30 August 2002

A Short Driven Foucault Pendulum

By John Dooley

Presented at the first annual Citizen Scientist Conference, Philadelphia, 29 June 2002.

The Foucault pendulum is sensitive to the rotation of the earth on its axis.

In the figure, point B is due north of point A. If the pendulum leaves point A, traveling in a northerly direction, it will arrive later at point B, provided that the earth does not rotate.

If the earth rotates to the east, then point A moves eastward with a greater velocity than does point B. (Both points make a complete circle during one day - one earth rotation - but A has a larger distance to travel, and thus must travel faster.)

If the earth rotates to the east, and the pendulum leaves point A, traveling north, the result is different. Because the pendulum has the same eastward velocity component as point A, it travels faster towards the east than does point B. The pendulum passes to the east of point B.

This argument can be extended. The result is that all horizontal motions are deflected in a clockwise direction, because of the rotation of the earth on its axis.

With every swing, Foucault's pendulum changes the direction of its swing slightly. Each path is deflected clockwise from the previous path.

We observe the plane of oscillation of the pendulum to rotate clockwise. At our latitude, the plane of oscillation changes by about 10 degrees per hour.

Foucault's famous pendulum in the Pantheon near Paris is 67 meters long, and swings most of the day after being started in the morning. (See http://visite.artsetmetiers.free.fr/site_anglais/pendulum_museum_a.html)

A shorter pendulum stops swinging more rapidly, and must be "pumped" with energy to keep it swinging. The pumping mechanism described by Crane (A Foucault Pendulum "Wall Clock,"by Richard H. Crane, American Journal of Physics 63, (1) 33-39 (1995)) uses a magnet, switched on and off, at the lowest point of the pendulum swing.

The pumping mechanism used here was discussed by Landau and Lifshitz (Mechanics, Pergamon Press, Oxford, 1975, page 84, problem 3). It is the same as the mechanism a child uses to "pump" a swing: The mass of the pendulum is raised each time the pendulum passes through the low point, and lowered each time the pendulum reaches its highest point.

 

 

The pivot point is driven vertically, with an amplitude of about 1 mm, using a loudspeaker. The driving voltage for the speaker is produced with the circuit below:

The square wave is derived from the signal sent to the flashing colon of the LED display of a digital clock, (flashing once per second).

In correct operation, the loudspeaker pulls up just before the pendulum swings through its lowest point. This happens twice during each period of a 1 meter pendulum.

The pendulum period is matched to the clock driver signal by changing the length of the pendulum. If the mass of the pendulum bob is raised by 1mm (for a 1 meter pendulum), the period changes by .001 second. Fine adjustment is made using the bob shown at the right.

The PVC pipe cap is threaded onto a piece of lamp hardware with a 1mm pitch. The cap is raised 1mm with a full turn. The cap provides about half the mass of the bob. With fractional turns, the period of the pendulum can be adjusted with precision of about .0001 second.

The adjustment process is comparable to the process of regulating an old fashioned grandfather clock.

 

The pendulum is mounted on the speaker by gluing the top portion of a roll-on deodorant bottle to the speaker. The ball is a quick-disconnect for experimental convenience. The pendulum string is held in a pin vise, which is fixed to the ball with a compression union, as shown at the right.

One version of the pendulum itself is sketched below. The upper part of the string is a nylon guitar string, 1 mm in diameter. The lower part is a cylindrical shoestring, which acts as a vibration damper.

The pendulum pivot must be cylindrically symmetrical so that the pendulum has no preferred direction of oscillation. (A common problem is for the pendulum to rotate clockwise until it stops at a preferred direction.) The pivot point is manufactured by cutting into the nylon guitar string, reducing its diameter from 1 mm to about 0.3 mm.

The pivot is cut with a Dremel grinder, moved against the string as it is turned by a drill press at about 1 revolution per minute.

The pendulum may be viewed live on line at

http://muweb.millersville.edu/~physics/exp.of.the.month/35/

The same site may be found at Google.com, using search phrase "live real time pendulum."