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Q & A: pendulum at pole

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Most recent answer: 06/13/2008
Say you place a pendulum over one of the earth's poles. You walk the weight out to the end of it's swing, lets say 10 meters, and are ready to let it go. At this position it is traveling about 60 meters per day in a circle. What happens to the rotational energy that the Earth has imparted to the weight? I am unable to travel to the pole to conduct my own experiment. Please Help!
- Ted Pendleton (age 13)
That's a great question.
 I think what you mean is the angular momentum associated with the Earth's rotation, not the energy, since the net energy includes the gravitational potential energy and the kinetic energy of the pendulum's swing, both quite large compared to the rotational kinetic energy. However, for a simple back-and-forth swing there would be NO other rotation (angular momentum about the line straight down from the pivot point) so we do have to figure out how that back-and-forth motion is modified due to the Earth's rotation.

It's a little easier to do the math if you have the pendulum extended only to a small angle, not straight out, but it's still essentially the same problem. In order to keep that initial angular momentum, the pendulum has to make a very slightly open elliptical swing, rather than straight back-and forth. So it's always rotating a tiny bit about its pivot line.  You wouldn't notice that unless you looked rather carefully. What is most noticeable is that, since the angular momentum goes into that slight modification of the oscillations rather than into a steady daily rotation, the Earth actually rotates beneath the fixed pendulum swing. Since we tend not to sense the Earth's rotation, what you see is that the plane of the pendulum swing rotates backward once per day, compared to reference points on the ground.

If you try the same experiment with a good pendulum at some point away from the equator but not at a pole, you'll find the pendulum plane rotates less than once per day.
I've seen that experiment work here in Illinois. It's called "Foucault's pendulum", and it was one of the first direct ways of showing that (assuming Newton's laws are right) the Earth rotates. You're not at the equator, so it should work a bit for you too.

Mike W.

There is a very nice article in Wikipedia on Foucault pendula complete with animations:

(published on 06/13/2008)

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