## Posts Tagged 'noninertial reference frames'

### Measuring the rotation of Earth

The Foucault pendulum at the Pantheon in Paris, France. [Credit: Allie Wilkinson, http://www.alliewilkinson.com/]

Yesterday — September 18, 2013 — marked the birthday of Jean Bernard Léon Foucault, usually known as Léon Foucault amongst his fans. Foucault is best remembered for his realization that a pendulum allowed to swing in any direction (instead of back and forth in a single plane of motion) on a sufficiently long chain could show the rotation of Earth. Known as a Foucault pendulum today, these devices are a staple of many science museums, including my local museum, the Science Museum of Virginia. Unfortunately, they’re a little too big for most of us to build in labs or at home, but we can still see how they work by using a simpler example — and that’s something you might be able to make yourself!

Jean Bernard Léon Foucault, the physicist who figured out how to show Earth rotates on its axis using a special pendulum.

By Foucault’s time, astronomy had established conclusively that Earth revolves around the Sun, with the associated idea that Earth rotates on its axis. In fact, the basic design of the Foucault pendulum dates back to the 17th century, but from what I can tell, Foucault didn’t know about that earlier work. In any case, by 1851 there was no controversy over the rotation of Earth, but Foucault scored a major public relations coup when he hung a heavy pendulum from the roof of the Panthéon in Paris, France, on a chain 67 meters long. The demonstration captured the imagination of the general public and helped secure Foucault’s reputation. (He also obtained some of the earliest accurate measurements of the speed of light, showed that light travels more slowly in water than air, and discovered eddy currents in metals, which are a topic I should write about someday.)

The Foucault pendulum is a simple design: it’s a heavy mass on the end of a very long chain, suspended so it can swing freely in any direction. That’s it! By Newton’s laws of motion, if you set the pendulum moving, it will swing slowly back and forth, passing through the point directly below the place where the chain is attached on each swing. The amount of time for the pendulum to complete one full swing depends only on the length of the chain (and the local gravitational field strength). However, because Earth is rotating, the room in which the pendulum resides is also rotating, which means the mass will appear to follow a curved path over time. By measuring the rate of precession — how far the mass deviates from two-dimensional motion — we can determine how fast Earth is rotating.

To demonstrate this, I created some simple videos. You can see them all on my YouTube channel, but let’s start with the first one. In all cases, we’re seeing the pendulum from above, as though we were sitting directly at the point where its chain attaches. Instead of the classic Foucault pendulum, imagine that the pendulum is on a frame attached to a turntable, which works on the same principle. If you look directly down on the turntable (without rotating yourself), you’d see the pendulum swinging back and forth as though the turntable wasn’t there at all. However, if you attached a camera to the frame holding the pendulum, so that the camera rotates with the turntable, the pendulum would appear to trace a lovely curved path.

The specific shape of the path — the number of leaves on the flower, if you will — depends on the ratio of the rate at which the pendulum swings to the speed at which the turntable rotates. In the example above, the pendulum swings twice as fast as the turntable rotates, and we end up with a four-petalled flower. If the rates are exactly the same, the pendulum traces a circle.

If the ratio is a whole number or simple fraction, the pattern repeats itself (though it may take a while). If the ratio is complicated, then the pattern is much more interesting (to me at least). However, the main thing is that if we know how long it takes the pendulum to swing through one full cycle, we can work backward to figure out how fast the turntable is spinning.

That’s where the Foucault pendulum comes in. Earth spins on its axis roughly once every 24 hours; any practical pendulum we construct will swing much faster than that, so to make things slow enough, we need to use a really long chain. (The heavy mass helps reduce the effects of air resistance, which inevitably will slow the pendulum to a halt. Even a well-constructed Foucault pendulum needs restarting every once in a while.) However, the turntable analogy only works perfectly if the Foucault pendulum is at the North or South Pole; at other latitudes, the axis of Earth doesn’t lie along the same line as “down”, so we don’t get the same flower patterns. That complicates the math, but the basic principle is still the same — we just have to take latitude into account before we can measure how fast Earth is rotating.

What’s great about Foucault pendulums or similar experiments is that they provide an independent means of measuring rotation, apart from the apparent motion of the Sun and stars. Since we and our labs rotate along with Earth (and that rotation is pretty slow by our standards), we aren’t aware of being in motion. The outcome of most experiments won’t be affected by Earth’s rotation, unless they cover a very large distance. A rocket or other projectile that travels many miles will follow a curved path with respect to Earth’s surface, a phenomenon that also occurs in the atmospheric circulation known as the Coriolis effect. (Toilets are too small for this effect, whatever the urban myth says; Phil Plait has the whole story.)

A major reason I like the Foucault pendulum and related experiments is that they demonstrate physics in non-inertial reference frames. That’s a mouthful, so let me explain: if you’re moving at a constant speed without changing direction, you’re in what’s called an inertial reference frame. From a physics point of view, any forces acting on you cancel out, resulting in motion as smooth as can be. If you are changing speed and/or direction, however, you’re accelerating. Strong acceleration is something you can feel in your body: a sudden burst of speed in a car, or taking a curve too fast, or cresting a hill too quickly. Anything you do while undergoing accelerated motion takes place in a non-inertial reference frame.

As we see in the videos, a pendulum in a non-inertial rotating reference frame looks like it has forces acting on it: it changes direction, as though it’s being pushed around. Sometimes people will refer to “ficticious forces”, but I don’t think that’s necessarily useful. What I care about is the motion itself, and what it tells us about the whole system involved — because that kind of insight leads to a deeper understanding of the underlying physics. General relativity, the modern theory of gravity, is based on connecting gravitation to non-inertial reference frames, and rotational motion near black holes reveals a lot about their structure.

The simple pendulum, old as it is, still provides a powerful way for us to learn a lot about our world and Universe.