The old planetarium adjoined the science hall of the museum and the center piece exhibit there was a Foucault pendulum. It was a wonderful demonstration of the rotation of the earth. The planetarium office was on the third floor and also attached to the side of the cylindrical section of the building containing the planetarium. It was my office until I moved into a brand new, below ground suite of staff offices beneath the exhibit halls. The interesting thing was that the pendulum was suspended from the ceiling of that third floor office. Yes, the room had an opening in the floor, appropriately surrounded by a rail for safety.

One of the Google Doodles for this past week was the Foucault pendulum. I looked at it a day or so after it was on the main search page. I didn’t look at any of the explanations it might have linked to, but it reminded me of trying to explain the pendulum to folks in the exhibit hall. The time it takes the pendulum to apparently rotate is a very difficult concept and I’ve been trying to think of a good way to explain it for many decades. Over the past few years, I think I finally hit upon a reasonable and understandable explanation.

Image credit: Foucault pendulum at Berry College in Rome, Georgia by Courney McGough CC BY-NC-ND 2.0. Note, this is not the particular pendulum discussed in this post.

The Easy Part

Basically, the Foucault pendulum demonstrates that the earth is rotating. A pendulum always tries to swing in the same direction. Imagine holding a pendulum over a rotating phonograph record (that’s a vinyl record, you know, music, for you young folk). Also imagine placing your favorite, tiny action figure near the outside of the record. To the little figure, it will look like the pendulum, swinging constantly in the same direction, is rotating, changing it’s direction. That proves the record is turning. This is how a Foucault pendulum would behave at the North Pole and it’s simple to understand.

However, things are different at the equator. Imagine starting a pendulum at the equator swinging north-south. Now as the earth rotates, the pendulum can continue to happily swing north and south without any change in it’s direction, all the way around the course of a day. The pendulum itself would be traveling around the earth, but not rotating, relative to the ground beneath it, as it went.

The Weird Part

So the pendulum would appear to rotate once every 24 hours at the pole, but at the equator it would never rotate, it would take infinite time. What’s worse, between the two extremes it takes longer and longer to rotate as one moves from the pole to lower latitudes.

In Atlanta, for example, at about 34 degrees latitude, the pendulum would take about 43 hours to go around once. How can that be since the earth rotates in 24 hours?

Try Driving

Imagine the earth as a smooth sphere, no mountains or oceans, like a globe. And imagine you’re going to drive around the earth in a car. To make it interesting, we’ll drive fast enough to make it around the earth in 24 hours. At the equator, this is easy. We point the car east and take off an 1000 miles per hour, or so.

What about 34 degrees above the equator? There, we want to drive along a parallel line, a parallel of latitude. If we point our car east and set off straight ahead, we won’t follow the parallel. Instead the car will follow a great circle (like a tilted equator). It will cross the real equator ¼ of the way around, hit 34 degrees south latitude on the opposite side, and arrive again at it’s starting point.

To follow the parallel at 34 degrees, we have to turn the steering wheel slightly to the left to stay on track. Aircraft and ship’s pilots understand this. Usually they like to travel along great circles.

So if we cut our steering wheel to the left a little and drove at an appropriate speed, we’d make it around the parallel in 24 hours. Now, what if we took the car off of the earth-sized sphere and set it down on an infinite plane? With the steering wheel cut to the left that same amount and traveling at the speed we used before, how long would it take the car to drive around in a circle? The answer is about the 43 hours it took the Foucault pendulum to go around.

Well, think about it. If we took our car from the equator, with it’s steering wheel pointed straight ahead, it wouldn’t go around in a circle at all. It would just drive off toward infinity.

For what it’s worth, our car at the north pole would have to turn the steering almost 90 degrees to the left and drive in donuts around the pole. In that case, just like the pendulum, the car would go around in about 24 hours on the plane, if we drove (in this case very slowly) at a speed to make it around the pole in 24 hours.

Moving vs. Turning

Notice that as the car is farther north, it’s driving more slowly but turning more. That’s what’s really going on here. At the equator, you really don’t turn about a vertical axis at all. You head off toward the east as if you’re standing on a moving sidewalk. Sure, you’re turning about the earth’s axis, but not around your own axis pointed straight up. At the equator it’s all moving and no turning.

At the pole, it’s all turning but no moving. You don’t move or you’d leave the pole. As you go from the equator to the pole, your motion around the earth becomes more a turning motion and less a moving motion. You can tell there’s less “moving” because the parallels are smaller and smaller. You don’t have as far to go.

For the mathematical types, the time it takes the pendulum to rotate is 24 hours divided by the sine of the latitude.

That Crazy World of Geometry

This is just one example of how much trouble we get into if we leave the comfortable world of Euclid’s plane geometry and try to do things in a crazy world of say a sphere or saddle-shape. And that’s just for a two-dimensional surface (of sorts). If you get into other geometries in three and more dimensions, then you’re in the crazy world that Einstein and his successors used to explain gravity purely in terms of geometry—no forces needed.

Back in the exhibit hall

Each morning I’d manually start the massive pendulum on it’s three-story cable swinging north-south. Later in the day, it’s direction would have visibly changed. Traditionally, an exhibit will have little blocks that the pendulum can knock over as it changes direction. It’s good enough to show that the earth truly is rotating without understanding all of the above. And it’s pretty clever, really, to be able to show such a large effect with such a simple experiment.