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School Foucault pendulum

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2014 Eur. J. Phys. 35 065023

(http://iopscience.iop.org/0143-0807/35/6/065023)

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School Foucault pendulum

Boris Lacsny, Igor Štubna and Aba Teleki

Department of Physics, Constantine the Philosopher University, A Hlinku 1, 949 74Nitra, Slovakia

E-mail: blacsny@ukf.sk

Received 18 August 2014, revised 2 September 2014Accepted for publication 8 September 2014Published 6 October 2014

AbstractA Foucault pendulum was assembled for a university/high-school physicscourse. The pendulum is 2.85 m long and the mass of the bob is 4.70 kg. Anew technique based on the spark burned points on a sheet of paper was usedto register the pendulum’s motion. A Ruhmkorff induction coil was used as ahigh-voltage source. Plots of the elliptical trajectories (except the first trajec-tory, which is a straight line) can be created and the angles between the majoraxes of the ellipses can be determined after assigning the coordinates to theburned points.

Keywords: Foucault pendulum, school experiment, pendulum trajectory

1. Introduction

A Foucault pendulum illustrates the Earth’s daily rotation without the need to observe the sky.The theory of the Foucault pendulum is based on the solution of the equation of the motion ofthe bob, which is subjected to both the force of the gravity and the Coriolis force [10, 11]. Anexplanation of the oscillations of the Foucault pendulum with almost no mathematics neededis given in [12, 13].

In the past, there have been some attempts to design and construct a Foucault pendulumfor school laboratories [2–7]. Registering the pendulum’s motion is an important task. Toevaluate the trajectory of the pendulum’s bob, it is desirable to have coordinates of its points.The registration of the pendulum’s motion with a laser beam is mentioned in [2, 3], andregistration of the motion using a Hall sensor is described in [1]. The simplest anchoring of asmall Foucault pendulum uses either a stylus on a conical support [8], a hardened steel ball onan alumina plate, [7] or a cobalt steel ball and a cobalt steel disk, which are both polished tomirror grade. [9] A Charron collar with a stabilization slit for the wire was used in [5, 6]. It isalso important to have a bob of sufficient mass, so it has enough mechanical energy tomaintain oscillations for the time during which an experiment is conducted [5–7]. If a

European Journal of Physics

Eur. J. Phys. 35 (2014) 065023 (5pp) doi:10.1088/0143-0807/35/6/065023

0143-0807/14/065023+05$33.00 © 2014 IOP Publishing Ltd Printed in the UK 1

Foucault pendulum is set in motion, initially the trajectory of the bob is a straight line in thehorizontal plane. This trajectory gradually changes into an ellipse.

This paper describes the design and construction of a Foucault pendulum suitable forlaboratory work in high schools and universities.

2. Design of the pendulum

As follows from theory (which assumes a small amplitude of the pendulum oscillation), theangular velocity of the rotation of the plane of the oscillation is ω ω φ= sinE , where ωE isthe angular velocity of the daily rotation of the Earth (relative to stars), and φ is the latitude atthe location of the experimental setup. For the Earth, ωE= 7.2921 × 10

−5 rad s−1, and in ourcase, φ = 8.4314 rad (48° 18.5’). Therefore, ω= 1.9603 × 10−5 rad s−1. The angular deflectionof the plane of the oscillation is 11.232° in 1 h.

The pendulum is suspended along the wall, where a cantilever for the anchorage is fixedboth to the wall and to the ceiling to make the anchorage firm. The anchorage is similar to thatshown in [7]. A steel rest with a mirror-polished alumina plate (20 × 20 × 5 mm), is fixed tothe cantilever with three leveling bolts, which can adjust the horizontal position of thealumina plate, as seen in figure 1. A brass ring with a hardened steel ball (∅6.0 mm) is put onthe alumina plate. The steel wire (∅1.0 mm) is fixed to the ring on its lower side. The bob is asteel cylinder of ∅8.0 × 12 cm, with a mass of 4.70 kg and a spike on its lower side. Thelength of the pendulum is 2.85 m, its period is T≈ 3.39 s, and the initial amplitude of theoscillation of the bob in the horizontal plane is 15 cm when the angular amplitude is 3°.

How the Foucault pendulum is set in motion is also important. The pendulum must hangfreely for a sufficient period of time in order to reach a steady position without revolving thebob. Then the bob is carefully deflected from equilibrium with a thread, and the thread isburned.

Figure 1. The anchorage of the pendulum. (1) Ring, (2) steel ball, (3) alumina plate,(4) steel rest, (5) leveling bolts, (6) cantilever, (7) spark.

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To register the trajectory of the bob, we used a new technique: sparks generated by ahigh-voltage source. A Ruhmkorff induction coil is a very suitable device for this purpose.The intensity of the spark can be adjusted with the output voltage from the direct currentpower supply. Using common leads, the output of the induction coil is connected both to ametallic plate on which the paper rests, and to the ring of the anchorage. A thin, soft wire mustbe used for the connection to the ring. Another way is using the spark, as displayed infigure 1.

When the pendulum swings, the sparks burn little holes into the sheet of paper that restson the metallic sheet, as seen in figure 2. These holes are very visible against light (e.g., on thewindow), where they can be marked. The coordinates of the holes serve as the derivation ofthe equation of the pattern burned into the paper. The first line pattern, which is straight, isburned immediately after the release of the pendulum. The next patterns, which are elliptical,are burned after selected time intervals. For example, the results obtained with the Foucaultpendulum in 30 min intervals are depicted in figure 3. We measured the revolution of10.8° ± 0.7° (obtained for five measurements), which is close to the theoretical value of 11.2°in 60 min (i.e., the relative error is ∼4%).

The electric field strength for an electric breakdown in the air depends on the distancebetween the electrodes. If the distance varies from 3–5 mm, the field strength varies from3.3 kVmm−1 to 3 kVmm−1. On the other hand, too long a distance requires higher voltageand can lead to an inaccuracy in the burned trajectory. Too short a distance also requires

Figure 2. The registration of the pendulum trajectory. (8) Fixing the wire, (9) steelcylinder (bob), (10) spike and spark, (11) sheet of paper on the metallic plate.

Figure 3. (a) The record of the pendulum trajectory from burned points (recorded in30 min intervals). (b) The angles between the major axis of the elliptical trajectories andthe initial straight trajectory (horizontal line).

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higher strengths (e.g., a 1 mm distance requires 4.5 kVmm−1 [14]). A distance of 3–5 mmbetween the spike and the paper sheet during the pendulum swing seems to be optimal.

3. Methodical notes

The experiment with a Foucault pendulum is suitable for a basic university physics course ora high-school physics course. The designed pendulum and recording apparatus can beassembled from common tools and devices. The length of the pendulum is given by the heightof the laboratory/classroom. The necessary duration of the experiment is more than 1 h, andthe trajectories should be recorded every 20–30 min. To obtain one trajectory, two or threeperiods are sufficient. We do not recommend changing the paper sheet after every registrationof the trajectory. It is clear that touching the bob or wire during the oscillation is stronglyprohibited, as it disturbs the oscillations. The high voltage is used in this experiment onlyduring the registration procedure, and the instructions that are valid for the Ruhmkorffinduction coil are valid here.

A simple graphical method can be used to process the results. The trajectory, which iscreated from burned points, can be scanned and then opened in a graphic program (e.g.,CorelDraw, InkScape, MS Paint, Adobe Illustrator). A straight line is put through the first setof points. An ellipse is put through the next set of points, which were obtained after20–30 min. The major axis of the ellipse and the measurement of the angle between the axisand the straight line are visible in figure 3. Every student does this independently, and thefinal angle is obtained with the help of the instructor. The student determines the coordinatesand then plots graphs of the point sets (which are ellipses, except for the first set, which is astraight line) in a spreadsheet program, such as MS Excel or Open Office Calc. Then, thedetermination of the angles between the linear trend lines of the ellipses can be completed.

4. Conclusions

A Foucault pendulum for a basic university physics course or a high-school physics coursewas assembled. The length of the pendulum is 2.85 m and the mass of the bob is 4.70 kg. Anew technique, based on the spark-burned points on a sheet of paper, was used to register thependulum’s motion. A Ruhmkorff induction coil is a very suitable device for this purpose. Ifthe pendulum trajectory is registered in 20 to 30 min increments, two or three sets of burnedpoints are obtained. After assigning coordinates to the points, plots of the elliptical trajectories(except the first, which is a straight line) can be completed, and the angles between the majoraxes of the ellipses can be determined. A difference between theoretical and experimentalvalues of the angular deflection of the plane of the oscillation is 0.4°, which represents arelative error of 4% in 1 h.

Acknowledgements

The design and construction of the pendulum was supported by the grants KEGA UKF61-4/2012 and UGA VI/30/2012. The authors also thank K Mitterpach for his technical help.

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