The Foucault pendulum

The Foucault pendulum: a simple explanation, some history about the ideas of inertial frames, some implications, correction of a few common misunderstandings and finally a detailed analysis of the motion.


Does The Earth Move For You Too?
The stars appear to move in circles about a line through the poles of the earth. An ancient explanation of this (probably predating classical Greece) is that the stars are attached to a sphere which rotates about the earth. Aristarchus of Samos (third century BC) explained the apparent motion of the stars and planets by proposing that the earth turns on its own axis and also travels around the sun. Hipparchus (second century BC) and Ptolemy (second century AD) rejected this view for two reasons. First, one cannot feel the rotation of the earth. Second, one cannot (without powerful telescopes) see annual changes in the relative position of the stars. The earth-centred view dominated European science until the seventeenth century.

Whether or not the earth moves used to be an important question for Christians, as well as cosmologists. Giordano Bruno taught that the earth moved. He held a range of heretical views and was charged by the Holy Inquisition in Venice. He was sentenced by Pope Clement VIII and was burned alive, with his tongue gagged, in 1600. Galileo also taught that the Earth moved. He was charged with the same heresy in 1633 but he was spared on condition that he renounced his views. (The Vatican has since changed its position: in 1992 Pope John Paul II officially expressed regret at the church's treatment of Galileo.)

Jean-Bernard-Leon FoucaultThe observation of Hipparchus and Ptolemy that one cannot feel the rotation of the earth is correct. However, the rate of rotation required for the heliocentric picture (0.0007 revolutions per minute) is so slow that one would not expect to feel it. How can one measure such a slow rotation? In 1851, Jean-Bernard-Leon Foucault suspended a 67 metre, 28 kilogram pendulum from the dome of the Pantheon in Paris. The plane of its motion, with respect to the earth, rotated slowly clockwise. This motion is most easily explained if the earth turns.

The Foucault Pendulum at the University of New South Wales (animations require flash 8 plugin)

The pendulum ball (above, left) has just been released from in front of the camera position. The close-up (middle) shows a fine cable under the handrail of the stairs. Behind this is a line on the wall. Together they define a reference plane in the North-South direction.
One hour later, the amplitude of the swing has diminished, and the plane has precessed about 9 degrees in the anti-clockwise direction (Sydney is in the Southern hemisphere*). We clearly see the pendulum crossing the reference plane.

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Why Does The Orbit Of Foucault's Pendulum Precess?

Suppose* someone put a pendulum above the South Pole and sets it swinging in a simple arc. To someone directly above the Pole and not turning with the earth, the pendulum would seem to trace repeatedly an arc in the same plane while the earth rotated slowly clockwise below it. To someone on the earth, however, the earth seems to be stationary, and the plane of the pendulum's motion would seem to move slowly anticlockwise, viewed from above. We say that the pendulum's motion precesses. The earth turns on its axis every 23.93 hours, so to the terrestrial observer at the pole, the plane of the pendulum seems to precess through 360 degrees in that time.

* When this paragraph was written, it was hypothetical. Town et al have since assembled and reported the motion of a pendulum at the South Pole.

The situation is more complicated for other latitudes. On the equator, the pendulum would not precess at all - its precession period is infinite. At intermediate latitudes the period has intermediate values. One can calculate that the precession period for an ideal pendulum and support system is 23.93 hours divided by the sine of the latitude (see details). For example, at Sydney's latitude of 34 degrees S, the period is about 43 hours, a precession rate of about one degree every seven minutes.

The same effect that causes a pendulum to appear to veer slightly to the left (in the Southern hemisphere) is also responsible for the apparent motion of the major ocean currents. In the Southern hemisphere these are anticlockwise (they appear to turn to the left) and conversely in the Northern hemisphere.

One might choose to say that the earth was stationary but that a mysterious force makes moving objects turn. Using Newton's laws and the known rotation rate of the earth, one can calculate the size of these "forces", which are called Coriolis forces. Because it is so convenient to measure motion with respect to the surface of the earth, these forces are often used in calculation.

The animation below shows a pendulum swinging above the South pole. At left is the view from a position high above the equator at midday on the equinox---an observer near the sun would see this. The view at right is the view from directly below the South pole. Note how the path, as seen from the Earth, curves always to the left. (To make the details easy to see, the pendulum is depicted as very large and very slow. the amplitude of its motion was chosen equal to the radius of the Earth, and its period 8 hours: these values have no special significance. The animation makes some approximations about the motion.



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Cosmological Questions

What does non-rotating mean? What is the frame of reference in which centrifugal and Coriolis forces vanish, the frame where Newton's laws work? Observationally, we find that this Newtonian or inertial frame is one in which the distant galaxies are not rotating. But if we removed everything in the universe except the earth, how would we know if the earth were turning or not? How would the pendulum know whether to precess or not? Or, to put the question formally, is it just a coincidence that the frame in which the distant galaxies do not rotate is an inertial frame? Ernst Mach thought not, and speculated that the distant stars must somehow affect inertia (Mach's Principle), but no-one has yet come up with a successful and elegant theory. The recent cosmological hypothesis of the inflationary universe offers hope of a different resolution: if the universe expanded exceedingly rapidly in its early phase, any initial rotation will have slowed down correspondingly and so the distant objects have almost no rotation. For more about inertial frames, see our Relativity Site.

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What of Ptolemy's objection?

So, if we are whizzing around the earth's axis, and around the sun. why don't we feel it? We do not feel uniform motion: we feel forces, and they are more closely associated with acceleration than with motion. (See An introduction to Galilean and Newtonian mechanics.) If the sea is smooth and the ship's motion also smooth, you may not even notice that it is moving, though you will notice the waves if present. The waves accelerate the ship up and down, so it exerts variable forces on your feet. They accelerate you, and you feel the variable forces doing that. The acceleration due to the Earth's rotation, at Sydney's latitude, is 28 mm.s−2. This requires a force that is 0.3% of your weight, and it doesn't vary quickly. From this calculation, you wouldn't expect to feel the Earth rotating. Due to its orbit around the sun, the acceleration is 7 mm.s−2. The speeds may be high, but the accelerations are trivial. In motion, you don't feel speed, you feel the forces associated with acceleration. (See Physclips' section on circular motion for the details of these calculations.)

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A Peculiar Domestic Question

Does the rotation of the earth affect the way in which water runs down the plug hole when you empty the bath? Some people say that the water goes down clockwise in the Southern hemisphere and anticlockwise in the Northern hemisphere. Such people have probably never, or very rarely, looked. In some bathtubs (basins, toilets etc) and under some conditions, the water runs out clockwise, in others it runs out anticlockwise. There is no correlation with the hemisphere. Other effects may lead to the direction of draining. For instance, some basins have separate cold and hot water taps that are positioned symmetrically left and right. If you fill the basin using the left hand tap, you set up a rotation in one direction, and this will determine the direction in which it drains. Using the other tap reverses the direction. Many basins and baths are sufficiently symmetrical that it is possible, with some care, to have the water drain with no observable rotation.

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Ocean currents and winds

The rotation of the Earth does, however, account for the direction of the major circulations of air and wind on the planet. A non-mathematical discussion of the
coriolis forces that give rise to these currents.

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The 'plane' of the pendulum's swing is not fixed in space

It is worthwhile correcting a common misunderstanding about Foucault's Pendulum. It is sometimes said (perhaps poetically) that the pendulum swings in a plane fixed with respect to the distant stars while the Earth rotates beneath it. This is true at the poles. (It is also true for a pendulum swinging East-West at the equator.) At all other latitudes, however, it is not true. At all other latitudes, the plane of the pendulum's motion rotates with respect to an inertial frame.

It is easy to deal with this misunderstanding. Consider a pendulum at the equator, swinging in a North South plane. It's obvious from symmetry that the plane of this pendulum doesn't rotate with respect to the earth and that, relative to an inertial frame, it rotates once every 24 hours.

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Alternatively, consider the motion of a point on the earth at a place that is neither at the poles or the equator. During a day, a vertical line at that place traces out a cone, as shown in the sketch at right. (If the earth were not turning, the half angle of the cone would be 90° minus the latitude.) During each cycle of the pendulum, when it reaches its lowest point its supporting wire passes very close to the vertical. So, at each lowest point of the pendulum, its wire is a different line in this cone. This cone is not a plane, so those lines do not all lie in the same plane!

For yet another argument, consider the motion of the pendulum after one rotation of the earth. With respect to the earth, the period of precession of the pendulum is 23.9 hours divided by the sine of the latitude. For most latitudes, this is considerably longer than a day. So, after the earth has turned once, the pendulum has not returned to its original plane with respect to the earth. For example, our pendulum in Sydney precesses at a rate of one degree every seven minutes, or one complete circle in 43 hours.

(I apologise for emphasising this rather obvious point. I only do so because a correspondent has pointed out to me that many web pages about the Foucault pendulum – and even, allegedly, a few old text books! – make the mistake of stating that the pendulum swings in a fixed plane while the earth rotates beneath it.)

So, what is the path of motion of the pendulum? Remember that the point of suspension of the pendulum is accelerating around Earth's axis. So the forces acting on the pendulum are a little complicated, and to describe its motion requires some mathematics. (Indeed, even talking of a 'plane' of motion on a short time scale is an approximation because even in half a cycle the supporting wire actually sweeps out a very slightly curved surface.)

 

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Our Foucault Pendulum

The Foucault Pendulum at the School of Physics of The University of New South Wales is a "hands-on" version. There is no electromagnetic drive but, because of its size once it is started it will swing for several hours. Visitors are invited to start it swinging in a plane that is accurately defined by a fixed vertical wire and a vertical line on the wall. (
See the animation above) The pendulum takes seven minutes to precess one degree, but even smaller angles than this can be seen by sighting along the reference plane.

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The Motion of Foucault's Pendulum

For more information about the motion of the Foucault pendulum see:

Most Foucault's pendulums are in the Northern hemisphere, from where the Earth appears to turn counterclockwise, or the plane of the pendulum appears to precess clockwise. I recently visited the Musée des Arts et Métiers in Paris, where two Americans, Mike and Hilary, took this film clip. You'll also see a guide explaining the pendulum – in sign – using a small model.

 

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Animations by George Hatsidimitris and Zara Pamboukhtchian


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