A Pendulum Day

“I have come to believe that the whole world is an enigma, a harmless enigma that is made terrible by our own mad attempt to interpret it as though it had an underlying truth.”
― Umberto Eco, Foucault’s Pendulum

But if fell later as they tried to move another piece. Note the rare “suspended section” of blocks. I’m not sure of the physics of leaving a few behind for a handful of microseconds.

Along with my Difficult Reading Book Club I’m plowing ahead through Umberto Eco’s Foucault’s Pendulum – ten pages or so a day. It’s enjoyable, though truly difficult. I feel I should be looking up every odd word – searching out details on every unique concept – but there are pages to get through so I soldier on. Have to come back later. I’d take notes – but they would be longer than the tome itself.

One concept that haunts my dreams is the eponymous swinging orb. I knew about the Foucault Pendulum, of course. I have even seen one – a big, famous one – at the Smithsonian in Washington (though it looks like it isn’t there any more). I knew the theory, that the pendulum is actually always going in the same plane, but the earth moves under it. The more I thought about it the more I realized it isn’t that simple.

What follows is some boring, technical crap. If that doesn’t interest you, here’s some cute cat photos.

Ok, I can imagine a Foucault Pendulum at the North Pole. I can see it moving around in 24 hours.

But, I thought, what about one at the equator? Wouldn’t it be stationary?

So I looked it up online and I was right. It would not move.

But what threw me off were the latitudes in between. Because there is an angle between the string of the pendulum and the rotation of the earth – it rotates, but slower. The closer to the equator, the longer it takes to go around. The precession period for an ideal pendulum and support system is 23.93 hours (a sidereal day) divided by the sine of the latitude. In the middle of the US, this is about 32 hours. This period of time is called a pendulum day.

sidereal day(23.93 hours)<solar day(24 hours)<pendulum day(varies by latitude) (though I guess there is a latitude near the north pole where the pendulum day is the same as the solar day….)

The problem that I have is this: imagine the pendulum at our latitude… it goes through a 24-hr. cycle… now the pendulum is in exactly (more or less) the same spot that it was at the beginning… yet the pendulum, because the pendulum day is longer than 24 hours, is not at the same spot.

If the pendulum is truly staying the same… and the earth moving beneath it… why doesn’t it return to the same relative spot in 24 hours?

I spent way too much time thinking about it. I kept thinking about cones.

I’m not sure I’ve completely worked it out – but this site helps. Here is the meat of the text:

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 apologize for emphasizing 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.)

Now my head hurts. Unfortunately I can’t relax. I have my reading to do.

Sweet dreams.

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