The pendulum's period against its amplitude

At its defaults it draws 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.

pendulum-period 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.

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 Moon's orbit seen with the Sun held off to the right, drawn as it would be if the Earth's binding energy fell towards the Sun more weakly than the rest of it. The Earth is bound by 4.5 × 10⁻¹⁰ of its mass-energy and the Moon by 1.9 × 10⁻¹¹, so with the Sun pulling at 5.93 mm/s² the Moon is pushed sunward relative to the Earth by 2.6 × 10⁻¹² m/s² for every unit of η. That push turns once a synodic month relative to the orbit, 29.53 days, and Hill's equations about a circular orbit — integrated from the forced solution and held on it to 4 × 10⁻¹¹ over twelve months — give a radial displacement of 8.0 m times η times the cosine of the lunar phase: outward at new moon, inward at full. The complete lunar theory, with the Sun's tide on the orbit included, gives 13.1 m. The displacement is drawn about 7 × 10⁶ times larger than it would be at η = 1.

The gravitational self-energy of seven bodies as a fraction of their mass-energy, |Ω|/Mc², on a logarithmic axis spanning twenty-seven decades. Each is integrated shell by shell over a density profile: uniform for the lead sphere and the Moon, an iron core inside a rock mantle for the Earth, and polytropes solved from the Lane–Emden equation for Jupiter, the Sun, a white dwarf and a neutron star, each of which reproduces its polytrope's 3/(5 − n) to a per cent. A 10 kg lead sphere: 7.5 × 10⁻²⁶; the Moon: 1.9 × 10⁻¹¹; the Earth: 4.5 × 10⁻¹⁰; Jupiter: 1.5 × 10⁻⁸; the Sun: 3.2 × 10⁻⁶; a white dwarf: 8.7 × 10⁻⁵; a neutron star: 1.3 × 10⁻¹. A laboratory mass is bound by a part in 10²⁵, which is why no composition test can ask whether binding energy falls like the rest of a body. The Earth and the Moon differ by 4.3 × 10⁻¹⁰, enough for an orbit to show it; a neutron star is bound by a tenth of itself, and the Newtonian estimate drawn for it is only that.

The radial response of a circular orbit to a small push whose direction turns at a rate Ω relative to the orbit, as a multiple of the push divided by n squared, where n is the orbital frequency, on a logarithmic axis. The closed form is (1 + 2n/Ω) divided by (1 − Ω squared over n squared), and Hill's equations integrated at 0.6, 0.925 and 1.5 of the orbital frequency stay on it. It has two factors: a resonance, because a push turning at nearly the orbital rate keeps pushing in step with the orbit's own radial oscillation, and a factor from the along-track part of the push, which the Coriolis term turns into more radial motion. A binding-energy push turns once a synodic month, at 0.9252 of the orbital frequency, where the response is 22.0 times the push divided by n squared — 3.16 of that from the Coriolis term and 6.94 from resonance. A push fixed relative to the stars would turn at exactly the orbital frequency and not settle at all.

Two sets of 6000 synthetic range residuals, with per-point errors of 20 mm and 2 mm, generated with no binding-energy signal and folded by lunar phase. Each is fitted by least squares for a constant and a cosine and sine of the phase. At 20 mm the synodic amplitude comes out at 0.075 ± 0.368 mm, the σ times the square root of 2/N that the fit is required to reproduce, which at 13.1 m per unit η bounds η below 5.6 × 10⁻⁵ at two standard errors; at 2 mm the synodic amplitude comes out at 0.000 ± 0.037 mm, the σ times the square root of 2/N that the fit is required to reproduce, which at 13.1 m per unit η bounds η below 5.6 × 10⁻⁶ at two standard errors. The dashed curves are the signal η = 10⁻³ would produce on the upper set and η = 10⁻⁴ on the lower. Published limits from lunar ranging sit near 10⁻⁴, well above what statistics alone would allow, because the real residuals carry thermal expansion of the reflectors, atmospheric delay, the Moon's interior and a synodic term in the Sun's own perturbation of the orbit, all of which the fit has to separate — and because the phases where the signal peaks, new and full moon, are the ones hardest to range at.

Three tests placed by the difference in gravitational self-energy between the two bodies they compare, and by the bound each sets on a difference in how those bodies fall. The diagonal lines are constant η, the ratio of the two. The bounds are rounded published values and the self-energy differences are the integrated ones. Torsion balance, two laboratory masses: 7.5 × 10⁻²⁶ of self-energy difference, falls alike to 2.0 × 10⁻¹³, so η below 2.7 × 10¹²; lunar ranging, the Earth against the Moon: 4.3 × 10⁻¹⁰ of self-energy difference, falls alike to 5.0 × 10⁻¹⁴, so η below 1.2 × 10⁻⁴; triple pulsar, neutron star against white dwarf: 1.3 × 10⁻¹ of self-energy difference, falls alike to 2.6 × 10⁻⁶, so η below 2.0 × 10⁻⁵. The torsion balance is the best differential accelerometer of the three and says nothing about binding energy at all. Lunar ranging moves to the right by 16 decades by using bodies that are bound; the pulsar moves another 8 by using a body that is a tenth binding energy, and it reaches a smaller η with a far worse bound on the acceleration difference — the same move from numerator to denominator that the composition tests made.

physicscheck asserts something about pendulum-period 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.