short foucault

arXiv:0902.1829v2 [physics.class-ph] 11 Feb 2009A Short Foucault Pendulum Free of Ellipsoidal Precession Reinhard A. Schumacher and Brandon Tarbet Department of Physics, Carnegie Mellon University, Pittsb urgh, PA 15213∗ (Dated: February 12, 2009) A quantitative method is presented for stopping the intrins ic precession of a spherical pendulum due to ellipsoidal motion. Removing this unwanted precessi on renders the Foucault precession due to the turning of the Earth readily observable. The method is insensitive to the size and direction of the perturbative forces leading to ellipsoidal motion. W e demonstrate that a short (three meter) pendulum can be pushed in a controlled way to make the Foucaul t precession dominant. The method makes room-height or table-top Foucault pendula more accur ate and practical to build. I. INTRODUCTION L´ eon Foucault built his first pendulum to demonstrate the tu rning of the Earth in the basement of a building, using a roughly two meter long fiber [1]. He also soon recogniz ed the problem arising from the intrinsic precession of a spherical pendulum caused by unwanted ellipsoidal moti on. Imperfections in the suspension or initial conditions of the pendulum generally cause this to quickly grow to the po int that the precession due to the Earth’s turning is overwhelmed. The pendulum can come to precess in either sens e (clockwise or counterclockwise) at almost any rate, or indeed even cease all precession. These practical proble ms are mitigated in pendula of great length, and so most are constructed to have lengths of tens of meters, starting w ith the celebrated 67 m long device built by Foucault in Paris in 1851. The precession of an ideal spherical pendulum with no ellips oidal motion is caused by the non-inertial nature of the reference frame tied to the surface of the Earth. The so-call ed Coriolis force advances the plane of the pendulum’s motion by an amount ΩF= Ω Earthsinθlatitude (1) where Ω Earth is the sidereal rate of rotation of the Earth, θlatitude is the latitude of the pendulum measured from the equator, and Ω Fis the Foucault precession rate; many textbooks treat this p roblem. Near 40onorthern latitude, this amounts to almost 10oof clockwise advance per hour, for an 18 hour half-rotation o f the pendulum, after which the motion repeats. This amount of precession is easily masked, however, by the intrinsic precession of a less than perfect pendulum that develops some non-planar ellipsoidal motion . The construction of a room-height or table-top version of a F oucault pendulum thus presents a technical challenge, first to minimize the amount of ellipsoidal motion that accru es as the pendulum swings, and secondly to compensate in some way for the irreducible amount of this motion that rem ains. In this paper we will first discuss the dynamics of the spherical pendulum that lead to the problem (Section II) . Then we introduce a method that stops the ellipsoidal motion of the pendulum from causing precession, and show tha t this immunity is, to first order, independent of the minor axis of the ellipse. The method hinges on the observati on that pushing the pendulum bob away from the origin after it passes, rather than either pulling it in or al ternately pulling and pushing it, acts in a way to counter the unwanted intrinsic precession (Section III). We then pr esent the design of a pendulum and a driving mechanism to exploit this method (Section IV), and demonstrate the val idity of this approach by discussing the supporting experimental results (Section V). Finally, we contrast our results with earlier published work on Foucault pendula and summarize how our method and design are new and unique (Se ction VI). II. THE PROBLEM OF INTRINSIC PRECESSION The dynamics of the idealized spherical pendulum are determ ined by the centrally-directed force of gravity and initial conditions, and lead to approximately elliptical m otion with a semi-major axis aand a semi-minor axis b, as shown in Fig. 1. It is somewhat counter-intuitive but true that the centrally-directed restoring force of gravity results in a constant intrinsic precession rate Ω about the zaxis. This precession arises from the symmetry-breaking ∗Electronic address: [email protected] 2 condition of having a finite minor axis, b. The rate is linearly proportional to b, and its sense is in the same direction as the ellipsoidal motion. An example of the exact path of a pe ndulum with a large ratio of b/ais given in Fig. 2, which illustrates a numerical solution of the equations of m otion of a spherical pendulum. FIG. 1: (color online) Planar view of the approximate path of a spherical pendulum with semi-major axis aand semi-minor axisbthat is moving in a counterclockwise ellipsoid. The suspens ion is centered on the zaxis above the origin. The pendulum is precessing at rate Ω, and in one full cycle the apex advance s by a distance ∆ y, as suggested by the light dotted and rotated ellipse. The impulsive driving force is applied at x=d, and it is resolved into components parallel and perpendicu lar to the major axis. The minor axis can be larger or smaller, resultin g in a b-dependent magnitude of the transverse force F⊥for a fixed longitudinal force F/bardbl. FIG. 2: (color online) Numerical simulation (using Mathema tica) of a spherical pendulum with a large ratio of semi-mino r to semi-major ellipse axes. The arrows indicate the countercl ockwise intrinsic precession, in the same sense as the motio n of the pendulum. Any ellipsoidal motion that develops in a pendulum will resu lt in an intrinsic precession rate Ω that is wholly unrelated to the Foucault precession Ω Fof Eq (1). This has been worked out in detail by Olsson [2, 3] an d Pippard [4], among others, and it appears in some textbooks, for example i n Ref [5]. The main result is Ω =3 8ω0ab L2, (2) 3 where ω0= 2π/T=/radicalbig g/Lis the pendulum’s angular frequency, Lis the length, Tis the period, and gis the acceleration due to gravity. The formula is the lowest-orde r term in a complicated motion, but it is easily sufficient for our purpose. The area of an ellipse, A, is given by A=πab, so another way to write the intrinsic precession rate is Ω =3 4A L2T. (3) From the first expression, note that the ratio of Ω (the very sl ow intrinsic precession) to ω0(the rapid pendular rate) is proportional to the ratio of the area of the ellipse to the a rea of a sphere of radius L, that is, Ω ω0=3 2(πab) (4πL2)(4) To minimize this ratio, Foucault pendula are generally made withLvery large compared to the axes aandb, since it is comparatively easy to keep bsmall while making Llarge. Besides causing precession, ellipsoidal motion changes th e central oscillation frequency of the pendulum from ω0to ω=ω0/parenleftbigg 1−1 16a2+b2 L2/parenrightbigg . (5) The fractional change in frequency due to a finite semi-minor axisbturns out to be of order 10−7, and therefore negligible for the present discussion. Every Foucault pendulum, no matter how carefully construct ed to avoid asymmetries in its suspension, and no matter how carefully “launched” to make ellipsoidal area Aas small as reasonably achievable will, over time, acquire an intrinsic precession Ω that can easily grow to overwhelm t he Coriolis-force induced Foucault precession Ω F. Near 40olatitude, for a pendulum of length L= 3.0 meters and semi-major axis a= 16. cm, the Foucault rate is equaled with a semi-minor axis of b= 3.9 mm. The unwanted intrinsic precession may add or subtra ct from the Foucault precession; when it is subtractive, then the pendulum stops all precession when the corresponding value of bis reached. In our experience, it is easy to launch such a pendulum with se mi-minor axis well under one millimeter, but under free oscillation, on the time scale of only two to three minut es,bgrows enough such that Ω dominates Ω F. When the pendulum at an extremum, at x=±ain Fig. 1, its motion is entirely transverse, with momentum m˙yas large as it gets. Preferential damping of this component o f the motion will reduce the unwanted ellipsoidal excursions. In previous work, this has been tried using a so- called Charron’s ring around the suspension wire near the top [6, 7], or letting a part of the pendulum bob scrape an a nnular disk [8, 9] at r = a , or using eddy current damping between a permanent magnet in the bob and a non-ferro us metal annular disk [10] located near x = a. We adopt the touch-free eddy-current damping method in the des ign we present later in this paper. Even in principle, none of these methods will stop ellipsoidal motion complete ly, so an additional method is needed to cope with any remaining intrinsic precession. Every pendulum suffers dissipative losses of energy, mainly due to air friction. Long, very massive, museum-type pendula can simply be relaunched once every day or so, but a sm all pendulum has a free exponential decay time of order one half hour, so a “driving” mechanism is needed to res tore the lost energy. Various mechanisms have been reported for this task [8, 9, 10, 11]. Like some others, we wil l use magnetic induction to sense the passage near the origin of a permanent magnet embedded in the pendulum bob . A carefully-timed electromagnetic impulse then imparts lost momentum to the bob. We will introduce a quantit ative method for using a magnetic push not only to compensate for dissipative losses, but also to compensate f or ellipsoidal precession. The perturbations that lead to ellipsoidal motion are, in ou r experience, in approximate decreasing order of severity, (1) internal stresses or other imperfections in the fiber sup porting the pendulum bob, (2) less than perfectly symmetric suspension of the fiber at its upper end, (3) nearby iron objec ts that result in asymmetric force on the drive magnet in the bob, (4) a driving coil that is not sufficiently level and centered under the pendulum. All but the first of these were straightforward to reduce to insignificance, but the fir st was persistent. This led to the necessity of finding a method to evade the problem of ellipsoidal motion rather tha n to remove it. There is no simple force law that leads to the intrinsic precession Ω, but rather it is an inescapabl e feature of the spherical pendulum. Nevertheless, we can apply a separate perturbative force to counteract, i.e. nul lify, the intrinsic precession. We discuss this in the next section. 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 consisting of components parallel and perpendicular to the major axis. The parallel component F/bardblis the larger one, and it is 4 adjusted to overcome the dissipative forces such as air resi stance. The perpendicular component of the pushing force F⊥counteracts the intrinsic precession, and we will now show t hat it is possible to select distance dat which the impulsive drive force is applied to stop the precession Ω. Th e crucial result will be that this distance is independent of semi-minor axis b. Starting from x=−a, in one half cycle of duration T/2, intrinsic precession advances the ellipse in angle by Ω T/2. The pendulum arrives at x≃+aand y≡∆y=1 2aΩT, (6) where ∆ yis the transverse displacement at the apex of the ellipse. Ho wever, we arrange to apply an impulsive momentum change at x=dthat is calibrated to move the pendulum bob by a distance −∆yas it traverses the remaining longitudinal distance a−d. If this is done, the bob will arrive at location x=a,y= 0, as desired. The impulse and momentum change are related by m∆vy=F⊥∆t, (7) where ∆ tis the duration of the applied force and mis the mass of the bob. It turns out that ∆ tis a few milliseconds, compared to hundreds of milliseconds for the duration of the swing, so it is appropriate to use the impulse formulation of Newton’s second law. We treat this perturbation in the ydirection as though the bob were free of other forces, which is reasonable since ∆ yis of order 10 microns compared to aof order 10 centimeters. The horizontal component of the motion is simply x(t) =asinω0t, (8) where ω0is the angular frequency characterizing the pendular oscil lations. To good accuracy, the time tdbetween passage closest to the origin and the instant the pendulum re achesx = d is thus td=1 ω0sin−1d a. (9) In one quarter of the full period Tthe pendulum reaches its apex. Therefore, one way to write a r elation between the distance ∆ ythat the pendulum advances and the transverse velocity ∆ vythat must be imparted by the driver is ∆y= ∆vy/parenleftbigg1 4T−td/parenrightbigg =∆vy ω0/parenleftbiggπ 2−sin−1d a/parenrightbigg =∆vy ω0cos−1d a(10) From the geometry of the situation shown in Fig. 1 we see that t hat the driving force components are related by tanθ=y d=F⊥ F/bardbl, (11) and, using the formula for an ellipse (x/a)2+ (y/b)2= 1, (12) we have F⊥=F/bardblb d/radicalBigg 1−/parenleftbiggd a/parenrightbigg2 . (13) Combining Eqs (10), (7), (13), (6), and (2) we arrive at ∆y=1 2a/parenleftbigg3 8ω0ab L2/parenrightbigg /parenleftbigg2π ω0/parenrightbigg =F/bardbl∆t mω0b d/radicalBigg 1−/parenleftbiggd a/parenrightbigg2 cos−1d a. (14) One sees in Eq (14) the crucial cancellation of the factor bthat occurs between the intrinsic precession rate and in the perpendicular component of the force. This leads to the r esult that follows being independent of the transverse size of the ellipse. This, in turn, means the result is insens itive to exactly what non-central forces may act on the pendulum to cause non-vanishing ellipsoidal motion. An app roximation is being made that F/bardbldoes not depend on b; this will be justified later. 5 It is now convenient to introduce some dimensionless scalin g parameters. Let Q≡π 2maω 0 F/bardbl∆t, (15) which is the ratio of the momentum of the pendulum as it crosse s the origin to the momentum kick it receives on each half oscillation. The denominator is the momentum the drive r must supply to compensate for the dissipative losses in order to maintain the full amplitude of the swing. The factor ofπ/2 stems from letting Qrepresent the conventional “quality factor” of an oscillator, specifically that Q= 2πTotal Energy Energy Loss per Cycle. (16) We expect this ratio to be quite large, on the order of 1000. Th en let α=a L(17) be the scaled amplitude of the pendulum, that is, the amplitu de divided by the length; this parameter is of order 0.1 for a practical pendulum. Next, let δ=d a(18) be the scaled distance from the origin at which the driving fo rce is applied, written as a fraction of the amplitude; this value can range from near zero to unity. With these dimen sionless variables, Eq (14) can be written 3 4Qα2=√ 1−δ2 δcos−1δ (19) Equation 19 is the main result of this model. It relates the sc aled distance δat which the impulsive driving force must be applied on each half oscillation, to the physical par ameters of the pendulum, specifically the scaled amplitude αand the quality of the oscillator Q. When Eq (19) is satisfied, the intrinsic precession is stopp ed or nullified, and the result is independent of the transverse size of the ellip se. Figure 3a is an illustration of this result for the cases o f three different lengths of pendulum, each with a maximum excu rsion of 0.15 meters, as a function of the oscillation parameter, Q. It is seen that a low-loss pendulum with a larger value of Qwill have to be pushed when it is closer to the origin, while a pendulum with more dissipative losses , and therefore lower Q, will have be driven further away from the origin. A high-quality, low-loss pendulum needs li ttle energy input. Therefore, the transverse kick, which scales in magnitude with the longitudinal kick at a given δ, must be applied early in the ellipsoidal swing, when the force has a larger transverse component. Part (b) of Fig. 3 il lustrates the relationship between the location of the impulse, δ, and the time, td, at which it is applied, as per Eq (9). Taken together, the two parts of the Fig. 3 enable design of a F oucault pendulum not plagued by intrinsic precession. For example, a 3.0 meter long pendulum with a 0.15 meter ampli tude ( α= 0.050) and a quality factor of Q= 1000 must receive its impulsive drive at δ=.49, which corresponds to time td= 284 msec and a drive location of d= 7.4 cm from the origin. On the other hand, a 1.0 meter long pendulu m with the same αandQmust receive its drive pulse at δ=.088, at a time of 28 msec and 1.3 cm from the origin. As introduced in Eq (15), Qis proportional to the ratio of the momentum of free oscillat ion at the origin to the damping/driving momentum that accrues with each half perio d of oscillation. In fact, it is quite easy to measure this parameter for an actual pendulum by measuring the period, T, and the exponential decay time, τ. The motion of the pendulum in the presence of velocity-dependent losses such as air friction is given, to a good approximation at low speeds, by the damped harmonic oscillator formula x(t) =ae−t/τsinω0t. (20) The momentum loss between times t= 0 and t = T /2 is m˙x(0)−m˙x(T/2) =maω 0(1−e−T/2τ)∼=maω 0T 2τ. (21) This leads to Q=πτ T, (22) 6 FIG. 3: (color online) (a) For three pendulum lengths ( L) with amplitude of 0.15 meter, the relationship of the scale d driving distance ( δ=d/a) versus the quality factor Qfor 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 t he same pendulum lengths as in (a), the distance versus time relationship for the driving pulse. from which Qcan be determined experimentally. For the actual three mete r pendulum discussed below, we found the decay time to be about 30 minutes and hence Qto be roughly 1600. Eq (22) can be used with Eq (19) in order to express Eq (19) in dimensioned variables as 3 8/radicalbiggg L/parenleftBiga L/parenrightBig2 τ=√ a2−d2 dcos−1(d/a). (23) One important approximation that was made needs to be examin ed. In going from Eq (14) to Eq (19), the cancellation of the semi-minor ellipse parameter bwas crucial to showing that the result is independent of inev itable changes in the size of the pendulum’s ellipsoidal motion. In fact,F/bardbldoes depend very slightly on bin the design we will discuss in the following sections. The pushing agent is a magnetic coil that produces a pulsed dipole field. It acts against a permanent magnetic dipole inside the pendulu m bob. Though we operate in the near field of the coil, we presume that the magnetic field has a distance dependence o f the dipole form B(r) =B0(r0/r)3, where B0and r0are suitable scales. The dipole-dipole repulsion that driv es the pendulum goes as the gradient of this field, so the interaction has the form /vectorF=/vectorF0(r0/r)4, (24) where the components of /vectorFare what we earlier called F⊥andF/bardbl. It is easy to expand this expression to show that F/bardbl=F0/parenleftBigr0 d/parenrightBig4/braceleftBigg 1−5 2/parenleftbiggb d/parenrightbigg2/parenleftbig 1−δ2/parenrightbig/bracerightBigg (25) As seen in the second term in the curly brackets, there is a qua dratic dependence on the semi-minor axis bthat we ignored earlier. This term is very small: for typical valu es of δ=d/a≃6.0cm/16.0cm= 0.375, and b/d≃ 0.5cm/6.0cm= 0.083, the second term is 0.015, which is much less than 1.0. Hen ce this approximation was justified. IV. EXPERIMENTAL SETUP To verify the mathematical model discussed above, a pendulu m of about three meter length was built. The mechanics and drive electronics will be described here. An i mage of the lower end of the setup is shown in Fig. 4. 7 FIG. 4: (color online) Image of the copper pendulum bob suspe nded above the sensing and driving coils (red wire on a white bobbin). The aluminum damping ring on brass legs is seen. The white band carries angular position markings. The driver bo x and an oscilloscope are visible in the background. The mass of the pendulum was in the form of a lathe-turned copp er cylinder, with a diameter of 2.75” (6.99 cm) and height 3.00” (7.62 cm). A 3/8” (.953 cm) axial hole was drille d and tapped to accept a small threaded set screw with a 0.040” (0.10 cm) hole in its center. The small hole was for th e fiber suspending the pendulum, and the threading was used to adjust the distance between the center of mass and the suspension point. In practice, the fiber was held flush with the top surface of the bob. The bottom of the cylindr ical bob was chamfered by 0.10” (0.25 cm) to allow closer approach to the damping ring. To hold the permanent ma gnet, 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 we re the dimensions of the magnet. The total mass of the pendulum bob, apart from the magnet, was 2.47 kg. This is a small mass compared to what has generally been reported 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 fib er was circular and the diameter was uniform to better than 1%. The polymeric material was thought to be advantageo us due to its lack of crystalline internal structure. It was found that the polymer material was very flexible, yet str ong, and did not suffer the bending fatigue and eventual failure of some metal fibers we tried. In practice, we found th at the growth of elliptical motion of the pendulum was dominated by an unobservable non-uniformity of this fiber, b ut it was less severe than with various metal fibers. Tests wherein the fiber was rotated without changing any other aspe ct of the pendulum showed this to be the case. Thermal expansion and contraction of the fiber was large enough to nec essitate occasional shimming of the pendulum’s length at the level of about one millimeter. The upper end of the fiber passed through a close-fitting drill ed hole in an aluminum plate. The hole had a sharp rim, and no special steps were taken to soften the bend of the fi ber as it exited the hole. On the upper side of the plate the fiber was clamped in a way that thin shims could be ins erted to fine tune the length. The plate was leveled and clamped to rigid brackets on the ceiling of the laborator y. The upper support of the fiber used in this setup, though carefully arranged, was certainly less exacting tha n what has been found to be necessary for other Foucault pendula reported in the literature. We view this as an advant age of our design. The permanent magnet placed inside the pendulum bob was a neo dymium-iron-boron disk magnet, placed with its dipole axis vertical. The axial field strength was 2.1 kGauss on contact, but it varied by ±10% from one edge of the disk to the other. Similarly, the radial field was 1.0 kG at the edge of the disk, but varied also by ±10% from one 8 side to the other. Surprisingly perhaps, these variations d id not affect the performance of the pendulum. One purpose of the magnet was to provide eddy-current dampin g when the bob passed over an aluminum ring near the extrema of its motion. This “damping ring” had an inner di ameter of 11 7/8” (30 cm), and was 1/4” (0.64 cm) thick. Three legs made of threaded brass rod were used to leve l the ring to a precision of less than half a millimeter of variation around the perimeter. The amplitude of the pend ulum was adjusted such that the magnet passed over the inner edge of the ring with a clearance of three to four mil limeters. This maximized damping of the unwanted transverse precessional speed. Since the eddy current damp ing force is velocity dependent, it can never entirely stop this motion, but our experience was that it limited the ellip soidal motion to a semi-minor radius of less than half a centimeter. As a side note, L´ eon Foucault first identified t he eddy-current phenomenon in 1851, thus we use two phenomena in this investigation that are attributed to him. Two concentric coils under the pendulum sensed and controll ed its motion. Both coils were wound from 22 AWG copper wire on polyethylene bobbins, and both had 240 turns. The inner coil was designated as the “drive” coil used for pulsing the pendulum on each half cycle. It had a mean radi us of 0.80” (2.0 cm) and a total length of 100’ (30 m) of wire. The outer coil was the “sense” coil for detecting t he approach of the pendulum. It had a mean radius of 1.5” (3.8 cm) and a total length of 187’ (57 m) of wire. The supp ly leads to both coils were twisted pair copper wire near the coils, and RG-174 coaxial cable at distances where t he steel ground braid of the latter no longer perturbed the motion. The concentric sense and drive coils were supported by a smal l aluminum platform that sat on the main table, supported by three brass screws with knurled heads. These we re used to level the coils: imperfect leveling caused the induced signal from the pendulum, described below, to be asymmetric. The clearance between the bottom of the pendulum and the top of the coils was 6 ±1 mm. The sense coil acted to detect the induced EMF of the swinging pendulum via Faraday induction. The typical waveform is shown in Fig. 5, showing that the approaching coi l induced a negative voltage that peaked at about V−=−34 mV, as viewed across 1 MΩ input impedance on an oscilloscope. When the pendulum was c entered over the coil, the voltage crossed upwards through zero, followe d by a positive excursion of inverted shape and the same magnitude V+as the negative part. The peak magnitudes of the waveform, V ±, depended on the speed of the pendulum (and hence the amplitude of the swing), as well as th e distance between the pendulum magnet and the coil. Monitoring V ±over time was a sensitive way to detect small changes in the pe ndulum’s amplitude and/or length. FIG. 5: Induced voltage at the input stage of the driver circu it, due to the magnet in the pendulum approaching, passing ov er, and then receding from the sense coil. Upper trace (1) shows a complete cycle of two passages. Lower zoomed trace shows how on the trailing end of the sense pulse (1) the drive pulse ( 2) causes a large bipolar induced pulse in the sense coil. Scr een capture was from a TDS 3032B oscilloscope. The pendulum driver circuit developed for this study is show n in Fig. 6, and operated as follows. The initial signal from the coil was filtered with a 1 µF capacitor that reduced RF noise to less than one millivolt. An op-amp circuit of type LM324N level-shifted and amplified the signal by a fac tor of 200. This signal was the input to a SN7414N Schmitt trigger that switched at 0.9 and 1.7 Volts. In effect, this was a discriminator with a TTL output “trigger” 9 pulse when the induced voltage went more negative than −20 mV; the trigger stayed high until the trailing end of the sense-coil signal shown in Fig. 5 fell. A NAND gate output was used to trigger two 555 timers called DELAY and INHIBIT; the second input of the NAND was an inhibit signal th at ensured that the DELAY timer was not restarted by the off-scale pulse induced by activation of the drive coil . The DELAY timer set the interval between the approach of the pendulum and the drive pulse in the second coil; it was a djustable in the range from near zero to several hundred milliseconds. The INHIBIT timer duration was set to be longe r 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 the delay timer. The output of the DELAY timer was passed via an in verter to the input of the third 555 timer called DRIVE. This timer was adjustable over a range of several tens of microseconds, and was used to set the duration of the current through the drive coil. The output of the DRIVE ti mer was used to control two high-current MOSFET relays that switched current to and from the drive coil. The c ircuit was designed to maintain a substantial current draw from the commercial 12V switching power supply at all ti mes, hence the dual opposite-acting relays. A Zener diode across the drive coil prevented back-emf from damagin g the relay and power supply during switching. The very brief drive current was estimated to be 7.5 Amps. The cir cuit was constructed using the wire-wrap method and housed with its power supply in a small box. External connect ions were made using LEMO-style coaxial connectors, as seen in Fig. 4. FIG. 6: Circuit diagram of the pendulum driver. The op-amp (L M324) and solid-state relays (Crydom CMX60D10) are supplied by +12V, while all other components used VCC= +5V. Unlabeled circuit elements use standard TTL chips. Th e “Out” connections are for monitoring the circuit on an oscil loscope. 10 V. RESULTS The pendulum described in the preceding section was used to d etermine the accuracy to the model that lead to Eq (19). The prediction that the Foucault precession rate sh ould be unaffected by the transverse size of the elliptical motion is to be confirmed. The measured pendulum parameters were T= 3.44±0.01sec, and a= 16.2±0.2cm. The free decay period was determined by making an exponential fit to amplitude-vs.-ti me data, leading to τ= 29.7±0.5min. From these values we computed L= 2.938±0.009m, α= 0.0551±0.0007, and Q= 1628 ±28. Solving Eq (19) numerically leads from these values to δ= 0.319±0.006. This value in turn can be converted using Eq (9) to an expe cted drive time td= 178±3ms. On the other hand, the actual value of tdwas adjusted such that a steady Foucault precession, as shown below, was observed. The direct measurement of this ti me using an oscilloscope gave td= 180±2ms. Thus, the model expectation as given in Eq (19) is found to be in excelle nt agreement with experiment with about 1% precision. Figure 7 shows the result of a set of measurements of the prece ssion rate spanning several days. The abscissa angles were measured from an origin arbitrarily picked to point nor th, with positive azimuthal angles to the counterclockwise side, as seen from above. Red circular data points are for cou nterclockwise and blue squares are for clockwise ellipsoid al excursions in the pendulum’s motion. Green diamond points a re for readings with no measurable sense, i.e. for b≃0. At the latitude of θlatitude = 40o26′26” (Pittsburgh, Pennsylvania), the expected Foucault pre cession rate in one sidereal day is ΩF=−2π 23.935 hourssinθ=−0.1703/hr→ −9.757o/hr, (26) and this is shown as the blue bar. The vertical error bars on th e points represent the estimated random measurement errors, and these were dominated by a ±1/2 degree precision of the angle measurements. The horizonta l error bars represent the span of angles over which the rate was measured , with a typical span being ten degrees. The scatter of the data points clearly clusters around the expected rate, w ith no trend in angle or sense of ellipsoidal motion. The simple average of the measured points gives Ω F=−9.85±.15, where the given uncertainty is the error on the mean, not the standard deviation, and this is the red-dotted band s hown in the figure. The weighted mean of the data gives ΩF=−9.69±.09, which is in very good agreement with the simple average, a nd both are in excellent agreement with the expectation of Eq (26). We prefer the former uncertainty because the wide scatter of the data points suggests that 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 a poorly-controlled systematic effect is the temperature-re lated changes in the length of the pendulum. Even at the sub-millimeter level, these affected the strength of the mag netic kick the pendulum received, and hence affected the value of α, which in turn could affect control of the precession rate. We fine-tuned the length of the pendulum with shims, as needed, to minimize this effect, but the compensati on was not perfect, and this probably led to some loss of reproducibility. Nevertheless, we have found the expected Foucault precession rate with an angle-integrated precisi on of 1.5%. As discussed in connection with Eq (14), if tdis correctly chosen, the rate of precession in this pendulum should not depend on the size of the semi-minor axis, b. We could not control bsince it depended on the small asymmetries of the fiber itself and any other perturbations of the pendulu m’s motion. Of course it was damped by the effect of eddy current braking against the damping ring, but there was always some irreducible amount of ellipsoidal motion to contend with. We assign clockwise motion to have negative bvalues and counterclockwise motion to have positive bvalues. Figure 8 shows our measured precession rates, the sa me data set as in the previous figure, as a function ofb. The size of bwas measured by watching the pendulum pass over a ruler place d under the pendulum, and the precision of doing this was no better than ±1mm. We plot the points at the mean value of the ellipse size du ring the measurement interval. It is clear from the figure that there i s no correlation between the two variables, thus proving the claim that they are in fact independent when the pendulum is properly arranged. The Foucault precession was clockwise in this northern-hemisphere setup, i.e.toward decreasing angles. Note that even when the pendulum w as moving in a counterclockwise ellipse (positive values of b) that would normally precess in a counterclockwise directi on, the action of the driver was such that the clockwise Foucault precession rate was unaffected. This again shows how our method nullifies the unwanted intrinsic precession. As further evidence that our method succeeds in controlling the motion of the pendulum, we intentionally reduced the value of the driving time by ≈20%, so that the transverse kick given by F⊥was larger than optimal. This means that any ellipsoidal precession was overcompensated by the drive system, resulting in forced precession in the opposite sense to what free oscillation would produce. Figu re 9 shows the precession rate as a function of angle for td= 147±1 msec. There is, in this new situation, substantial systema tic deviation from the expected rate as a function of azimuthal angle. The angle-integrated average rate is now Ω F=−9.53±.57, which is still in agreement with the expected Foucault rate, but now with a much wider err or band, as shown. One also sees the propensity for 11 FIG. 7: (color online) Precession rate as a function of azimu thal angle of the pendulum. Red circles are for counterclock wise, while blue squares are for clockwise ellipsoidal perturbat ion. Green diamond points were for measurements with no disc ernible elliptical motion. The expected rate is shown as the thick bl ue line. The measured average precession rate (solid) and it s uncertainty band (dotted) are shown as red lines. clockwise ellipsoidal motion (blue squares) to decrease th e magnitude of the precession rate, while counterclockwise motion (red circles) increases the magnitude of the precess ion rate. This is as expected in our mathematical model. There are again some cases of irreproducibility of the data p oints at a given angle. We believe this is the result of occasional shimming of the pendulum’s length during a multi -day run; it leads to uncontrolled small variations in the performance of the system. The heavy black dashed line in Fig. 9 is a guide to the eye to ill ustrate what happens when the ellipsoidal precession is not perfectly canceled. There is a preferred axis near −67o, where the measured rate crosses the Foucault rate, because here the non-central forces present in the system happen to v anish. For more negative azimuthal angles (which then wrap to the positive side of the diagram) the precession rate is slower. This is because the unwanted forces act to cause clockwise ellipsoidal motion, but the drive system is overc ompensating and adding a component of counterclockwise precession. At about +23o, which is 90oaway from the preferred axis, there is a tipping point where t he non-central forces act to push the pendulum in the other direction, count erclockwise. But here the overcompensating drive system makes the pendulum precess more quickly clockwise. We do not claim that the departure from the Foucault rate is strictly linear, as shown, only that the trends are consiste nt with our understanding of the physical process. Finally, in Fig. 10 we show the effect of reducing the drive tim etdby≈20% on the relationship between precession rate Ω and the semi-minor axis b. One sees clearly a correlation between them, whereas in Fig . 8 there was no correlation. For positive values of b(counterclockwise ellipses), the precession rate has incr eased magnitude since the overcompensating drive system “adds” to the Foucault rate. For negative bvalues (clockwise ellipses) the precession rate is decreased in magnitude since the overcompensating d rive system “subtracts” from the Foucault rate. Thus, we have shown that when deviating from the prediction of Eq (19) for the correct drive distance, and therefore the correct 12 FIG. 8: (color online) Precession rate as a function of the si ze of the semi-minor axis bof elliptical motion. Positive values of bare for counterclockwise ellipses, negative values for clo ckwise ellipses. The horizontal lines are the same as in the p revious figure. drive time, the pendulum shows marked departure from the con stant Foucault precession rate that is expected. VI. FURTHER DISCUSSION AND CONCLUSIONS In the previous work of Crane [9], he recognized the importan ce of nullifying the intrinsic precession that remains after damping as well as possible. However, he used a “push-p ull” drive system to mitigate problems of alignment of the coils with the pendulum. This led him to introduce a caref ully-placed fixed permanent magnet at the origin to provide the desired stopping of intrinsic precession. No qu antitative understanding of how to predict the placement of this 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 an d more direct, in that it does not require this additional magnet. Alignment of the driving and sense coils was not found to be a problem, and the results were not sensitive to the alignment at the level of about a millimeter . This was because the method we have introduced is in a sense self-correcting: if the drive coil causes some small a mount of ellipsoidal motion, the very action of the method prevents this motion from causing unwanted precession. The work of Mastner et al. [10] made note of the benefits of a “push only” driving force, but they did not offer the quantit ative explanation as to how and why it worked. They built a traditional, very long, very massive pendulum, taki ng 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 ellip soidal 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 the 13 FIG. 9: (color online) Precession rate as a function of azimu thal angle of the pendulum when the driving time tdis reduced by ≈20% from the ideal value. As in Fig. 7, red circles are for coun terclockwise, while blue squares are for clockwise ellipso idal motion. Green diamond points were for measurements with no d iscernible elliptical motion. The expected rate is shown as the thick blue line. The measured average precession rate (soli d) and its uncertainty band (dotted) are shown as red lines. T he heavy black dashed line is a guide to the eye, as discussed in t he text. ellipse. Our driver system used Faraday induction and magne tic repulsion to control the pendulum, using a circuit based on simple op-amp, logic, and timer chips. Eddy current damping was used to reduce the ellipse size, but active compensation did the rest. The design is immune to small non- central forces that are difficult to control in a short pendulum. We plan to further test this method on even shorter pendula, since there is no lower limit at which the model given here should apply. VII. ACKNOWLEDGMENTS We thank Mr. Gary Wilkin for his expert help in the machine sho p. We thank Mr. Michael Vahey for construction of the pendulum driver, and we thank Mr. Chen Ling for help wit h exploratory initial trials in construction of a Foucault pendulum. [1] M. L. Foucault, “D´ emonstration physique du mouvement d e rotation de la terre au moyen du pendule” Comptes Rendus Acad. Sci. 32, 135-138 (1851). [2] M. G. Olsson, “The precessing spherical pendulum,” Am. J . Phys. 46, 1118-1119 (1978). 14 FIG. 10: (color online) Precession rate as a function of the s ize of the semi-minor axis bwhen the driving time tdis reduced by≈20% from the predicted value. Positive values of bare for counterclockwise ellipses, negative values for clo ckwise ellipses. This figure is to be compared with Fig. 8. [3] M. G. Olsson, “Spherical pendulum revisited,” Am. J. Phy s.49, 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 Fouca ult et sur l’entretien des oscillations,” Comptes Rendus Ac ad. Sci.192, 208-210 (1931). [7] See for example C.F. Moppert and W. J. Bonwick, “The New Fo ucault Pendulum at Monash University”, Q. Jl. R. Astr. Soc.21, 108-118 (1980), and references therein. Also Haym Kruglak et al, “A short Foucault pendulum for a hallway exhibit”, Am. J. Phys. 46, 438-440 (1978). [8] H. Richard Crane, “The Foucault Pendulum as a murder weap on and a physicist’s delight”, Phys. Teach. 264-269 (May 1990). [9] H. Richard Crane, “Foucault pendulum “wall clock””, Am. J. Phys. 63, 33-39 (1995); “Short Foucault Pendulum: A Way to Eliminate the Precession due to Ellipticity”, Am. J. Phys .49, 1004-1006 (1981). [10] G. Mastner et al, “Foucault pendulum with eddy-current damping of the ellip tical motion”, Rev. Sci. Inst. 55, 1533-1538 (1984). [11] Joseph Priest and Michael Pechan, “The driving mechani sm for a Foucault pendulum (revisited)”, Am. J. Phys. 76, 188-188 (2008).