seconds a day, since one part per million is 0.086 s a day. On a wrist, at about 31 °C, the watch runs 1.2 ppm slow, 0.11 s a day; in a drawer at 10 °C, 7.7 ppm, 0.66 s a day; at 0 °C 1.8 s a day; at 60 °C 3.6. Every departure from the turnover, hotter or colder, makes it lose. The turnover is set by the angle at which the fork is cut from the crystal, and is chosen near the temperature of the place watches are worn — the parabola is a fact about quartz, and its vertex is a fact about people.

The fork inside a watch

Every quartz watch keeps time with a tuning fork three millimetres long, cut from a crystal of quartz and ringing thirty-two thousand seven hundred and sixty-eight times a second. Each number in that sentence is a decision. The frequency is two to the fifteenth, so that fifteen halvings make one tick a second, and it sits just above the range of human hearing. The shape is a fork because two tines swinging against each other leave their base still and lose almost no energy into it. And the fork's rate drifts with temperature along a parabola whose vertex is set near twenty-five degrees, which is why a watch worn on a wrist keeps better time than one left in a cold drawer.

waves · Resonance

Named alongside it

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

ResonanceEffective massFrequencyFrequency divisionHarmonic oscillatorInertial massMeasurementMode shapeOscillatorPiezoelectricityQuality factorTemperature coefficient

All concepts