Coriolis Effect and Foucault's Pendulum

Choose a latitude and the page swings Foucault's pendulum over a floor that turns with the Earth, integrated live, at a time-lapse you set. Below it a ball crosses a turning roundabout, seen from the ground and from the roundabout, and a stone falls down a mine shaft. Every figure is worked out from the Earth's spin Ω.

Worked example — Coriolis Effect and Foucault's Pendulum

How fast the plane turns against the floor	11.32°/h
Time for one full turn of the plane	31.79 h
How far it turns in 24 hours	271.76°
One swing, there and back	6.65 s
Coriolis acceleration on water moving at 0.1 m/s	1.10×10⁻⁵ m/s²
The same, in units of g	1.12×10⁻⁶ g
Sideways drift while crossing a 0.2 m basin	0.022 mm
Time to fall the depth of the shaft, no air	5.686 s

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Coriolis Effect and Foucault's Pendulum

The Coriolis acceleration is the term −2Ω × v that appears when motion is described from a frame turning at angular velocity Ω, here the ground. It is at right angles to the velocity, so it changes a body's direction and never its speed, and no force produces it. For motion along the ground only the part of the Earth's spin about the local vertical counts, Ω sin φ: it turns a Foucault pendulum's plane at that rate and gives the Coriolis parameter f = 2Ω sin φ. The part along the ground, Ω cos φ, pushes vertical motion east or west.

What each symbol means

Ω: the Earth's spin, 7.292115×10⁻⁵ rad/s: one turn against the stars in 23.93 h
φ: latitude, negative south of the equator
f: Coriolis parameter, 2Ω sin φ: the sideways acceleration of a body moving along the ground, per metre per second of its speed
Ro: Rossby number U/(fL): the acceleration a motion has of its own over its Coriolis acceleration

Assumes: Small swings, so the pendulum's motion is two oscillations at right angles. The centrifugal part of the Earth's turn is folded into g, which points along the plumb line, and g = 9.80665 m/s² everywhere. The stone falls in vacuum. The roundabout is flat and frictionless, so nothing pushes the ball once it is rolled.
Breaks when: A real pendulum whose swing has opened into a slight ellipse precesses by itself, at a rate that grows with the width of the ellipse, which is why Foucault released his bob by burning the thread that held it aside, giving it no sideways push. Air slows a falling ball: Reich's took about 6.01 s to fall, against 5.69 s in vacuum. Over hundreds of kilometres f changes with latitude, which this page holds fixed.

In Paris the plane takes 31.79 hours to go round, not 24 🖖

Press Paris Observatory, 1851 and let it run at an hour per second. Nothing turns the swing about the vertical. The floor turns under it, and a floor turns only with the share of the Earth's spin that lies along its own vertical: at latitude φ, Ω sin φ. At the North Pole that is all of it, and the plane goes round once in 23.93 h, keeping pace with the dashed once-a-day line. At 48.836° N it is 11.32°/h instead of 15.04°/h, so a full turn takes 31.79 h, and after 24 hours the plane has gone only 271.76°. Léon Foucault said as much in the note printed in the Comptes rendus of 3 February 1851: he gave the sine law, and reported that at “our latitudes” the plane does not make a whole turn in twenty-four hours. On the equator the floor does not turn about the vertical at all, and the plane stays where it was released for as long as the bob swings. South of it the floor turns the other way, and Sydney gives 8.38°/h anticlockwise.

Measured against its own motion, water in a sink feels more Coriolis than Foucault's 11 m pendulum 🖖

On the 11 m pendulum at Paris the sideways Coriolis force never exceeds 1/8,600 of the pull that swings the bob. Water crossing a basin at 0.1 m/s gets a Coriolis acceleration of 1.12×10⁻⁶ g and drifts 0.022 mm on its way across 0.2 m; set against the water's own motion that is a Rossby number of 4,550, beside the pendulum's 8,600 in the table below. What separates them is everything else. Nothing but the Earth turns a pendulum's plane, so every swing adds the same small turn and an hour adds 11.32°. Water in a basin keeps whatever swirl it was poured in with, and draining concentrates it, the way a skater spins faster with arms pulled in, so the filling, a draught or a warm wall decides long before the Earth does. In 1962 Ascher Shapiro of MIT made a tank drain counterclockwise by filling it with a clockwise swirl, covering it, holding the room's temperature steady and waiting 24 hours; the float over the drain did not begin to turn until 12 to 15 minutes into a 20-minute drain. A weather system 1,000 km across, moving at 10 m/s, sits at the other end of the table with a Rossby number of 0.0911, and there the rotation is in charge.

A stone dropped down a mine shaft lands east, not west 🖖

If the ground turned away beneath a falling stone, one let go 158.5407 m above the bottom of a shaft at Freiberg would land 1.67 km to the west, which is how far the ground there travels east in the 5.686 s the fall lasts. Press Freiberg shaft, 1831: it lands 27.65 mm to the east. The stone was moving with the top of the shaft when it was let go, and the top is further from the Earth's axis than the bottom, so it starts with a little more eastward speed than the ground it falls towards. Described from the turning ground, that is the Coriolis acceleration, 2Ω cos φ times the stone's downward speed, and it points east in both hemispheres. Between 19 August and early September 1831 Ferdinand Reich dropped 106 balls down the Drei-Brüder-Schacht, a mean fall of 158.5407 m at 50°53′22.81″ N, and measured 28.396 mm east. He also measured 4.374 mm south and did not trust it; every southward term in the theory is of order Ω²ht², about 0.03 mm here. The stone answers to the part of the spin along the ground, Ω cos φ, the reverse of the pendulum: nothing at the pole, most on the equator.

Problem solved in full

Fire a ball straight up the shaft: where does it come down? 5 steps
At Freiberg a ball is fired straight up from the bottom of Reich's shaft, just fast enough to reach the top, 158.5407 m above, and falls back. Where does it land, relative to where it was fired from? Press Freiberg shaft, 1831 to check the first step.

Open this problem in the tool
1. The page's stone, dropped from the top, falls for 5.686 s and lands 27.65 mm east. The ball sees the same latitude and the same g:
  
  $t = \frac{2 h}{g} = \frac{2 \times 158 . 5407}{9 . 80665} = 5.686 s, x = \frac{1}{3} g Ω t^{3} cos φ = 27.65 mm east$
2. To rise 158.5407 m it must leave at v₀ = √(2gh), and it is in the air for 2v₀/g:
  
  $v_{0} = 2 g h = 2 \times 9.80665 \times 158.5407 = 55.76 m/s, T = \frac{2 v _{0}}{g} = 11.37 s$
3. In the turning frame the Coriolis acceleration is −2Ω × v. Its eastward part is −2Ω cos φ times the upward speed, v₀ − gt: west on the way up, east on the way down.
  
  $a_{E} = - 2Ω cos φ (v_{0} - g t)$
4. Integrated once, that gives the sideways speed. At the top, t = v₀/g, the ball is drifting west at Ω cos φ v₀²/g; the way down takes exactly as long, and its eastward push brings that speed back to zero just as the ball lands, without bringing the ball back.
  
  $v_{E} (t) = - 2Ω cos φ (v_{0} t - \frac{1}{2} g t^{2}) : v_{E} (\frac{v _{0}}{g}) = - \frac{Ω cos φ v _{0}^{2}}{g} = - 14.59 mm/s, v_{E} (\frac{2 v _{0}}{g}) = 0$
5. Integrated again over the whole flight, T = 2v₀/g:
  
  $x_{E} (T) = - 2Ω cos φ (\frac{1}{2} v_{0} T^{2} - \frac{1}{6} g T^{3}) = - \frac{4}{3} Ω cos φ \frac{v _{0}^{3}}{g ^{2}} = - 110.58 mm$
Answer

It lands 110.58 mm west of where it was fired, four times as far as the dropped stone lands east, and with no sideways speed left. The ratio is exactly 4 at any height and latitude, since both distances go as Ω cos φ h^3/2/√g.

$x_{E} = - 110.58 mm = - 4 \times 27.646 mm (west)$

Learning path

Rotation and angular momentum

References (3)

Foucault's own announcement: the pendulum's plane turns against the floor at the Earth's rotation times the sine of the latitude, so that at Paris it does not make a whole turn in twenty-four hours; first a 2 m pendulum in a cellar, then 11 m in the Meridian Room of the Paris Observatory: L. Foucault, "Démonstration physique du mouvement de rotation de la terre au moyen du pendule." Comptes rendus hebdomadaires des séances de l'Académie des sciences 32 (1851), 135–138.
The eastward fall measured: 106 balls dropped down the Drei-Brüder-Schacht at Freiberg in August and September 1831, a mean fall of 158.5407 m at 50°53′22.81″ N, landing 28.396 mm east (and 4.374 mm south, which Reich doubted), against his own computed 27.512 mm: F. Reich, Fallversuche über die Umdrehung der Erde, angestellt auf hohe oberbergamtliche Anordnung in dem Drei-Brüderschachte bei Freiberg. Freiberg: J. G. Engelhardt, 1832, pp. 46–47.
A sink's drain direction is set by how it was filled, unless everything else is removed: Shapiro filled a tank with a clockwise swirl, covered it, kept the room's temperature steady, let it settle for 24 hours, and saw it drain counterclockwise: A. H. Shapiro, "Bath-Tub Vortex." Nature 196 (1962), 1080–1081. doi:10.1038/1961080b0

By the 42Math team · Reviewed against published sources. Corrections welcome.