How Do We Know the Earth Is Spinning?
– A Pendulum's Swing Slowly Turned by Itself
We are taught that the Earth spins, yet we feel no movement under our feet. The first person to show it in a way anyone could see was a man with a single pendulum in Paris in 1851. A pendulum does not change its swing direction on its own. And yet, watch it for a few hours and the direction drifts. It was not the pendulum that had moved. It was the floor we stand on.
In the atrium of a science museum, a large metal ball hangs from the ceiling on a long wire. It swings slowly back and forth along the same line.
On the floor, small pins stand in a ring. A sign says, "When the pins fall over, it is proof that the Earth is rotating."
But the ball seems to be going back and forth along one line only. So what is knocking the pins over, one after another?
Only two reasons the swing direction drifts
Pull a hanging ball aside and let go, and it keeps swinging in the same plane. With no sideways push, there is no reason for that plane to twist.
The floor under the pendulum slowly changes direction as the Earth rotates. To someone standing on it, the pendulum's swing seems to turn instead.
The idea is simple. But showing it on a scale large enough to see, and in a way nobody could argue with, was very hard.
An experiment that began in a cellar at midnight
The experiment was devised by a Frenchman named Léon Foucault. He first studied medicine, but he is said to have disliked the sight of blood and turned to physics experiments instead.
One night in January 1851, Foucault hung a heavy weight on a wire about 2 metres long in his cellar. He chose an hour when carriages in the street were shaking the ground least. After a few swings, he saw the direction of the swing drift slightly clockwise.
The hard part was the release. Let go by hand, and a tiny sideways push gets in. The pendulum then traces an ellipse, and that ellipse itself begins to turn. So Foucault is said to have tied the weight to the wall with a string and burned the string through to let it go. No human hand touched it.
In March of the same year, Foucault gave a public demonstration at the Panthéon, a large building in Paris. The wire was about 67 metres long and the weight about 28 kilograms. A point under the weight drew a line in sand spread on the floor. With each swing, the line shifted sideways by a few millimetres. The crowd saw the Earth rotate with their own eyes.
The puzzle of not making a full circle in a day
The Earth turns once in about a day. So you might expect the swing direction to make one full circle in a day, too. In Paris, however, a full circle took about 32 hours. Foucault had predicted even this gap correctly.
The key is latitude. Directly above the North Pole, the floor turns once a day as seen from above. The pendulum does not change direction by itself, so it appears to make one circle a day. At the equator, though, the floor stays facing the sky and is simply carried along with the Earth. The floor itself is not twisted. So the swing direction does not turn at all.
At latitudes in between, the turning rate falls in between, too. Try changing the latitude with the slider in Figure 1. In the Northern Hemisphere, the swing drifts clockwise as seen from above. In the Southern Hemisphere, the direction is reversed, and it drifts anticlockwise.
Set Figure 1 to the North Pole (90 degrees) and the swing turns exactly 90 degrees, a right angle, in 6 hours, so a full circle takes about 24 hours. At Tokyo's latitude, a full circle takes about 41 hours. How much the floor is twisted relative to the sky depends on latitude.
Records are said to show that around 1661, the Italian scientist Viviani noticed that a pendulum's swing direction drifts slowly. But Foucault was the first to link it to the Earth's rotation. Seeing something is one thing. Grasping what it means takes another step.
Before his experiment at the Paris Observatory, Foucault is said to have sent out an invitation along the lines of "Come and see the Earth turn." Something debated for centuries, he claimed, he could now simply "show" with one wire and one ball.
Summary
A pendulum does not change its swing direction by itself. It looks as though it does because the floor beneath it turns with the Earth. How much the floor is twisted depends on latitude, so the swing goes round once a day at the North Pole, in about 41 hours in Tokyo, and not at all at the equator. Foucault started with a 2-metre pendulum in a cellar, then used a 67-metre pendulum to show it to the public.
The pendulum did not turn.
It was we, the watchers, who were turning.
Why the Earth's spin goes unnoticed in daily life is explained in "Why Do You Land in the Same Spot When You Jump?", and how the spin affects winds and whirlpools is covered in "Is It True That Bathtub Drains Swirl the Other Way in the Southern Hemisphere?"
- Tie a few coins to the end of a string and hang it from a coat hanger or similar. The longer the string, the slower and easier to watch.
- Pull the coins aside a little and release them gently. Then slowly turn the hand holding the hanger by half a turn.
- Even as the string's anchor point turns with your hand, the coins' swing direction should barely change relative to the room. Think of your turning hand as the "Earth" and the room as "space", and you have the same relationship as Foucault's pendulum.
If you push sideways as you release, the coins will trace an ellipse. You will feel why Foucault burned the string to let go. Some science museums display a real Foucault pendulum.
For those who want more – terms, formulas and links to textbooksEach part is labelled by level, from junior high science to university courses
- Junior highCovered in junior high school science
- High schoolCovered in high school physics or earth science
- High school+Advanced high school material, or textbook sidebars
- UniversityUniversity courses not taught in high school (mechanics, rotating frames)
- ResearchNot yet settled textbook knowledge; topics researchers are still investigating
Junior highTerms: this phenomenon has names
- Foucault pendulum: a long pendulum kept swinging for a long time. The turning of its swing plane shows the Earth's rotation.
- Inertia: the tendency of an object to keep its current motion unless a force acts on it from outside. This is why the pendulum's swing plane is preserved.
- Sidereal day: the time for one turn of the Earth measured against distant stars. It is about 23 hours 56 minutes, roughly 4 minutes shorter than the 24-hour day measured by the Sun.
Junior highHigh schoolChecking with a formula: the time for the swing to go round in Paris and Tokyo
The rate at which the floor is twisted relative to the sky equals the Earth's rotation rate multiplied by the sine of the latitude. We use this relation to find the time for a full circle.
| In symbols | T ＝ T₀ ÷ sin φ |
| In words | Time for the swing to go round once ＝ time for one turn of the Earth (sidereal day) ÷ sine of the latitude |
| Where it comes from | Split the Earth's rotation at a given place into a "vertical" part and a "horizontal" part. Only the vertical part twists the floor. Its size shrinks by the sine of the latitude (Foucault's sine law). |
| T | Time for the swing direction to go round once (in hours) |
| T₀ | Time for the Earth to turn once relative to the stars = sidereal day (in hours) |
| φ | Latitude of the pendulum site (in degrees) |
| Sidereal day T₀ | about 23.93 hours |
| Sine of Paris's latitude (about 48.85° N) | approx. 0.753 |
| Sine of Tokyo's latitude (about 35.7° N) | approx. 0.584 |
| Swing amplitude of the Panthéon pendulum (centre to edge) | said to be about 3 metres |
| Time for a full circle in Paris (hours) | 23.93 ÷ 0.753 ≒ 31.8 |
| Time for a full circle in Tokyo (hours) | 23.93 ÷ 0.584 ≒ 41.0 |
| Angle turned per hour in Paris (degrees) | 360 ÷ 31.8 ≒ 11.3 |
| Angle turned per hour at the North Pole (degrees) | 360 ÷ 23.93 ≒ 15.0 |
| Time for one swing of the Panthéon pendulum (seconds), from length 67 metres and gravitational acceleration 9.8 | 67 ÷ 9.8 ≒ 6.84 |
| The square root of 6.84 is said to be about 2.62. Multiply by 2 × pi (6.28) | 6.28 × 2.62 ≒ 16.5 |
| Angle turned in one second (degrees). One hour is 3600 seconds | 11.3 ÷ 3600 ≒ 0.00314 |
| Angle turned in one swing (16.5 seconds) (degrees) | 0.00314 × 16.5 ≒ 0.052 |
| Circumference of a circle of radius 3 metres (metres) | 2 × 3.14 × 3 ≒ 18.8 |
| Shift of the sand line per swing (metres) | 18.8 × 0.052 ÷ 360 ≒ 0.0027 |
In Paris, the swing goes round in about 32 hours, turning about 11 degrees an hour. At the pendulum's edge, the sand line shifts by about 3 millimetres per swing. The shift is tiny, but anyone watching for half an hour can notice it. Do the same experiment in Tokyo and a full circle takes about 41 hours.
High schoolHigh school+Pendulum length and the difficulty of the experiment
High schoolThe period of a simple pendulum does not depend on the weight of the bob or (for small swings) on the amplitude. It is set only by the string length and gravitational acceleration. The period is "2 × pi × the square root of (length ÷ gravitational acceleration)". A 67-metre pendulum takes more than 16 seconds per swing, so it moves slowly and the drift is easier to observe.
High school+A long string and a heavy bob are used to delay the shrinking of the swing caused by air resistance, so the pendulum can keep swinging for hours. Also, if the pendulum traces even a slight ellipse, the ellipse itself slowly turns. That effect mixes with the turning caused by the Earth's rotation. This is why a straight release mattered above all.
UniversityMotion seen from a rotating floor: the Coriolis force
When we describe motion on a floor that turns with the Earth, apparent forces appear: the centrifugal force and the Coriolis force. Put the Coriolis force into the pendulum's equation of motion and solve it, and you find that the swing plane turns at an angular speed equal to "the Earth's rotation rate × the sine of the latitude". This is Foucault's sine law. It is one of the most famous examples in the mechanics of rotating frames (non-inertial frames). Effects that cause errors in real pendulums, such as the precession caused by elliptical paths (Airy precession), can be treated in the same framework.
📖 Derivation of the formula and further reading: Foucault pendulum (Wikipedia) ／ Coriolis force (Wikipedia)
ResearchWhat is still not clear
- Is the "anomaly" during eclipses real? In 1954, the Frenchman Allais reported that a pendulum's motion was disturbed during a solar eclipse. Follow-up experiments have been done since, but some report reproducing it and some do not, and no conclusion is said to have been reached.
- Why the length of a day fluctuates. Today the Earth's rotation is measured precisely with devices using light and lasers, not pendulums. The length of a day fluctuates by a few thousandths of a second, and part of the cause (such as exchanges between the Earth's core and its outer layers) is still being investigated.
- How accurately can a small pendulum measure it? Even in a museum, slight asymmetries in the suspension or air currents can make the turning rate depart from the sine law. Ways to suppress such errors in short pendulums are still being tried.
In other words, this article too is "an explanation within what is known today". That the Earth rotates is beyond doubt, but the fine fluctuations in how it rotates are still being measured.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Junior high | Science: Earth's rotation and motion of celestial bodies / inertia | Why the pendulum keeps its direction, why the floor turns |
| High school | Physics: period of a simple pendulum / Earth science: Earth's rotation | Period of the 67-metre pendulum, differences by latitude |
| High school+ | Physics: inertial forces (advanced) | Why tracing an ellipse spoils the experiment |
| University | Mechanics: rotating frames and the Coriolis force | Derivation of Foucault's sine law |
| Research | Geodesy: precise measurement of Earth's rotation | Eclipse anomaly, fluctuations in day length |
| ― | Links to daily life | How to view a museum pendulum, direction of typhoon swirls |
- Wikipedia (Japanese), "Foucault pendulum"
- Wikipedia (Japanese), "Coriolis force"
- L. Foucault, "Démonstration physique du mouvement de rotation de la Terre au moyen du pendule", Comptes Rendus de l'Académie des Sciences, 32, 1851.
- W. Tobin, "The Life and Science of Léon Foucault: The Man Who Proved the Earth Rotates", Cambridge University Press, 2003.
- A. D. Aczel, "Pendulum: Léon Foucault and the Triumph of Science", Atria Books, 2003.
※This article is a general-audience science explainer. The figures given are rough guides to help you understand how things work. Historical accounts vary, and details differ between sources.