One puck, two frames

At its defaults it draws one puck, two frames. A puck slides outward from the centre of a turntable at 1.4 m/s across a table of radius 1 m, which turns through 0.55 of a revolution in the 0.71 seconds the crossing takes. On the left, in the room: a straight line, because no force acts along it. On the right, on the turntable: a spiral, because the coordinates turned underneath it. The two panels are the same numbers with the same clock — the second is the first with the angle rotated by −Ωt — so whatever is bending the path on the right is arithmetic, and it is still worth a name, because on a turning planet the right-hand panel is the one everybody lives in.

rotating-frame 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.

A puck slides outward from the centre of a turntable at 1.4 m/s across a table of radius 1 m, which turns through 0.55 of a revolution in the 0.71 seconds the crossing takes. On the left, in the room: a straight line, because no force acts along it. On the right, on the turntable: a spiral, because the coordinates turned underneath it. The two panels are the same numbers with the same clock — the second is the first with the angle rotated by −Ωt — so whatever is bending the path on the right is arithmetic, and it is still worth a name, because on a turning planet the right-hand panel is the one everybody lives in.

Trajectories integrated from Newton's law in a rotating frame with the Coriolis term and nothing else — no pressure gradient, no friction, no force of any kind. 0.5 m/s at 45°, 0.3 m/s at 30°. Each path closes on itself after one inertial period, checked to a millionth of its own radius, and the radius is the speed divided by the Coriolis parameter: 4.85 km, 4.11 km. The deflection usually described as a curving of the path is a complete circle, traversed clockwise in the northern hemisphere in half a pendulum day, and a body left alone in a rotating frame goes nowhere at all.

The period of an inertial oscillation against latitude, which is π divided by the Earth's rotation rate and the sine of the latitude. It is exactly half a pendulum day — checked here at five latitudes against the Foucault period computed independently — so a free parcel's velocity vector and a swinging pendulum's plane turn at the same rate, which is a coincidence only until the two are recognised as the same rotation. At the pole the period is 11.97 hours; at 45° it is 16.93; at four degrees of latitude it is over ten days and rising, because the Coriolis parameter vanishes at the equator. Peaks at exactly this frequency are the largest feature in almost every long record of ocean current, and they were first noticed in Nansen's drift measurements.

The same integration as the closed circle, with one change: the Coriolis parameter is allowed to vary with latitude, as it does. Over 30 loops at 30° the parcel drifts 3.6 kilometres west and 0.4 south, because the northern half of each circle happens where f is larger and is therefore tighter than the southern half. With β set to zero the identical integrator returns to the origin, which is how the drift is shown to be the physics rather than the timestep. Surface drifters in the real ocean draw exactly this: loops at the local inertial period, slowly migrating, and the migration is a direct measurement of the gradient of the Earth's rotation.

The Coriolis deflection of something moving horizontally at 20 m/s for 40 seconds — a range of 0.80 km — against the latitude it does it at. The term is 2Ω×v, so the horizontal part carries sin φ and vanishes at the equator; at the pole the drift is 2.3 m, which is 0.29% of the range. It is quadratic in the time, so over a few seconds it is nothing and over a day it is the whole circulation of the atmosphere.

The time for the swing plane of a Foucault pendulum to turn once, against latitude. The rate is Ω sin φ — the component of the planet's rotation about the local vertical — so the period is a sidereal day divided by sin φ: 23.9 hours at the pole, 31.8 hours at 48.85°, and longer than a week below 8.2°. The curve rises without limit toward the equator, where the plane never turns at all. The axis starts at 12° for that reason rather than at zero.

physicscheck asserts something about rotating-frame 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.