The Foucault Pendulum and the Schuler Pendulum

by Walt Arnstein

In an earlier posting, we examined the behavior of a simple pendulum in the presence of varying temperature, gravity, and swing amplitude. In this follow-up, we will inspect two special pendulums that tell us much about our planet’s celestial mechanics: The Foucault Pendulum and the Schuler Pendulum.

Because the Earth rotates about its own axis but we intuitively consider our immediate surroundings to be stationary, we are forced to live with certain ‘fictitious’ physical phenomena that characterize observations from rotating frames of reference. For example, if you start from the Equator and begin to move rapidly northward, you may think that you are moving steadily and in a straight line, but in fact, you are moving on a curved rotating surface and are in need of slowing down in order to stay on the same meridian. That is, standing on the Equator, you were being carried eastward at some 1000 mph by the Earth’s rotation. As you move northward along your starting longitude line, you will be tending to maintain the eastward velocity you started with as you enter latitudes where surface speed is slower. What you will ‘feel’ is an Eastward-directed ‘force’ that will keep increasing, reaching a maximum near the North Pole. This fictitious ‘force’ is called the Coriolis force and acts on air masses and solid bodies alike when they move in a direction that forces them to change their velocity’s component tangential to the surface of the Earth. In the case of air masses, it is the source of the clockwise-directed cyclones in the Northern Hemisphere and counterclockwise cyclones that circulate in the Southern Hemisphere. It causes currents to move clockwise around high pressure areas and counterclockwise around low pressure areas on your weather map. It also causes water in bathroom sinks to drain clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere.

The Coriolis effect has an interesting influence on a pendulum swinging from a pivot with no restrictions as to direction of swing. The effect is increasingly graphic as the pendulum gets longer while its swing amplitude remains low. What we have in effect is a graphic manifestation of the Earth’s rotation. The device is called a Foucault Pendulum, named after the 19th Century French physicist Jean Bernard Leon Foucault, (pronounced Foo-ko’). If properly started without sideways thrust or rotation, the pendulum will precess in its swinging, its plane of oscillation slowly rotating clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere. The effect would be most dramatic at the North Pole, where its plane of oscillation would describe a full turn in 24 hours. At lower latitudes, it would precess more slowly, the rate being proportional to the sine of the latitude, finally vanishing at the Equator.

What we are actually seeing is the Earth (and ourselves) turning counterclockwise under the pendulum while the pendulum continues to swing in its original plane. But remember, we are looking at the phenomenon from a rotating frame of reference, so that to us, it is the pendulum that seems to precess.

The Foucault pendulum seems to have struck the fancy of humanity, as a mystic symbol of our existence on Earth. A beautiful Foucault pendulum is located in the entrance hall of the Science Museum in South Kensington, UK. One almost as famous is in the lobby of the UN Secretariat Building in New York City. Visitors can watch its 2-foot brass sphere, kept swinging from its thin cable by synchronized electromagnetic impulses at the base, progressively knocking over golf tees placed at regular intervals in a circle within its range of motion. Many other good Foucault pendulums are to be found all over the world — but probably not in Singapore, Quito or Galapagos. Too close to the Equator. Also not on either Pole. Who in his right mind would go there to watch it?

A Foucault pendulum, incidentally, is probably the only way (not counting a GPS receiver) to find your latitude without looking at the Sun and stars. Using your trusty Speedy Pro, Flieger, or Chronomat, time the pendulum’s precession over a measured angle and compare it with the corresponding time it would take at 1 turn per day. The ratio of the latter to the former is the sine of your latitude, which you can get from your scientific calculator.

A useful form of pendulum is the Schuler pendulum. To gain an insight into its usefulness, consider the problem of maintaining a horizontal platform in flight. Let’s assume there is no GPS, no celestial navigation to help us, how can we keep the floor beneath us horizontal? One way would be to track the local vertical. If we hung a pendulum from our vehicle and started from a stop, the pendulum would tend to follow the local vertical. Of course, every time we accelerated and reached a new speed or direction, the pendulum would swing and we would have a vertical that oscillated about the true vertical. The magnitude and frequency of oscillation would vary with the length of the pendulum. The longer the pendulum, the lower the magnitude and frequency of its oscillation. So, we might ask ourselves, is there any length of pendulum that would track the vertical without starting to oscillate? The answer is yes: It would be a pendulum equal in length to our distance from the center of the Earth. This is called a Schuler pendulum and its characteristic would be that as we accelerated forward, it would swing backward in reaction in such a way that the vertical would always be maintained.

Obviously, we can’t build a physical pendulum whose length is approximately the radius of the Earth, but we can build a computer controlled emulation of such a pendulum. What we can do is simply sense any acceleration of our ship, with sensitive accelerometers and gyros and immediately force our platform to turn in the appropriate direction. Our platform will in effect be a Schuler pendulum and this technique has, in fact, been used for stable platforms in space vehicles.

But there is one interesting characteristic of the above pendulum that is not intuitively obvious: Suppose we begin our trip with a slight error in our vertical? What will happen? Our local vertical will oscillate about the true vertical with this error (hopefully insignificant) and we won’t have any way to observe or eliminate the error. This will be the case with either the physical pendulum or the electronic pendulum. The period of the oscillation? 84.4 minutes if we are near the surface of the Earth — exactly the same as the orbital period of a satellite orbiting the Earth at our altitude! This equivalence will hold for any planet and at any distance from its center. It all comes out plainly, right there in the math.

Isn’t physics wonderful?