The Foucault pendulum: seeing the Earth turn with your own eyes

We know the Earth turns: the Sun rises and sets, and the stars drift across the sky through the night. But all of that could also be explained by the sky turning around us. For centuries the question was this: standing on the ground, inside a closed building, without looking at the sky at all, can we see that the Earth turns? In 1851 the French physicist Léon Foucault answered “yes” with a pendulum.

Paris, 1851: “Come and see the Earth turn”

Léon Foucault (1819–1868) was a physicist also known for his experiments measuring the speed of light and for his work on the gyroscope. In January 1851 he made his first attempt in the cellar of his house, with a pendulum about 2 metres long, and saw its plane of swing slowly turn. In early February he showed the experiment to scientists at the Paris Observatory, with a longer pendulum of about 11 metres. The line on the invitation is famous: “You are invited to come and see the Earth turn.”

The same month he published the result in Comptes rendus, the journal of the French Academy of Sciences. The demonstration drew so much interest that, at the request of the president of the day, Louis-Napoléon Bonaparte, the experiment was moved in March 1851 under the dome of the Panthéon in Paris.

The Panthéon set-up: a 67-metre wire and a 28-kilogram ball

The Panthéon pendulum was a steel wire about 67 metres long, fixed to the top of the dome, with a brass-coated lead ball of 28 kilograms at its end. The ball had a sharp point underneath. Sand was spread on the floor, and at every swing the point drew a small mark in it.

A pendulum this long swings back and forth slowly. For small angles, the period of a simple pendulum is

With $L = 67$ m and $g \approx 9.81$ m/s², this gives $T \approx 16.4$ seconds. The long wire and the heavy ball let the pendulum keep swinging for hours: air resistance slows a heavy ball less.

Within a few minutes the audience noticed something curious: the marks in the sand did not stay on the same line. Each swing landed a little beside the one before, and the plane turned slowly clockwise. Today many science museums set small pins in a circle around the pendulum; as the plane turns, the ball knocks them over one by one.

Why does the pendulum keep its plane?

The forces on the pendulum are the tension in the wire and the weight of the ball. Both lie in the vertical plane in which the ball swings. No force acts on the pendulum to turn it about the local vertical axis; the floor, however, turns about that axis by sin φ times the Earth’s rotation (φ: latitude). Only at the pole does this mean “the plane stays fixed relative to the stars”. It hangs from a single point that lets it swing in any direction, so even if the ceiling turns it cannot twist the pendulum out of its plane.

What turns is not the pendulum but the floor beneath it. This is easiest to picture at the North Pole. A pendulum set up there swings in a plane fixed in space while the Earth turns once beneath it in one sidereal day, about 23 hours 56 minutes. To an observer on the ground, the floor stays still and the plane of the pendulum seems to make one turn the other way in the same time.

A day at the pole, never at the equator: the role of latitude

Paris is not at the pole. The Panthéon pendulum takes about 32 hours to make one full turn. The reason for the difference is latitude.

We can think of the Earth’s spin as a rotation vector pointing along its axis. Where you stand, this vector splits into two parts: one along your local vertical, the other horizontal. Only the vertical part turns the floor around itself like a record turntable. Its size is proportional to the sine of the latitude. Foucault himself proposed in his first report that the turning rate of the plane is proportional to the sine of the latitude:

Here $\varphi$ is the latitude. Three results follow:

• At the poles $\sin 90^\circ = 1$, so the plane turns about 15° an hour and completes a turn in one sidereal day (23 h 56 min).
• At the equator $\sin 0^\circ = 0$, so the plane does not turn at all. There the floor does not spin around itself under the plane of the pendulum; it is only carried along with the Earth.
• In the Southern Hemisphere the latitude is taken as negative and the plane turns the other way: anticlockwise, seen from above.

City by city

Let us apply the same rule to two cities in Türkiye. To find the turning per hour we multiply 15.04° by the sine of the latitude. For the time of a full turn we divide 360° by that number.

Istanbul (about 41.0° north): $\sin 41.0^\circ \approx 0.656$. The plane turns $15.04^\circ \times 0.656 \approx 9.87^\circ$ an hour. A full turn takes $360 / 9.87 \approx 36.5$ hours, or about 36 hours 28 minutes.

Ankara (about 39.9° north): $\sin 39.9^\circ \approx 0.642$. The plane turns about $9.65^\circ$ an hour. A full turn takes about $37.3$ hours, or 37 hours 19 minutes.

Paris (about 48.85° north): $\sin 48.85^\circ \approx 0.753$. The plane turns about $11.33^\circ$ an hour. A full turn takes about $31.8$ hours.

Two things stand out. First, the length of the pendulum does not appear in the calculation at all. Whether the wire is short or long, the plane turns at the same rate; the length only changes how long one swing takes. Second, after turning 180° the plane is back on its starting line, because the pendulum swings to both ends of that line. So all the pins around it are down after half a full turn: in Istanbul that takes about 18 hours.

In the model above you can change the latitude, the length of the pendulum and how fast time runs. “From the ground” keeps you on the floor and shows the plane turning. “Turning with the pendulum” keeps the plane fixed and shows the floor turning beneath it by sin φ times the Earth’s rotation; only at the pole is this the same as a view fixed to the stars. Both views describe the same event.

The Coriolis effect: the same sin φ

An observer standing on a turning floor sees the paths of moving objects slowly bend. The effect used to explain this bending is called the Coriolis effect, after Gaspard-Gustave de Coriolis, who studied it mathematically in 1835. It is not a real force; it comes from measuring relative to a turning floor.

The part of the Coriolis effect that deflects horizontal motion is also proportional to the sine of the latitude. It bends motion to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. In a Foucault pendulum the ball is deflected very slightly to the right at each swing, and these small deflections add up to the turning of the plane. On a large scale the same effect plays a part in the direction in which hurricanes and large weather systems spin. It does not, however, decide which way the water in a sink swirls: at that scale the shape of the sink and the water’s initial motion matter far more.

The difficulties of a real pendulum

On paper the Foucault pendulum is simple, but in practice it has to be set up with care:

• A clean release: if the ball is given even the slightest sideways push, it traces a thin ellipse instead of a straight line, and that ellipse starts turning on its own. Foucault tied the ball to one side with a thread and burned the thread to release it, so the ball started swinging without being pushed to either side.
• The suspension point: the wire must be able to swing equally easily in every direction. A mounting that favours one direction pulls the plane towards it.
• Damping: air resistance shrinks the swing within a few hours. Most museum pendulums have a small electromagnet that keeps the swing going without disturbing its plane.

Foucault’s original ball is kept today at the Musée des Arts et Métiers in Paris; a copy swings in the Panthéon.

Why does this idea matter?

The Foucault pendulum is one of the first experiments to show that the Earth turns using only a measurement made on the ground, without looking at the sky. You can try out how day and night come about in our lesson on the Earth’s rotation. In our predict, observe, explain post we use a classroom activity to show why the period of a pendulum does not depend on its mass.

Summary

• In 1851 Foucault showed that the Earth turns, using a 28-kilogram ball hung on a 67-metre wire in the Panthéon.
• No force acts on the pendulum to turn it about the local vertical axis; the floor turns about that axis by sin φ times the Earth’s rotation. Only at the pole does this mean “the plane stays fixed relative to the stars”.
• The plane turns about $15.04^\circ \times \sin\varphi$ an hour: at the pole one turn takes 23 h 56 min, at the equator the plane does not turn, and in the Southern Hemisphere it turns the other way.
• In Istanbul a full turn takes about 36.5 hours, in Ankara about 37.3 hours. The length of the pendulum does not change the turning rate.