Foucault Pendulum in Middle Latitudes
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.
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.
This web site does the best job that I know of to explain why this works:
Todd Clark, Office of Science
US Department of Energy
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
Dr. Ken Mellendorf
Illinois Central College
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
Click here to return to the Physics Archives
Update: June 2012