A Short Foucault Pendulum Free of Ellipsoidal Precession
aa r X i v : . [ phy s i c s . c l a ss - ph ] F e b A Short Foucault Pendulum Free of Ellipsoidal Precession
Reinhard A. Schumacher and Brandon Tarbet
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 ∗ (Dated: February 12, 2009)A quantitative method is presented for stopping the intrinsic precession of a spherical pendulumdue to ellipsoidal motion. Removing this unwanted precession renders the Foucault precession dueto the turning of the Earth readily observable. The method is insensitive to the size and directionof the perturbative forces leading to ellipsoidal motion. We demonstrate that a short (three meter)pendulum can be pushed in a controlled way to make the Foucault precession dominant. The methodmakes room-height or table-top Foucault pendula more accurate and practical to build. I. INTRODUCTION
L´eon Foucault built his first pendulum to demonstrate the turning of the Earth in the basement of a building,using a roughly two meter long fiber [1]. He also soon recognized the problem arising from the intrinsic precessionof a spherical pendulum caused by unwanted ellipsoidal motion. Imperfections in the suspension or initial conditionsof the pendulum generally cause this to quickly grow to the point that the precession due to the Earth’s turning isoverwhelmed. The pendulum can come to precess in either sense (clockwise or counterclockwise) at almost any rate,or indeed even cease all precession. These practical problems are mitigated in pendula of great length, and so mostare constructed to have lengths of tens of meters, starting with the celebrated 67 m long device built by Foucault inParis in 1851.The precession of an ideal spherical pendulum with no ellipsoidal motion is caused by the non-inertial nature of thereference frame tied to the surface of the Earth. The so-called Coriolis force advances the plane of the pendulum’smotion by an amount Ω F = Ω Earth sin θ latitude (1)where Ω Earth is the sidereal rate of rotation of the Earth, θ latitude is the latitude of the pendulum measured from theequator, and Ω F is the Foucault precession rate; many textbooks treat this problem. Near 40 o northern latitude, thisamounts to almost 10 o of clockwise advance per hour, for an 18 hour half-rotation of the pendulum, after which themotion repeats. This amount of precession is easily masked, however, by the intrinsic precession of a less than perfectpendulum that develops some non-planar ellipsoidal motion.The construction of a room-height or table-top version of a Foucault pendulum thus presents a technical challenge,first to minimize the amount of ellipsoidal motion that accrues as the pendulum swings, and secondly to compensatein some way for the irreducible amount of this motion that remains. In this paper we will first discuss the dynamics ofthe spherical pendulum that lead to the problem (Section II). Then we introduce a method that stops the ellipsoidalmotion of the pendulum from causing precession, and show that this immunity is, to first order, independent of theminor axis of the ellipse. The method hinges on the observation that pushing the pendulum bob away from theorigin after it passes, rather than either pulling it in or alternately pulling and pushing it, acts in a way to counterthe unwanted intrinsic precession (Section III). We then present the design of a pendulum and a driving mechanismto exploit this method (Section IV), and demonstrate the validity of this approach by discussing the supportingexperimental results (Section V). Finally, we contrast our results with earlier published work on Foucault pendulaand summarize how our method and design are new and unique (Section VI). II. THE PROBLEM OF INTRINSIC PRECESSION
The dynamics of the idealized spherical pendulum are determined by the centrally-directed force of gravity andinitial conditions, and lead to approximately elliptical motion with a semi-major axis a and a semi-minor axis b ,as shown in Fig. 1. It is somewhat counter-intuitive but true that the centrally-directed restoring force of gravityresults in a constant intrinsic precession rate Ω about the z axis. This precession arises from the symmetry-breaking ∗ Electronic address: [email protected] condition of having a finite minor axis, b . The rate is linearly proportional to b , and its sense is in the same directionas the ellipsoidal motion. An example of the exact path of a pendulum with a large ratio of b/a is given in Fig. 2,which illustrates a numerical solution of the equations of motion of a spherical pendulum. FIG. 1: (color online) Planar view of the approximate path of a spherical pendulum with semi-major axis a and semi-minoraxis b that is moving in a counterclockwise ellipsoid. The suspension is centered on the z axis above the origin. The pendulumis precessing at rate Ω, and in one full cycle the apex advances by a distance ∆ y , as suggested by the light dotted and rotatedellipse. The impulsive driving force is applied at x = d , and it is resolved into components parallel and perpendicular to themajor axis. The minor axis can be larger or smaller, resulting in a b -dependent magnitude of the transverse force F ⊥ for afixed longitudinal force F k .FIG. 2: (color online) Numerical simulation (using Mathematica) of a spherical pendulum with a large ratio of semi-minor tosemi-major ellipse axes. The arrows indicate the counterclockwise intrinsic precession, in the same sense as the motion of thependulum. Any ellipsoidal motion that develops in a pendulum will result in an intrinsic precession rate Ω that is whollyunrelated to the Foucault precession Ω F of Eq (1). This has been worked out in detail by Olsson [2, 3] and Pippard [4],among others, and it appears in some textbooks, for example in Ref [5]. The main result isΩ = 38 ω abL , (2)where ω = 2 π/T = p g/L is the pendulum’s angular frequency, L is the length, T is the period, and g is theacceleration due to gravity. The formula is the lowest-order term in a complicated motion, but it is easily sufficientfor our purpose. The area of an ellipse, A , is given by A = πab , so another way to write the intrinsic precession rateis Ω = 34 AL T . (3)From the first expression, note that the ratio of Ω (the very slow intrinsic precession) to ω (the rapid pendular rate)is proportional to the ratio of the area of the ellipse to the area of a sphere of radius L , that is,Ω ω = 32 ( πab )(4 πL ) (4)To minimize this ratio, Foucault pendula are generally made with L very large compared to the axes a and b , sinceit is comparatively easy to keep b small while making L large.Besides causing precession, ellipsoidal motion changes the central oscillation frequency of the pendulum from ω to ω = ω (cid:18) − a + b L (cid:19) . (5)The fractional change in frequency due to a finite semi-minor axis b turns out to be of order 10 − , and thereforenegligible for the present discussion.Every Foucault pendulum, no matter how carefully constructed to avoid asymmetries in its suspension, and nomatter how carefully “launched” to make ellipsoidal area A as small as reasonably achievable will, over time, acquirean intrinsic precession Ω that can easily grow to overwhelm the Coriolis-force induced Foucault precession Ω F . Near40 o latitude, for a pendulum of length L = 3.0 meters and semi-major axis a = 16. cm, the Foucault rate is equaledwith a semi-minor axis of b = 3.9 mm. The unwanted intrinsic precession may add or subtract from the Foucaultprecession; when it is subtractive, then the pendulum stops all precession when the corresponding value of b is reached.In our experience, it is easy to launch such a pendulum with semi-minor axis well under one millimeter, but underfree oscillation, on the time scale of only two to three minutes, b grows enough such that Ω dominates Ω F .When the pendulum at an extremum, at x = ± a in Fig. 1, its motion is entirely transverse, with momentum m ˙ y as large as it gets. Preferential damping of this component of the motion will reduce the unwanted ellipsoidalexcursions. In previous work, this has been tried using a so-called Charron’s ring around the suspension wire nearthe top [6, 7], or letting a part of the pendulum bob scrape an annular disk [8, 9] at r = a , or using eddy currentdamping between a permanent magnet in the bob and a non-ferrous metal annular disk [10] located near x = a . Weadopt the touch-free eddy-current damping method in the design we present later in this paper. Even in principle,none of these methods will stop ellipsoidal motion completely, so an additional method is needed to cope with anyremaining intrinsic precession.Every pendulum suffers dissipative losses of energy, mainly due to air friction. Long, very massive, museum-typependula can simply be relaunched once every day or so, but a small pendulum has a free exponential decay time oforder one half hour, so a “driving” mechanism is needed to restore the lost energy. Various mechanisms have beenreported for this task [8, 9, 10, 11]. Like some others, we will use magnetic induction to sense the passage nearthe origin of a permanent magnet embedded in the pendulum bob. A carefully-timed electromagnetic impulse thenimparts lost momentum to the bob. We will introduce a quantitative method for using a magnetic push not only tocompensate for dissipative losses, but also to compensate for ellipsoidal precession.The perturbations that lead to ellipsoidal motion are, in our experience, in approximate decreasing order of severity,(1) internal stresses or other imperfections in the fiber supporting the pendulum bob, (2) less than perfectly symmetricsuspension of the fiber at its upper end, (3) nearby iron objects that result in asymmetric force on the drive magnetin the bob, (4) a driving coil that is not sufficiently level and centered under the pendulum. All but the first of thesewere straightforward to reduce to insignificance, but the first was persistent. This led to the necessity of finding amethod to evade the problem of ellipsoidal motion rather than to remove it. There is no simple force law that leadsto the intrinsic precession Ω, but rather it is an inescapable feature of the spherical pendulum. Nevertheless, we canapply a separate perturbative force to counteract, i.e. nullify, the intrinsic precession. We discuss this in the nextsection. III. METHOD TO NULLIFY INTRINSIC PRECESSION
As shown in Fig. 1, the driving force that pushes the pendulum away from the origin can be thought of as consistingof components parallel and perpendicular to the major axis. The parallel component F k is the larger one, and it isadjusted to overcome the dissipative forces such as air resistance. The perpendicular component of the pushing force F ⊥ counteracts the intrinsic precession, and we will now show that it is possible to select distance d at which theimpulsive drive force is applied to stop the precession Ω. The crucial result will be that this distance is independentof semi-minor axis b . Starting from x = − a , in one half cycle of duration T /2, intrinsic precession advances the ellipsein angle by Ω T /2. The pendulum arrives at x ≃ + a and y ≡ ∆ y = 12 a Ω T, (6)where ∆ y is the transverse displacement at the apex of the ellipse. However, we arrange to apply an impulsivemomentum change at x = d that is calibrated to move the pendulum bob by a distance − ∆ y as it traverses theremaining longitudinal distance a − d . If this is done, the bob will arrive at location x = a , y = 0, as desired. Theimpulse and momentum change are related by m ∆ v y = F ⊥ ∆ t, (7)where ∆ t is the duration of the applied force and m is the mass of the bob. It turns out that ∆ t is a few milliseconds,compared to hundreds of milliseconds for the duration of the swing, so it is appropriate to use the impulse formulationof Newton’s second law. We treat this perturbation in the y direction as though the bob were free of other forces,which is reasonable since ∆ y is of order 10 microns compared to a of order 10 centimeters. The horizontal componentof the motion is simply x ( t ) = a sin ω t, (8)where ω is the angular frequency characterizing the pendular oscillations. To good accuracy, the time t d betweenpassage closest to the origin and the instant the pendulum reaches x = d is thus t d = 1 ω sin − da . (9)In one quarter of the full period T the pendulum reaches its apex. Therefore, one way to write a relation betweenthe distance ∆ y that the pendulum advances and the transverse velocity ∆ v y that must be imparted by the driver is∆ y = ∆ v y (cid:18) T − t d (cid:19) = ∆ v y ω (cid:18) π − sin − da (cid:19) = ∆ v y ω cos − da (10)From the geometry of the situation shown in Fig. 1 we see that that the driving force components are related bytan θ = yd = F ⊥ F k , (11)and, using the formula for an ellipse ( x/a ) + ( y/b ) = 1 , (12)we have F ⊥ = F k bd s − (cid:18) da (cid:19) . (13)Combining Eqs (10), (7), (13), (6), and (2) we arrive at∆ y = 12 a (cid:18) ω abL (cid:19) (cid:18) πω (cid:19) = F k ∆ tmω bd s − (cid:18) da (cid:19) cos − da . (14)One sees in Eq (14) the crucial cancellation of the factor b that occurs between the intrinsic precession rate and inthe perpendicular component of the force. This leads to the result that follows being independent of the transversesize of the ellipse. This, in turn, means the result is insensitive to exactly what non-central forces may act on thependulum to cause non-vanishing ellipsoidal motion. An approximation is being made that F k does not depend on b ;this will be justified later.It is now convenient to introduce some dimensionless scaling parameters. Let Q ≡ π maω F k ∆ t , (15)which is the ratio of the momentum of the pendulum as it crosses the origin to the momentum kick it receives on eachhalf oscillation. The denominator is the momentum the driver must supply to compensate for the dissipative losses inorder to maintain the full amplitude of the swing. The factor of π/ Q represent the conventional“quality factor” of an oscillator, specifically that Q = 2 π Total EnergyEnergy Loss per Cycle . (16)We expect this ratio to be quite large, on the order of 1000. Then let α = aL (17)be the scaled amplitude of the pendulum, that is, the amplitude divided by the length; this parameter is of order 0.1for a practical pendulum. Next, let δ = da (18)be the scaled distance from the origin at which the driving force is applied, written as a fraction of the amplitude;this value can range from near zero to unity. With these dimensionless variables, Eq (14) can be written34 Qα = √ − δ δ cos − δ (19)Equation 19 is the main result of this model. It relates the scaled distance δ at which the impulsive driving forcemust be applied on each half oscillation, to the physical parameters of the pendulum, specifically the scaled amplitude α and the quality of the oscillator Q . When Eq (19) is satisfied, the intrinsic precession is stopped or nullified, andthe result is independent of the transverse size of the ellipse. Figure 3a is an illustration of this result for the cases ofthree different lengths of pendulum, each with a maximum excursion of 0.15 meters, as a function of the oscillationparameter, Q . It is seen that a low-loss pendulum with a larger value of Q will have to be pushed when it is closerto the origin, while a pendulum with more dissipative losses, and therefore lower Q , will have be driven further awayfrom the origin. A high-quality, low-loss pendulum needs little energy input. Therefore, the transverse kick, whichscales in magnitude with the longitudinal kick at a given δ , must be applied early in the ellipsoidal swing, when theforce has a larger transverse component. Part (b) of Fig. 3 illustrates the relationship between the location of theimpulse, δ , and the time, t d , at which it is applied, as per Eq (9).Taken together, the two parts of the Fig. 3 enable design of a Foucault pendulum not plagued by intrinsic precession.For example, a 3.0 meter long pendulum with a 0.15 meter amplitude ( α = 0 . Q = 1000must receive its impulsive drive at δ = .
49, which corresponds to time t d = 284 msec and a drive location of d = 7.4cm from the origin. On the other hand, a 1.0 meter long pendulum with the same α and Q must receive its drivepulse at δ = . Q is proportional to the ratio of the momentum of free oscillation at the origin to thedamping/driving momentum that accrues with each half period of oscillation. In fact, it is quite easy to measure thisparameter for an actual pendulum by measuring the period, T , and the exponential decay time, τ . The motion of thependulum in the presence of velocity-dependent losses such as air friction is given, to a good approximation at lowspeeds, by the damped harmonic oscillator formula x ( t ) = ae − t/τ sin ω t. (20)The momentum loss between times t = 0 and t = T /2 is m ˙ x (0) − m ˙ x ( T /
2) = maω (1 − e − T/ τ ) ∼ = maω T τ . (21)This leads to Q = π τT , (22) FIG. 3: (color online) (a) For three pendulum lengths ( L ) with amplitude of 0.15 meter, the relationship of the scaled drivingdistance ( δ = d/a ) versus the quality factor Q for the oscillation. Curves are for an L = 3.0 meter (solid blue), 2.0 meter(dashed red) and 1.0 meter pendulum (dotted green). (b) For the same pendulum lengths as in (a), the distance versus timerelationship for the driving pulse. from which Q can be determined experimentally. For the actual three meter pendulum discussed below, we foundthe decay time to be about 30 minutes and hence Q to be roughly 1600. Eq (22) can be used with Eq (19) in orderto express Eq (19) in dimensioned variables as38 r gL (cid:16) aL (cid:17) τ = √ a − d d cos − ( d / a ) . (23)One important approximation that was made needs to be examined. In going from Eq (14) to Eq (19), thecancellation of the semi-minor ellipse parameter b was crucial to showing that the result is independent of inevitablechanges in the size of the pendulum’s ellipsoidal motion. In fact, F k does depend very slightly on b in the design wewill discuss in the following sections. The pushing agent is a magnetic coil that produces a pulsed dipole field. Itacts against a permanent magnetic dipole inside the pendulum bob. Though we operate in the near field of the coil,we presume that the magnetic field has a distance dependence of the dipole form B ( r ) = B ( r /r ) , where B and r are suitable scales. The dipole-dipole repulsion that drives the pendulum goes as the gradient of this field, so theinteraction has the form ~F = ~F ( r /r ) , (24)where the components of ~F are what we earlier called F ⊥ and F k . It is easy to expand this expression to show that F k = F (cid:16) r d (cid:17) ( − (cid:18) bd (cid:19) (cid:0) − δ (cid:1)) (25)As seen in the second term in the curly brackets, there is a quadratic dependence on the semi-minor axis b thatwe ignored earlier. This term is very small: for typical values of δ = d/a ≃ . cm/ . cm = 0 . b/d ≃ . cm/ . cm = 0 . IV. EXPERIMENTAL SETUP
To verify the mathematical model discussed above, a pendulum of about three meter length was built. Themechanics and drive electronics will be described here. An image of the lower end of the setup is shown in Fig. 4.
FIG. 4: (color online) Image of the copper pendulum bob suspended above the sensing and driving coils (red wire on a whitebobbin). The aluminum damping ring on brass legs is seen. The white band carries angular position markings. The driver boxand an oscilloscope are visible in the background.
The mass of the pendulum was in the form of a lathe-turned copper cylinder, with a diameter of 2.75” (6.99 cm) andheight 3.00” (7.62 cm). A 3/8” (.953 cm) axial hole was drilled and tapped to accept a small threaded set screw witha 0.040” (0.10 cm) hole in its center. The small hole was for the fiber suspending the pendulum, and the threadingwas used to adjust the distance between the center of mass and the suspension point. In practice, the fiber was heldflush with the top surface of the bob. The bottom of the cylindrical bob was chamfered by 0.10” (0.25 cm) to allowcloser approach to the damping ring. To hold the permanent magnet, a recess was milled into the bottom of the mass,of diameter 3/4” (1.90 cm) and depth 1/4” (0.635 cm), which were the dimensions of the magnet. The total mass ofthe pendulum bob, apart from the magnet, was 2.47 kg. This is a small mass compared to what has generally beenreported for Foucault pendula.The fiber suspending the mass was an ordinary and inexpensive polymer material of the type used for yard trimmers.The diameter was 0.040” (0.10 cm). The cross section of the fiber was circular and the diameter was uniform to betterthan 1%. The polymeric material was thought to be advantageous due to its lack of crystalline internal structure. Itwas found that the polymer material was very flexible, yet strong, and did not suffer the bending fatigue and eventualfailure of some metal fibers we tried. In practice, we found that the growth of elliptical motion of the pendulum wasdominated by an unobservable non-uniformity of this fiber, but it was less severe than with various metal fibers. Testswherein the fiber was rotated without changing any other aspect of the pendulum showed this to be the case. Thermalexpansion and contraction of the fiber was large enough to necessitate occasional shimming of the pendulum’s lengthat the level of about one millimeter.The upper end of the fiber passed through a close-fitting drilled hole in an aluminum plate. The hole had a sharprim, and no special steps were taken to soften the bend of the fiber as it exited the hole. On the upper side of theplate the fiber was clamped in a way that thin shims could be inserted to fine tune the length. The plate was leveledand clamped to rigid brackets on the ceiling of the laboratory. The upper support of the fiber used in this setup,though carefully arranged, was certainly less exacting than what has been found to be necessary for other Foucaultpendula reported in the literature. We view this as an advantage of our design.The permanent magnet placed inside the pendulum bob was a neodymium-iron-boron disk magnet, placed with itsdipole axis vertical. The axial field strength was 2.1 kGauss on contact, but it varied by ±
10% from one edge of thedisk to the other. Similarly, the radial field was 1.0 kG at the edge of the disk, but varied also by ±
10% from oneside to the other. Surprisingly perhaps, these variations did not affect the performance of the pendulum.One purpose of the magnet was to provide eddy-current damping when the bob passed over an aluminum ring nearthe extrema of its motion. This “damping ring” had an inner diameter of 11 7/8” (30 cm), and was 1/4” (0.64 cm)thick. Three legs made of threaded brass rod were used to level the ring to a precision of less than half a millimeterof variation around the perimeter. The amplitude of the pendulum was adjusted such that the magnet passed overthe inner edge of the ring with a clearance of three to four millimeters. This maximized damping of the unwantedtransverse precessional speed. Since the eddy current damping force is velocity dependent, it can never entirely stopthis motion, but our experience was that it limited the ellipsoidal motion to a semi-minor radius of less than halfa centimeter. As a side note, L´eon Foucault first identified the eddy-current phenomenon in 1851, thus we use twophenomena in this investigation that are attributed to him.Two concentric coils under the pendulum sensed and controlled its motion. Both coils were wound from 22 AWGcopper wire on polyethylene bobbins, and both had 240 turns. The inner coil was designated as the “drive” coil usedfor pulsing the pendulum on each half cycle. It had a mean radius of 0.80” (2.0 cm) and a total length of 100’ (30m) of wire. The outer coil was the “sense” coil for detecting the approach of the pendulum. It had a mean radius of1.5” (3.8 cm) and a total length of 187’ (57 m) of wire. The supply leads to both coils were twisted pair copper wirenear the coils, and RG-174 coaxial cable at distances where the steel ground braid of the latter no longer perturbedthe motion.The concentric sense and drive coils were supported by a small aluminum platform that sat on the main table,supported by three brass screws with knurled heads. These were used to level the coils: imperfect leveling causedthe induced signal from the pendulum, described below, to be asymmetric. The clearance between the bottom of thependulum and the top of the coils was 6 ± V − = −
34 mV, as viewed across 1 M Ω input impedance on an oscilloscope. When the pendulum was centered overthe coil, the voltage crossed upwards through zero, followed by a positive excursion of inverted shape and the samemagnitude V + as the negative part. The peak magnitudes of the waveform, V ± , depended on the speed of thependulum (and hence the amplitude of the swing), as well as the distance between the pendulum magnet and the coil.Monitoring V ± over time was a sensitive way to detect small changes in the pendulum’s amplitude and/or length. FIG. 5: Induced voltage at the input stage of the driver circuit, due to the magnet in the pendulum approaching, passing over,and then receding from the sense coil. Upper trace (1) shows a complete cycle of two passages. Lower zoomed trace showshow on the trailing end of the sense pulse (1) the drive pulse (2) causes a large bipolar induced pulse in the sense coil. Screencapture was from a TDS 3032B oscilloscope.
The pendulum driver circuit developed for this study is shown in Fig. 6, and operated as follows. The initial signalfrom the coil was filtered with a 1 µ F capacitor that reduced RF noise to less than one millivolt. An op-amp circuitof type LM324N level-shifted and amplified the signal by a factor of 200. This signal was the input to a SN7414NSchmitt trigger that switched at 0.9 and 1.7 Volts. In effect, this was a discriminator with a TTL output “trigger”pulse when the induced voltage went more negative than −
20 mV; the trigger stayed high until the trailing end ofthe sense-coil signal shown in Fig. 5 fell. A NAND gate output was used to trigger two 555 timers called DELAY andINHIBIT; the second input of the NAND was an inhibit signal that ensured that the DELAY timer was not restartedby the off-scale pulse induced by activation of the drive coil. The DELAY timer set the interval between the approachof the pendulum and the drive pulse in the second coil; it was adjustable in the range from near zero to several hundredmilliseconds. The INHIBIT timer duration was set to be longer than the maximum possible delay plus drive times.The output of the INHIBIT timer was fed via an inverter back to the NAND, so the drive pulse could not re-fire thedelay timer. The output of the DELAY timer was passed via an inverter to the input of the third 555 timer calledDRIVE. This timer was adjustable over a range of several tens of microseconds, and was used to set the duration ofthe current through the drive coil. The output of the DRIVE timer was used to control two high-current MOSFETrelays that switched current to and from the drive coil. The circuit was designed to maintain a substantial currentdraw from the commercial 12V switching power supply at all times, hence the dual opposite-acting relays. A Zenerdiode across the drive coil prevented back-emf from damaging the relay and power supply during switching. Thevery brief drive current was estimated to be 7.5 Amps. The circuit was constructed using the wire-wrap method andhoused with its power supply in a small box. External connections were made using LEMO-style coaxial connectors,as seen in Fig. 4.
FIG. 6: Circuit diagram of the pendulum driver. The op-amp (LM324) and solid-state relays (Crydom CMX60D10) aresupplied by +12V, while all other components used V CC = +5V. Unlabeled circuit elements use standard TTL chips. The“Out” connections are for monitoring the circuit on an oscilloscope. V. RESULTS
The pendulum described in the preceding section was used to determine the accuracy to the model that lead toEq (19). The prediction that the Foucault precession rate should be unaffected by the transverse size of the ellipticalmotion is to be confirmed.The measured pendulum parameters were T = 3 . ± . a = 16 . ± . τ = 29 . ± . L = 2 . ± . α = 0 . ± . Q = 1628 ±
28. Solving Eq (19) numerically leadsfrom these values to δ = 0 . ± . t d = 178 ± t d was adjusted such that a steady Foucault precession, asshown below, was observed. The direct measurement of this time using an oscilloscope gave t d = 180 ± b ≃ θ latitude = 40 o ′
26” (Pittsburgh, Pennsylvania), the expected Foucault precession rate in onesidereal day is Ω F = − π .
935 hours sin θ = − . / hr → − . o / hr , (26)and this is shown as the blue bar. The vertical error bars on the points represent the estimated random measurementerrors, and these were dominated by a ± / F = − . ± .
15, where the given uncertainty is the error on the mean,not the standard deviation, and this is the red-dotted band shown in the figure. The weighted mean of the data givesΩ F = − . ± .
09, which is in very good agreement with the simple average, and both are in excellent agreement withthe expectation of Eq (26). We prefer the former uncertainty because the wide scatter of the data points suggeststhat perhaps there are some small systematic effects that are not captured using the weighted mean’s uncertainty.Hence the former uncertainty is a better estimate of the true uncertainty of the result. The main candidate for apoorly-controlled systematic effect is the temperature-related changes in the length of the pendulum. Even at thesub-millimeter level, these affected the strength of the magnetic kick the pendulum received, and hence affected thevalue of α , which in turn could affect control of the precession rate. We fine-tuned the length of the pendulum withshims, as needed, to minimize this effect, but the compensation was not perfect, and this probably led to some loss ofreproducibility. Nevertheless, we have found the expected Foucault precession rate with an angle-integrated precisionof 1.5%.As discussed in connection with Eq (14), if t d is correctly chosen, the rate of precession in this pendulum shouldnot depend on the size of the semi-minor axis, b . We could not control b since it depended on the small asymmetriesof the fiber itself and any other perturbations of the pendulum’s motion. Of course it was damped by the effect ofeddy current braking against the damping ring, but there was always some irreducible amount of ellipsoidal motionto contend with. We assign clockwise motion to have negative b values and counterclockwise motion to have positive b values. Figure 8 shows our measured precession rates, the same data set as in the previous figure, as a functionof b . The size of b was measured by watching the pendulum pass over a ruler placed under the pendulum, and theprecision of doing this was no better than ± i.e. toward decreasing angles. Note that even when the pendulum wasmoving in a counterclockwise ellipse (positive values of b ) that would normally precess in a counterclockwise direction,the action of the driver was such that the clockwise Foucault precession rate was unaffected. This again shows howour method nullifies the unwanted intrinsic precession.As further evidence that our method succeeds in controlling the motion of the pendulum, we intentionally reducedthe value of the driving time by ≈ F ⊥ was larger than optimal. Thismeans that any ellipsoidal precession was overcompensated by the drive system, resulting in forced precession in theopposite sense to what free oscillation would produce. Figure 9 shows the precession rate as a function of angle for t d = 147 ± F = − . ± .
57, which is still in agreementwith the expected Foucault rate, but now with a much wider error band, as shown. One also sees the propensity for1
FIG. 7: (color online) Precession rate as a function of azimuthal angle of the pendulum. Red circles are for counterclockwise,while blue squares are for clockwise ellipsoidal perturbation. Green diamond points were for measurements with no discernibleelliptical motion. The expected rate is shown as the thick blue line. The measured average precession rate (solid) and itsuncertainty band (dotted) are shown as red lines. clockwise ellipsoidal motion (blue squares) to decrease the magnitude of the precession rate, while counterclockwisemotion (red circles) increases the magnitude of the precession rate. This is as expected in our mathematical model.There are again some cases of irreproducibility of the data points at a given angle. We believe this is the result ofoccasional shimming of the pendulum’s length during a multi-day run; it leads to uncontrolled small variations in theperformance of the system.The heavy black dashed line in Fig. 9 is a guide to the eye to illustrate what happens when the ellipsoidal precession isnot perfectly canceled. There is a preferred axis near − o , where the measured rate crosses the Foucault rate, becausehere the non-central forces present in the system happen to vanish. For more negative azimuthal angles (which thenwrap to the positive side of the diagram) the precession rate is slower. This is because the unwanted forces act to causeclockwise ellipsoidal motion, but the drive system is overcompensating and adding a component of counterclockwiseprecession. At about +23 o , which is 90 o away from the preferred axis, there is a tipping point where the non-centralforces act to push the pendulum in the other direction, counterclockwise. But here the overcompensating drive systemmakes the pendulum precess more quickly clockwise. We do not claim that the departure from the Foucault rate isstrictly linear, as shown, only that the trends are consistent with our understanding of the physical process.Finally, in Fig. 10 we show the effect of reducing the drive time t d by ≈
20% on the relationship between precessionrate Ω and the semi-minor axis b . One sees clearly a correlation between them, whereas in Fig. 8 there was nocorrelation. For positive values of b (counterclockwise ellipses), the precession rate has increased magnitude since theovercompensating drive system “adds” to the Foucault rate. For negative b values (clockwise ellipses) the precessionrate is decreased in magnitude since the overcompensating drive system “subtracts” from the Foucault rate. Thus, wehave shown that when deviating from the prediction of Eq (19) for the correct drive distance, and therefore the correct2 FIG. 8: (color online) Precession rate as a function of the size of the semi-minor axis b of elliptical motion. Positive values of b are for counterclockwise ellipses, negative values for clockwise ellipses. The horizontal lines are the same as in the previousfigure. drive time, the pendulum shows marked departure from the constant Foucault precession rate that is expected. VI. FURTHER DISCUSSION AND CONCLUSIONS
In the previous work of Crane [9], he recognized the importance of nullifying the intrinsic precession that remainsafter damping as well as possible. However, he used a “push-pull” drive system to mitigate problems of alignment ofthe coils with the pendulum. This led him to introduce a carefully-placed fixed permanent magnet at the origin toprovide the desired stopping of intrinsic precession. No quantitative understanding of how to predict the placement ofthis magnet was offered. It seems to us that his very delicate ad hoc adjustment of this auxiliary magnet is difficult,and, as we have shown, not necessary. Our method is simpler and more direct, in that it does not require thisadditional magnet. Alignment of the driving and sense coils was not found to be a problem, and the results were notsensitive to the alignment at the level of about a millimeter. This was because the method we have introduced is in asense self-correcting: if the drive coil causes some small amount of ellipsoidal motion, the very action of the methodprevents this motion from causing unwanted precession. The work of Mastner et al. [10] made note of the benefits ofa “push only” driving force, but they did not offer the quantitative explanation as to how and why it worked. Theybuilt a traditional, very long, very massive pendulum, taking great care to minimize asymmetries.In conclusion, we have demonstrated the workings of a “short” Foucault pendulum that was designed on a quan-titative basis to avoid the unwanted precession due to ellipsoidal motion. The design formula we have derived, Eq(19), was shown to agree with measured values in our setup. We have shown that a driving mechanism that pushes,not pulls, the pendulum is the key to canceling the intrinsic precession for all values of the semi-minor axis of the3
FIG. 9: (color online) Precession rate as a function of azimuthal angle of the pendulum when the driving time t d is reduced by ≈
20% from the ideal value. As in Fig. 7, red circles are for counterclockwise, while blue squares are for clockwise ellipsoidalmotion. Green diamond points were for measurements with no discernible elliptical motion. The expected rate is shown as thethick blue line. The measured average precession rate (solid) and its uncertainty band (dotted) are shown as red lines. Theheavy black dashed line is a guide to the eye, as discussed in the text. ellipse. Our driver system used Faraday induction and magnetic repulsion to control the pendulum, using a circuitbased on simple op-amp, logic, and timer chips. Eddy current damping was used to reduce the ellipse size, but activecompensation did the rest. The design is immune to small non-central forces that are difficult to control in a shortpendulum. We plan to further test this method on even shorter pendula, since there is no lower limit at which themodel given here should apply.
VII. ACKNOWLEDGMENTS
We thank Mr. Gary Wilkin for his expert help in the machine shop. We thank Mr. Michael Vahey for constructionof the pendulum driver, and we thank Mr. Chen Ling for help with exploratory initial trials in construction of aFoucault pendulum. [1] M. L. Foucault, “D´emonstration physique du mouvement de rotation de la terre au moyen du pendule” Comptes RendusAcad. Sci. , 135-138 (1851).[2] M. G. Olsson, “The precessing spherical pendulum,” Am. J. Phys. , 1118-1119 (1978). FIG. 10: (color online) Precession rate as a function of the size of the semi-minor axis b when the driving time t d is reducedby ≈
20% from the predicted value. Positive values of b are for counterclockwise ellipses, negative values for clockwise ellipses.This figure is to be compared with Fig. 8.[3] M. G. Olsson, “Spherical pendulum revisited,” Am. J. Phys. , 531-534 (1981).[4] A. B. Pippard, “The parametrically maintained Foucault pendulum and its perturbations,” Proc. R. Soc. Lond. A420 ,81-91 (1988).[5] J. Synge and B. Griffith,
Principles of Mechanics (McGraw Hill, 1959), 3rd ed., pp 335-342.[6] M. Charron, “Sur un perfectionnement du pendule de Foucault et sur l’entretien des oscillations,” Comptes Rendus Acad.Sci. , 208-210 (1931).[7] See for example C.F. Moppert and W. J. Bonwick, “The New Foucault Pendulum at Monash University”, Q. Jl. R. Astr.Soc. , 108-118 (1980), and references therein. Also Haym Kruglak et al , “A short Foucault pendulum for a hallwayexhibit”, Am. J. Phys. , 438-440 (1978).[8] H. Richard Crane, “The Foucault Pendulum as a murder weapon and a physicist’s delight”, Phys. Teach. 264-269 (May1990).[9] H. Richard Crane, “Foucault pendulum “wall clock””, Am. J. Phys. , 33-39 (1995); “Short Foucault Pendulum: A Wayto Eliminate the Precession due to Ellipticity”, Am. J. Phys. , 1004-1006 (1981).[10] G. Mastner et al , “Foucault pendulum with eddy-current damping of the elliptical motion”, Rev. Sci. Inst. , 1533-1538(1984).[11] Joseph Priest and Michael Pechan, “The driving mechanism for a Foucault pendulum (revisited)”, Am. J. Phys.76