One parabola, several wells

At its defaults it draws one parabola, several wells. Unlike potential wells, each divided by its own curvature at the bottom, against the single parabola ½x² drawn through all of them. They agree near the minimum because a function with a minimum has no linear term there, so the quadratic term is the first thing it has. The labels give where each well departs from the parabola by more than 1% of the parabola's own value there: a pendulum at 0.35, a chemical bond at 0.01, a pair of atoms at 0.0015. A symmetric well has no cubic term and stays close for a long way; a well that is steeper on one side than the other has one, and leaves the parabola almost at once — which is why those numbers differ by factors of hundreds and not by a few per cent.

harmonic-compare is one function in lib/figures/mechanics.js — motion, force, energy and rotation. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Unlike potential wells, each divided by its own curvature at the bottom, against the single parabola ½x² drawn through all of them. They agree near the minimum because a function with a minimum has no linear term there, so the quadratic term is the first thing it has. The labels give where each well departs from the parabola by more than 1% of the parabola's own value there: a pendulum at 0.35, a chemical bond at 0.01, a pair of atoms at 0.0015. A symmetric well has no cubic term and stays close for a long way; a well that is steeper on one side than the other has one, and leaves the parabola almost at once — which is why those numbers differ by factors of hundreds and not by a few per cent.

Unlike potential wells, each divided by its own curvature at the bottom, against the single parabola ½x² drawn through all of them. They agree near the minimum because a function with a minimum has no linear term there, so the quadratic term is the first thing it has. The labels give where each well departs from the parabola by more than 1% of the parabola's own value there: a pendulum at 0.35, a chemical bond at 0.01, a pair of atoms at 0.0015. A symmetric well has no cubic term and stays close for a long way; a well that is steeper on one side than the other has one, and leaves the parabola almost at once — which is why those numbers differ by factors of hundreds and not by a few per cent.

The mean displacement of an oscillator against temperature, taken as a Boltzmann average over each well rather than from any expansion of it, with the temperature measured against each well's own depth so that unlike bonds can share an axis. A symmetric well gives exactly zero at every temperature — heating a harmonic solid makes it vibrate harder and does not make it larger. The others drift outward, because the outward side is the shallower one, and the measured slopes are a pendulum 0.000, a chemical bond 0.766, a pair of atoms 0.150. Thermal expansion is not a property a spring has; it is one a spring lacks.

The true period of an oscillation, obtained by integrating dx/√(2(E−V)) between the turning points, divided by the harmonic period the curvature at the bottom predicts. A parabola gives exactly one at every amplitude, which is the property that makes a clock possible. Nothing else does: a pendulum reaches 1.560 at an amplitude of 2.40, a chemical bond reaches 2.141 at an amplitude of 2.16, a pair of atoms reaches 3.455 at an amplitude of 0.54. A pendulum swinging 90° from vertical takes 1.1803 times its small-swing period, which is the tabulated value of the elliptic integral and is not built into this figure anywhere — the curve is the integral itself, evaluated between turning points found by bisection.

The angular frequency of six oscillators, each computed as the square root of a curvature over a mass, drawn on a logarithmic scale because they span 15 decades. A ship, a pendulum and a spring are metres and kilograms; a tuning fork is a shaped piece of quartz; a copper atom sits in a well whose stiffness follows from its Debye temperature — 213 N/m, which is the same order as a laboratory spring; and a carbon–monoxide bond is 1902 N/m over a reduced mass of seven atomic units. The formula does not change anywhere along the row.

Unlike potential wells, each divided by its own curvature at the bottom, against the single parabola ½x² drawn through all of them. They agree near the minimum because a function with a minimum has no linear term there, so the quadratic term is the first thing it has. The labels give where each well departs from the parabola by more than 2% of the parabola's own value there: a spring never leaves it at all, a pendulum at 0.49. A symmetric well has no cubic term and stays close for a long way; a well that is steeper on one side than the other has one, and leaves the parabola almost at once — which is why those numbers differ by factors of hundreds and not by a few per cent.

physicscheck asserts something about harmonic-compare that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.