Illustrated Physics
physics, drawn
Concept

Foucault pendulum — where it appears

A pendulum free to swing in any vertical plane, whose plane turns relative to the ground at the Earth's rotation rate times the sine of the latitude. Its period is a pendulum day, exactly twice the inertial period, the factor of two arising because a plane returns to itself after half a turn and a velocity does not.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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.

The forces that are not there

Writing Newton's law in a frame that is turning produces three extra terms. Nothing was added to the world to make them appear and nothing is removed by calling them fictitious — one of them flattens the planet, one of them turns the weather, and both are computable to four figures.

mechanics · Circular motion The deflection is a circle, not a bend. 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 deflection that closes on itself

The Coriolis term is usually described as bending a path to the right. Integrated rather than described, it does not bend the path — it closes it. A body left alone in a rotating frame travels a circle of radius U/f and comes back to where it started in half a pendulum day, having gone nowhere at all, and drifting buoys in every ocean draw exactly that.

mechanics · Circular motion A pendulum swung 30° with a sideways push, seen from above. The path of the bob of a 1 m pendulum free to swing in any direction, seen from directly above, released 30° from the vertical with a sideways push that would, for a small swing, make an ellipse 0.3 as wide as it is long. The path is an ellipse that turns: its long axis advances 5.30° every half swing, in the same sense as the bob goes round, where Airy's small-amplitude rate (3/8) ωab/L² gives 5.06°. Nothing outside the pendulum turns it — the pivot is fixed and there is no Earth rotation in this calculation. A pendulum on a turning Earth that is meant to show the Earth's rotation has this turning to contend with too.

The 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.

mechanics · Pendulum

Named alongside it

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

Rotating frameAngular velocityCoriolis forceFictitious forceLatitudeAmplitude dependenceAnharmonicityBeta planeCentrifugal forceCentripetal accelerationConservation lawsInertial frame

All concepts