# The Foucault Pendulum in Lesson Plans

This June, the department conducted a summer workshop for 9 physics teachers on experimental evidence for the rotation of the earth on its axis. In this workshop, each teacher built and took home a driven Foucault pendulum, whose plane of oscillation rotates because the earth rotates under the pendulum. In the photograph above, the participants have built their pendulums and have begun "tweaking" the length to match the 1Hz driver frequency. 5 pendulum bobs are visible in the photo.

This is the third in a series of workshops with the theme that students should believe physics evidence rather than physics authorities. The first two, on quantum mechanics and relativity, were the outgrowth of a National Science Foundation instructional laboratory instrumentation grant. This year's was not part of the NSF proposal for two reasons:

1. The cost of the Foucault pendulum that each teacher built was less than \$200; an insignificant cost as grants go.
2. At the time of writing the NSF proposal we had no indication that we could successfully build a low cost Foucault pendulum. In fact we could cite good reasons why it was not possible.

As part of their work, the teachers presented lesson plans that require students think critically about why they believe that the earth turns on its axis once per day. The Foucault pendulum is the only ordinary experiment that provides evidence in support of this model. (It was mentioned that this topic addresses PA standards 3.4.12.C and 3.1.12.C.) The following is a summary of the participants' ideas.

• Simple Harmonic Motion and Driven Oscillators
Students are given a tour of simple harmonic motion with mass on spring and the pendulum, with and without damping. The idea of pumping a swing is used to understand how the driven Foucault pendulum works. The idea of latitude is introduced. The rate of precession is estimated and compared to the students' calculation for the predicted result.
• How to drive a Foucault Pendulum
Each student hangs a pendulum from his/her neck and walks around the room, seeking the kind of motion that makes the pendulum swing. Translate the discussion to the pumping mechanism for the demonstration pendulum. Test a model in which the sun rises and sets because the earth rotates. Calculate the rate of rotation of the earth according the this model (about 0.00007 radian/second or .26 radian/hour or about 15 degrees/hour). Invite students to stop by the physics room throughout the day to monitor the precession of the pendulum and estimate its rate. Details to be discussed in a follow-on lesson.
• Damping Coefficient and Resonance Width
For calculus based physics, students test the exponentially damped pendulum position solution by plugging it into the differential equation of motion, including a viscous damping force proportional to velocity. They then measure the amplitude of the (undriven) demonstration pendulum as a function of time, to determine a best vale for the exponential decay constant, b. The driven resonance amplitude (as a function of driver frequency) is presented and used to establish the idea of the width of the resonance. The width is measured for the (driven) demonstration pendulum by varying the driver frequency and measuring the pendulum response amplitude. The width is compared to the decay constant. (Roughly, the amplitude falls to half its maximum when the frequency differs from the resonant frequency by b/2 . This experiment was performed during the workshop.) Students interested in a project would measure the variation of b with the mass of the bob. (Increasing the mass while keeping the volume and shape the same will decrease b .)
• The Ideas of the Foucault Pendulum
For students taking conceptual physics: The pendulum is set up with the driving mechanism hidden by a cover. Student expectation that the pendulum will die down is discussed, and students are invited to speculate on the nature of the driving mechanism. (Horizontal motion of the pivot will work, provided that the drive frequency equals the pendulum frequency. Vertical motion of the pivot, as used here, works provided that the drive frequency is twice the pendulum frequency.) Demonstrate the idea of the Foucault pendulum with a small pendulum mounted on a rotating platform. Invite observation of the driven pendulum over a long time period, to see if evidence appears for the earth's rotation on its axis.
• Joint Project for a Physics Team and a Technology Education Team
The Physics Team studies the problem of detecting the earth's rotation and the solution developed by Foucault (a 67 meter long pendulum in the French Pantheon - click for 0.7MegaByte photograph by Dr. Grosh). They define a list of needs, such as smaller size, a driver to compensate for wind resistance, and "perfect" cylindrical symmetry to avoid preferred direction of oscillation. The TE team finds solutions to these needs, and iterates them with the physics team, asking for example just how "perfect" the cylindrical symmetry must be. After construction the two teams work together to tweak the system into reliable performance.
• The Foucault Pendulum as a Problem in (Classical) Relativity
Individual pendulums are swung above circular maps representing the north pole region of the earth. The maps are rotated and students describe the motion of the pendulum as seen in the laboratory, and as seen by an observer rotating with the map. Imagine repeating the experiment at the equator (no precession) rather than at the north pole (1 full rotation of the plane of the pendulum in 24 hours). The class will track the motion of the driven demonstration Foucault pendulum for one week. Students critically evaluate the proposition that they have "proved" that the earth rotates on its axis. They have a model which works, but that fails to prove that another (unknown) force is not the cause of the precession. It is fair to say that the model can be accepted until it is shown to fail.
• Resonance and the Driven Pendulum.
Students, already experienced with pendulums, observe the driven demonstration pendulum and speculate on why it does not die down. The driving mechanism is discussed, and the issue of choosing the correct driver frequency is raised. Students set up four pendulums with three different lengths (The first two pendulums are identical). If the first pendulum sets the driver frequency, students can see that the driver will succeed at pumping energy into the identical pendulum, but will fail (in the long run) to pump energy into the different pendulums. The driver in this case should have the same frequency as the pendulum. The unusual nature of the demonstration drive is discussed: Because it pumps vertically (so as not to give the pendulum a preferred horizontal direction), this driver must run at twice the pendulum frequency. Some of the mystery is removed by discussion of how to pump a swing.
• Elementary Concepts of Motion.
After examples and simulations of periodic motion, discuss frequency, period, amplitude, acceleration and force. Emphasize the remarkable nature of the pendulum period by dropping two balls, first from the same height and then from much different heights. The sounds indicate that the shorter fall leads to shorter time. Repeat with two pendulums tacked to a wall, pulled back and released at the same time. The sound indicates that the shorter swing takes just the same time as the long swing to reach the wall. The time of travel is independent of the travel distance, for a pendulum. The forces involved in normal and driven pendulum oscillation are discussed. Note that 1 meter is the time for a pendulum clock with a 1 second "tick."
• A Pendulum in a Rotating House.
Students imagine a pendulum hung from a crane into a hole in their roof at home. Extending the fantasy, the house begins to rotate slowly. In what way would the pendulum behavior differ from pendulums studied in class? They then try the experiment in miniature, with rotating paper under the pendulum to represent the house. Repeat with the pendulum mounted at the center of a rotating platform. Use these images to interpret the behavior of the demonstration Foucault pendulum.
• Instructor's Favorite.
In the northern hemisphere, a Foucault pendulum plane of oscillation rotates clockwise because the earth rotates counterclockwise (as viewed from above the north pole). In addition to the Foucault precession, another is possible: elliptical precession. If the pendulum path (as projected on to the floor) is elliptical, then the ellipse precesses. If the pendulum travels clockwise around its elliptical path, the precession is also clockwise. The driving mechanism used here couples to the Coriolis force so as to force the elliptical motion into a clockwise path. Thus the elliptical precession always augments the Foucault precession. The effect may be demonstrated by forcing the pendulum into a counterclockwise elliptical path, by hand, when the pendulum is first set into motion. The motion becomes planar, because (absent the Coriolis force) the pumping mechanism drains energy from the minor axis motion at the same time that it adds energy to the major axis motion. Finally, after the better part of an hour, the motion is controlled by the Coriolis force and is seen to precess clockwise.