## Some Fun and Intuition about the Foucault Pendulum

The Foucault Pendulum is a beautiful demonstration of the fact that the earth rotates once every 24 hours, and that it is not flat. There are of course other ways to infer these facts. However, watching a Foucault pendulum follow its stately path when you understand what causes it to do what it does provides a deep and satisfying sense of connection between the local, here-and-now, human scale of things and a larger view of the universe.

First, for those who haven't visited a science museum for a while, here is a description of a typical Foucault Pendulum display. A shiny metal sphere of about one foot in diameter is hanging near the floor, suspended by a thin wire from a high ceiling, perhaps three stories up. It swings slowly back and forth, pendulum-style. There is a small pointer attached to the bottom of the sphere that points straight down and sweeps along just above the floor as the pendulum swings back and forth.

There is a sequence of dominoes standing in a circle of about eight feet in diameter on the ground under the pendulum, with the dominoes spaced about six inches apart around the circle. Occasionally, the pointer on the pendulum touches a domino. After a few swings during which it hits the domino progressively more directly, the pointer eventually knocks over the domino. Although it is happening too slowly to see directly, you eventually realize that the pendulum is slowly changing the direction in which it is swinging, and that it will eventually reach and knock down the next domino over, and will eventually move all the way around the circle and knock over all of the dominoes one at a time. It might take as long as 20 hours for the pendulum to move around enough to knock over all of the dominoes. (Dominoes 180 degrees apart get knocked down at the same time; it would take about 40 hours for the plane the pendulum swings in to go all the way around 360 degrees.)

There are two very interesting things the pendulum illustrates. The first is that the earth is rotating about once per 24 hours. To see this, imagine that the pendulum is set up exactly on the north pole. It swings back and forth. Underneath it, the earth is rotating. The pendulum continues to swing back and forth undisturbed in its own plane of rotation that is fixed relative to the stars, while the earth rotates underneath it. If you are standing on the earth next to the pendulum, you will be carried in 24 hours all of the way around it, back to your starting place. By a trick of perspective, it is natural to assume that you and the earth are standing still, and it is the pendulum that is slowly changing its plane of oscillation.

A couple of other examples are fun to think about. Imagine you are in the passenger seat of a car sitting out in a big open parking lot, and holding a yo-yo up by its string, with a friend in the driver seat. You start swinging the yo-yo back and forth, and then hold the string as still as you can, so that the back and forth motion continues undisturbed. Now, say your friend starts the car, and drives you slowly in a large circle. What will happen to your yo-yo? It will continue to swing in the initial plane you started in (say north-south). As you drive in the big circle, it will look, at least from inside the car, like the yo-yo is changing the direction in which it is swinging. But in fact, the yo-yo is just continuing to swing in the north-south direction, and it is you who are turning. If your friend continues to drive all the way around a big circle (in a flat parking lot), by the time he is done and has come back to his starting place the plane of oscillation of the yo-yo will appear to have gone through an entire 360-degree rotation.

This is just what happens as you stand there in your parka feeding the polar bears watching the pendulum swing back and forth at the north pole. After about 24 hours, the pendulum will appear to have rotated an entire 360 degrees.

But if you watch the stars (say it's winter so you can watch the stars for the entire 24 hours), you may notice that the pendulum is maintaining its orientation relative to the stars.

Now, what happens if we move the swinging pendulum away from the north pole?

For example, what if we look at the Foucault pendulum at the Griffith Observatory, which is about 30 degrees north latitude? (It is a little bit higher than that, but let's round off to 30 degrees.)

We will approach this problem by considering a huge cone sitting on the earth that has the right size and the right angle at it point so that it just tangentially touches the earth all the way around at 30 degrees North latitude. So, you could get in your car and drive all the way around the spherical earth at 30 degrees North latitude, or you could drive all the way around the circle on the huge cone that is tangent to the earth at 30 degrees north latitude, and you couldn't tell the difference between the earth and the cone.

Thinking again about the car in the flat parking lot, if your friend drives around in a larger circle, after he has gone 360 degrees the pendulum will again have gone around 360 degrees. But let's try something different. Let us conjure up a large hill for you and your friend to drive around. We will make it a perfect (or perhaps boring) hill, that is a perfect cone. The slope of the hill is, say 10 degrees. Now, let's say your friend drives all the way around the hill, steering so as to maintain a constant altitude. He drives in a big circle, around the side of the hill, and eventually comes back to his starting place. If you watch your pendulum yo-yo very carefully, you will notice something interesting: by the time you've gone all the way around the hill, your yo-yo has only gone about 355 degrees, and it is no longer oscillating in the north-south plane when you get back to your starting place!

Why might this be? It would seem that, appearances to the contrary, you have not managed to go a full 360 degrees. In fact you have gone a full 360 degrees, but the key thing is that your turn was accomplished partly by turning to the left, and partly by having your car's nose dip down as you moved around the side of the cone.

For intuition on this, imagine an 'extreme cone' of infinite height, namely a cylinder. If you could get your car to stick to the side of the cylinder and you could drive in a circle all the way around the cylinder, your driver friend would not have to turn to the left or right at all. The entire time he is driving straight ahead, and the traversal of the circle is accomplished entirely by the nose of the car dipping as it follows the side of the cylinder. It is a bit like going up and over the top of a hill. Initially your car's nose is pointing uphill, but it levels out as you get to the top of the hill, and then follows through and ends up pointing down hill as you get to the other side. Driving around the cylinder, you are constantly having the nose of your car 'lower' the direction it is pointing, as though you were going over the curved top of the hill, but all the way around.

For a cone of a given steepness, if you drive all the way around it at constant altitude, how will we figure out the total amount you have turned using the steering wheel versus the total amount you turned via the 'nose pitch-down' effect?

If you imagine a small paper cone with sides that make an angle of 10 degrees from horizontal, and draw a circle around the cone, the circle is like the path your car followed. If you take a pair of scissors and make a vertical cut from the rim of the cone to its apex, and then smooth the paper flat on the table, you will find a circle with a thin pie-shaped wedge missing. And, the circle you drew on the cone will be a partial circle with an arc missing. The missing pie wedge will be about 5 degrees wide, just the amount missing from your yo-yo's rotation as your friend drove all the way around the hill.

Now, say we set up a Foucault pendulum at a latitude of, say 30 degrees north. As the earth rotates, we will trace out a circle that goes the whole way around in about 24 hours. Our circle is a good distance from the north pole. If the earth were flat, our pendulum would appear to go all the way around in 24 hours. But in fact it does not! It goes much more slowly than that. In fact, it takes our pendulum about two days to go all the way around. This is a proof that the earth is not flat.

So, our wonderful Foucault pendulum gives us direct, local, observable evidence that our earth rotates on its axis (the earth is doing the rotating, not the stars and the sun around it), and also that the earth is not flat.

Let's try to be a bit more precise about the rate at which the pendulum's plane of oscillation rotates as a function of latitude.

Pick some fraction between 0 and 1. Make a circular disk of paper. Trim out of your paper circle a pie slice of the selected fraction. (If your fraction is, say, .25, you will trim a 90-degree pie slice out of the circle, to remove 25% of the area of the circle.) How long is the perimeter that remains? Assume for convenience that your original circle was of radius one. It will then have had a perimeter of 2 pi (by the definition of pi). Having lost one quarter of its perimeter, it will now have a perimeter of 1.5 pi.

Let us now take this partial circle, fold it into a cone, and apply tape where the two edges meet. How steep are the sides of this cone?

Well, the base of the cone is a circle of circumference 1.5 pi. The radius of that circle will be 0.75. That radius of 0.75 can be viewed as one of the legs of a right triangle whose hypotenuse is 1, the radius of the original circle we started with. The angle opposite is then a little more than 45 degrees, the angle whose sine is .707, or sqrt(2) / 2. So, the side of the cone will be 90 degrees minus this angle, or a bit under 45 degrees.

In general, the 'missing fraction' of the circle of paper used to create the cone will be the sine of the steepness of the side of the cone. If the 'cone' is actually a cylinder, the sides will be 90 degrees, and sine(90) is 1.0, so the 'missing fraction' of the circle is the entire circle. If the cone has sides that make a 30-degree angle, sine(30) is one half, so half of the circle would be missing.

This same fraction is the amount of the perimeter that is missing, and so if you drove a car around that perimeter, that is the fraction of an entire 360-degree turn you would make as you went all the way around.

The steepness of a tangent cone is 90 degrees minus latitude. So, sine(latitude) is the fraction of the circle that remains after the above trimming process, and that is how much you turn if you go all the way around the earth at a given latitude. And, consequently that is how much the Foucault Pendulum will appear to rotate in the opposite direction during one 24-hour period.