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The Anisosphere As a New Tool for Interpreting Foucault Pendulum Experiments

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The Anisosphere As a New Tool for Interpreting Foucault Pendulum Experiments Eur. Phys. J. Appl. Phys. (2017) 79: 31001 HE UROPEAN DOI: 10.1051/epjap/2017160337 T E PHYSICAL JOURNAL APPLIED PHYSICS Regular Article The anisosphere as a new tool for interpreting Foucault pendulum experiments. Part I: harmonic oscillators Ren´e Verreaulta Universit´eduQu´ebec `a Chicoutimi, 555 Boulevard de l’Universit´e, Saguenay, QC G7H 2B1, Canada Received: 2 September 2016 / Received in final form: 30 June 2017 / Accepted: 30 June 2017 c The Author(s) 2017 Abstract. In an attempt to explain the tendency of Foucault pendula to develop elliptical orbits, Kamer- lingh Onnes derived equations of motion that suggest the use of great circles on a spherical surface as a graphical illustration for an anisotropic bi-dimensional harmonic oscillator, although he did not himself exploit the idea any further. The concept of anisosphere is introduced in this work as a new means of interpreting pendulum motion. It can be generalized to the case of any two-dimensional (2-D) oscillating system, linear or nonlinear, including the case where coupling between the 2 degrees of freedom is present. Earlier pendulum experiments in the literature are revisited and reanalyzed as a test for the anisosphere approach. While that graphical method can be applied to strongly nonlinear cases with great simplicity, this part I is illustrated through a revisit of Kamerlingh Onnes’ dissertation, where a high performance pen- dulum skillfully emulates a 2-D harmonic oscillator. Anisotropy due to damping is also described. A novel experiment strategy based on the anisosphere approach is proposed. Finally, recent original results with a long pendulum using an electronic recording alidade are presented. A gain in precision over traditional methods by 2–3 orders of magnitude is achieved. Nomenclature (X, X0,X1,X2) KO: azimuths of rectilinear states (...) KO’s original symbols in this in general and at the times (T0,T1,T2) nomenclature list X, Y a, b Semi-major and semi-minor axes of Fast and slow rectilinear eigenstates elliptical orbit; b>0: counterclockwise on the anisosphere, respectively b< Xs,Ys Stereographic projection coordinates (ccw) ellipse; 0: clockwise (cw) ψ, χ ellipse respectively corresponding to 2 2 D t on the anisosphere ( t) KO: phase lag after elapsed time x, y L, R ccw and cw circular states on the ( ) KO: azimuths of rectilinear slow and fast eigenstates, respectively anisosphere δ M,N Fast and slow elliptical eigenstates ( ) KO: initial phase lag between (P LZ) KO: spherical triangle in the thesis the slow and fast components of a between longitude great circle, time great rectilinear oscillation δ˙ circle and characteristic great circle Rate of increase of phase difference (p,q) KO: slow and fast rectilinear eigenstate between fast and slow rectilinear X Y frequencies eigenstates and γ (T0,T1,T2) KO: times at start, after first sub-period, ( ) KO: precession rate of a rectilinear after second sub-period state due to Foucault effect ρ, ρ T0,T1 This work: average value and first ˙ ˙F Precession rate of a rectilinear state harmonic amplitude of in general, or due to Foucault effect ρ the variation of swinging period 2˙ Rate of increase of phase difference vs. swinging azimuth between fast and slow circular R L TX ,TY Swinging period of pendulum for eigenstates and χ −1 b/a rectilinear eigenstates X and Y ( ) KO: tan ( ) χ tan−1 (b/a); half-latitude on the a e-mail: [email protected] anisosphere 31001-p1 The European Physical Journal Applied Physics (ψ),η KO and this work: cot−1 δ˙ ; At this point, a word should be said about nomencla- ture. Linear/nonlinear pertaining to pendulum differen- KO: (= χmax + χmin); This work: latitude of slow elliptic tial equations designates oscillators. A 2-D linear oscillator eigenstate N on the anisosphere is a harmonic oscillator which normally has two distinct ψ Azimuth of major axis of an elliptical orbit eigenfrequencies and may have two distinct viscous damp- ing constants. Pendula operating at atmospheric pressure measured ccw from North; ◦ half-longitude on the anisosphere with amplitudes over 1 are definitely not linear oscilla- measured ccw about the upper pole L tors. Both the restoring torque and the main damping torque involve higher powers of the angular coordinates (ψD) KO: azimuth of the center of a sub-period, being at the same time and their derivatives. However the pendulum studied by the azimuth of an anisotropy axis Kamerlingh Onnes in his dissertation [16]wasagood example of a linear oscillator, since it was operated at ψs Azimuth of slow eigenaxis of a rectilinear ◦ orbit in Figure 17; corresponds to Y on amplitudes not exceeding 1 , and at low enough pressures the anisosphere to guarantee viscous damping, originating principally from the deformation of the suspension knives instead of from ωX ,ωY Frequencies of rectilinear eigenstates X and Y the surrounding gas. Linear/elliptic/circular pertaining to bob orbit desig- nates orbit polarization, in analogy with linear/elliptic/ circular polarization of light. Since the bob of a spherical 1 Introduction pendulum lies on a spherical surface, a stereographic pro- jection of bob orbit on a horizontal plane will conserve all New interest in a nicely functioning Foucault pendulum the angles already on the spherical surface containing the may be revived in preparation for the Great American bob trajectory. Hence, a trajectory in a vertical plane will Eclipse on 21st August 2017. Anomalies in the response of appear as a straight line in the stereographic projection, a many types of pendula, particularly during solar eclipses, small circle on the spherical surface will be projected as a have already been reported by different researchers dur- circle, and any trajectory in between will be very close to ing the last 60 years [1–12]. However, most attempts to an ellipse in the projection, assuming that the pole of that reproduce those observations have failed. Needless to say, stereographic projection is the anti-rest point, one pendu- successive eclipses always occur under different conditions lum length vertically above the suspension point. With of celestial environment. This makes the task of replicating those definitions, by analogy with optics where a propa- experiments more complicated. On the other hand, there gating medium may show linear birefringence (e.g., quartz has not yet appeared a Foucault pendulum standardized for propagation of light along a-axis), circular birefrin- in such a way that its response to the surrounding poten- gence (e.g., quartz for propagation of light along c-axis) tial could be prescribed. Most of the time, the physical and or in general elliptic birefringence when both are present environmental parameters recorded differ so much that it (e.g., quartz for propagation of light along any arbi- is not generally possible to conclude on the characteristics trary direction), Foucault pendula may also show linear of the pendulum used, especially the inherent anisotropy anisotropy (two rectilinear eigenstates swinging at 90◦ to parameters which drastically govern the response. one another), circular anisotropy (two circular eigenstates From Huygens’ time, the one-dimensional (1-D) pen- respectively with ccw and cw circular orbits as seen from dulum and coupling between neighboring 1-D pendula the suspension point) or in general elliptical anisotropy have extensively been studied [13]. In two-dimensional when both types are present. In this latter case, the eigen- (2-D) pendula, a new problem arises from the inherent states are two elliptical orbits with the same ellipticity, anisotropy of the system, since the eigenfrequencies in with their major axes at 90◦ to each other and with each dimension are in general different. It all boils down opposite senses of bob travel along the ellipse. to an intricate combination of two types of anisotropy: Among the numerous serious experimenters who have circular and linear. The ideal Foucault pendulum is claimed anomalous behavior of their pendula, none of assumed to possess pure circular anisotropy in the form of them succeeded in convincing the scientific community two different translation speeds along the two clockwise that they had sufficiently controlled all the parameters (cw) and counter clockwise (ccw) circular orbits which that presumably influence a spherical pendulum. In 1879, correspond to the eigenstates of the system [14,15]. despite the very high skill demonstrated in implementing However, when set into rectilinear oscillation within his pendulum setup, Kamerlingh Onnes [15] could not, vertical planes at various azimuths, a real 2-D pendulum by design, reproduce his anisotropy axes to better than always shows, for any given amplitude, a period of oscil- ∼10◦ between successive experiments. In aftermath analy- lation that varies with azimuth (linear anisotropy). If a sis, those axes appear to wander during the day. In the Foucault pendulum is set into rectilinear oscillation, pure fifties, Allais [4,5] designed two high quality pendula, the circular anisotropy causes the swinging azimuth to precess first one anisotropic by design (linear anisotropy) and the cw or ccw, the oscillation remaining essentially rectilinear other one allegedly isotropic. After experimenting for 6 to the naked eye, while pure linear anisotropy will show up years and finding statistically significant anomalies related as a tendency for the rectilinear orbit to become elliptic to alignment of celestial bodies, his work was brought into and even possibly circular. controversy and finally dismissed by the
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