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.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
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.
Which way the atoms leave, against the bias
The options are the ones The antiatom that had to be weighed anyway passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A toy version of the 2023 ALPHA-g measurement: the fraction of trapped antihydrogen atoms that leave through the bottom mirror as both mirrors are lowered over twenty seconds, against the magnetic bias, in units of the gravitational tilt, for three assumed accelerations. Each atom is given an energy between 0.02 and 0.5 K and a random starting phase, bounces between the mirrors, and leaves through the first one it reaches that has dropped below its energy. With no bias, 100 per cent leave downward if antihydrogen falls normally, 49 per cent if it does not fall and 0 per cent if it falls up. Each curve crosses one half exactly where the bias cancels the assumed gravity, and the crossing point is what the experiment measures. The curves are not steps — each falls from nine-tenths to one-tenth over about 0.6 g of bias — because the mirrors keep falling during the few milliseconds an atom takes to cross the trap: an atom that happens to arrive at the higher mirror just after it has dropped below its energy leaves that way. ALPHA-g's measured balance gave an acceleration of 0.75 g, with an uncertainty of about 0.2 g, ruling out falling upwards.
Gravity against everything else that pushes an antiparticle
The options are the ones The antiatom that had to be weighed anyway passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Forces on a single antiparticle, each as a multiple of its weight at the Earth's surface, on a logarithmic scale. An antiproton weighs 1.64·10⁻²⁶ N. An electric field of one microvolt per metre pushes it 9.8 times as hard, one millivolt per metre 9,768 times, one volt per metre ten million times; the field that exactly balances its weight is 1.02·10⁻⁷ V/m. Its own image charge in a metal wall a centimetre away pulls it 35 times its weight. A neutral antihydrogen atom escapes all of that, and is pushed instead by magnetic field gradients through its magnetic moment: 5.7 times its weight in a gradient of one gauss per centimetre, 565 times in one tesla per metre. Measuring a fall means controlling every other force to well below one.
The engine antigravity would build
The options are the ones The antiatom that had to be weighed anyway passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A cycle proposed by Philip Morrison in 1958, in units of mgh, with m an electron's mass and h the height of a tower. At the foot, a pair of gamma rays makes an electron and a positron; the pair is lifted to the top and annihilated; the two gamma rays fall back down, gaining energy as light does when it falls, 2mgh in all, and make the next pair. If the positron falls like the electron, lifting the pair costs 2mgh and the cycle returns exactly what was spent. If the positron falls upwards, it lifts itself and helps lift its partner, the lift costs nothing, and every cycle leaves 2mgh — a perpetual-motion machine. The only escape is for the falling light to gain nothing, and a photon is its own antiparticle: it cannot fall both like matter and like antimatter.
How much of a nucleus's mass is electrical energy
The options are the ones The antiatom that had to be weighed anyway passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The electrostatic energy of a nucleus's protons, computed for a uniformly charged sphere of the standard nuclear radius, as a fraction of the nucleus's rest energy, for eight elements against their atomic number. The share rises from 0.50 parts in a thousand for beryllium to 4.11 for platinum and 4.39 for uranium; titanium and platinum, the pair compared in space by the MICROSCOPE satellite, differ by 2.06 parts in a thousand. Part of that energy is carried, in quantum electrodynamics, by the virtual electron–positron pairs that polarise the vacuum round the nucleus. If positrons fell upwards, that part would weigh nothing or less, and two elements with different shares would fall at different rates. MICROSCOPE found titanium and platinum falling alike to about a part in 10¹⁵.
A tilt of a third of a millikelvin
The options are the ones The antiatom that had to be weighed anyway passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The energy of an antihydrogen atom along the vertical axis of a magnetic trap 25.6 cm long, late in the release, when the two mirror coils at its ends have been lowered to barriers of 1.0 mK, in temperature units. Gravity adds a straight slope of 0.30 mK across the trap: if antihydrogen falls down the bottom barrier is the lower, at 0.85 against 1.15 mK; if it falls up the top one is. A magnetic field gradient added along the axis tilts the trap as well, by an amount that can be set and reversed at will, and a bias equal to one g of gravity (dashed) makes the two barriers level again. Atoms leak out over whichever barrier is lower as the coils are ramped down.
What checks it
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.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
The antiatom that had to be weighed anyway
Every test of the equivalence principle for three centuries compared two lumps of ordinary matter. Whether antimatter falls down was settled on paper in the 1950s, by an engine that would run for ever if it fell up and by the electrostatic energy of nuclei, part of which is carried by fleeting positrons. It was not measured until 2023, because on a charged antiparticle gravity is ten million times weaker than a field of one volt per metre. The measurement needed neutral antihydrogen, held at half a kelvin in a magnetic bottle a quarter of a metre tall and let out over twenty seconds. It falls down, at 0.75 g with an uncertainty of a fifth of a g.
AstrophysicsThe binding energy that has to fall too
Every laboratory test of the equivalence principle compares bodies whose own gravity is a part in 10²⁵ of their mass, so none of them can ask whether gravitational binding energy falls like everything else. The Earth is bound by five parts in ten billion and the Moon by twenty times less, and if that difference fell differently the Moon's orbit would lean towards the Sun once a month — by a distance lasers have been measuring since 1969.
MechanicsThe 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.
AstrophysicsThe 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.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe pendulum, and the small lie that makes it simple
A pendulum's period does not depend on how far it swings. This is one of the most useful false statements in physics, and it is worth knowing exactly how false.
MechanicsThe 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.
MechanicsThe period that depends on the swing, computed exactly
A pendulum's period is not independent of amplitude. The exact answer is an elliptic integral, it has no elementary form, and plotting it shows precisely where the famous approximation earns its keep and where it collapses.
MechanicsThe point that takes the blow without a jolt
Hit a stick held at one end and the hand feels a jolt — forwards if the blow lands near the hand, backwards if it lands near the tip, and nothing at all if it lands two-thirds of the way along. That point is the centre of percussion, and it is exactly where a simple pendulum with the same period would end: two different questions about a swinging body with one answer.
AstrophysicsThe pull that has to equal the pull back
Every test of the equivalence principle drops two things and asks whether gravity pulls them equally. None of them asks whether they pull equally. A body's mass enters gravity twice — once as what is pulled and once as what does the pulling — and if the two ratios differed from one material to another, Newton's third law would fail and a lopsided body could push itself through space. The first test of this used a plastic cylinder floating in a liquid. The best uses the Moon, whose light crust is thicker on the side that never faces the Earth.