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Name: Donn
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Question:
I volunteer at a local museum. Near me is a Foucault pendulum. My question seems simple: Why does the pendulum seem to rotate with reference to the museum floor? I have visited a number of sites on the web and find the usual answer "because of the rotation of the earth" not satisfactory. I understand the polar case, where the earth rotates under the pendulum, but I do not understand the other cases, where both the museum and the pendulum would be joined together and riding along a particular latitude. It would seem that, if they are both moving together, then the pendulum would not change with reference to the museum floor. Such is not the case, and explanations that say blithely "it is because of the rotation of the earth" are not really explanations. I do not doubt that they are true, but I cannot make the leap from the explanation to the phenomenon.



Replies:
Donn -

I have viewed a Foucault pendulum with students over many years. Here is the way I visualize the pendulum problem... As you say the mechanics work perfectly at the poles and as you move from that center of rotation the problem and its solution become less perfect. However I find two thoughts useful. First, I need to stop using the earth as a frame of reference and visualize the problem from a point in space.

Secondly I combine this with a model of the problem. I recall the old 33 1/3 rpm record player. If I were to turn the record and mount my pendulum in the exact middle, it would work as if I were on the pole of the earth. If I were to move the pendulum half a radius from the center, it would be like moving to 45 degrees latitude. From my vantage point (much like viewing the earth from space) the pendulum would appear to continue to swing in the same plane. But if I can project myself onto the surface of the record, it would appear to move relative to my frame of reference... the record itself.

This model is imperfect... as are the pendulums at lower latitudes. We also do not have a means of including the gravitational factor in the model. Yet, even with these imperfections, I can understand the basic reason for the apparent motion of the museum pendulum.

It would be interesting to attempt this demonstration on a larger scale than an old record player. I wonder where one could find a turntable of some size. The size of a football field? It would be interesting to test my demonstration on that large of a scale.

Hope that helps.

Larry Krengel


This web site does the best job that I know of to explain why this works:

http://www.calacademy.org/products/pendulum/

Regards,

Todd Clark, Office of Science
US Department of Energy


Donn,

Consider the two extremes, at a pole and at the equator:

At a pole, turning the Earth has no effect on the pendulum. The pendulum is not even "aware" that the Earth has turned. As a result, the Foucault pendulum keeps moving along its original axis while the Earth rotates beneath it. All of the Earth's rotation is around the pendulum's base.

At the equator, turning the Earth completely changes the orientation of the pendulum's base. There is no rotation "around" the base at all. The pendulum will not rotate.

Most locations are in between a pole and the equator. Some of the rotation changes the orientation of the pendulum's base and some rotation is around the base. It is "between" the two extremes. The pendulum will rotate. The closer the Foucault pendulum is to the equator, the easier it is to disrupt the rotation. When the pendulum is as far north as the United States, the rotation effect is fairly strong.

An example to show how this works can be seen through astronomy. When at a pole, a star above you appears to still or makes a small circle. When at the equator, stars above you appear to follow a straight line from east to west. When away from the equator, stars directly above you appear to follow a curved path. In the Northern Hemisphere, a star will appear a little north of east, rise directly above you, and then set a little north of west. The further away you are from the equator, the more visible is this curvature.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College


Dear Donn,

A good question! First, remember that the pendulum and museum would be just as "joined together" if they were at the north pole as at any other latitude. Then remember that the pendulum would rotate in the opposite direction at the south pole and does not rotate at all at the equator. At other latitudes it rotates at an intermediate speed, given as 2pi sin (L) radians per day where L is the latitude. Note that this is the component of the angular speed of the earth along the vertical at latitude L since the angular velocity vector points up out of the geographical north pole.

I checked my graduate mechanics text from the University of Chicago (Classical Mechanics by Herbert Goldstein -- a beautiful book). On page 139 he mentions the case with the pendulum at the north pole and then says, "At other latitudes the result is more complicated...and detailed calculation will be left as an exercise." I take this to mean that there is no simple handwaving argument for all latitudes.

I hope this is helpful.

Best, Dick Plano, Professor of Physics emeritus, Rutgers University



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