Note of 4 November 1957
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MECHANICS — Observation of the movements of the paraconical pendulum
Note (*) by Mr. Maurice Allais, presented by Mr. Albert Caquot.
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The movements of a pendulum suspended by a support ball include periodic components. The experimental setup utilized and the operational technique are described. The observed movements result from four linked effects: the Foucault effect, an effect due to the suspension, random influences due to the support balls, and finally a periodic influence.
1. The object of the present Note is to give an account of the experimental arrangements which I utilized in various series of experiments performed in my laboratory at Saint-Germain-en-Laye from 1953 to 1957, for observing the movement with three degrees of freedom of a "paraconical pendulum" oscillating with an amplitude of the order of a tenth of a radian. I use the term "paraconical" for a pendulum which is suspended via a small ball.
The observed movement was characterized by remarkable periodicities of the order of 24h and 25h which I will describe subsequently.
2. Although I have used various different types of pendulum in turn, I will limit myself to description of the arrangement used during the continuous series of observations over 30 days in June-July 1955.
3. The paraconical pendulum which was used was an asymmetric pendulum consisting of a vertical bronze disk of 7.500kg, fixed on a shaft of bronze, suspended from a bronze bracket E, and hanging on a steel ball 6.5mm in diameter capable of rolling in all directions upon a plane horizontal surface S. This surface was itself supported by a circular support S' of aluminum, of thickness 4.5cm, cut away and formed with a support protrusion A. The cutaway allowed rotation of the pendulum during its movement through a total angle of 210 grades. This support S' was supported by three micrometer screws V. With the shaft of the pendulum and its bracket weighing 4.5kg, the total mass of the pendulum was 12kg and the length of the equivalent simple pendulum was about 83cm.
The steel balls were high precision S.K.F. ball bearings, and their bearing surfaces consisted oftungsten carbide and cobaltcarbide.
The experiments took place in a basement, and the center of gravity of thependulum was located about 1.5m below the natural surface of the soil. The support S" was bolted to a beam, which itself was clamped against the ceiling by a system of smaller beams.
4. By burning a thread, the pendulum was releasedfrom a position of rest every 20m, with an initial amplitude of about 0.11radian. The movement of the pendulum was observed for about 14m by sighting upon a needle provided at its lower extremity.
In general, this point described a curve which approximated to a flattened ellipse, of which the plane of the major axis was observed with a system of sights placed upon a circle C, centered upon the axis of the pendulum at rest, and graduated in grades and carrying a vernier. This system made it possible to determine the position of the plane of oscillation with an accuracy of the order of a tenth of a grade.
Furthermore, a system of two movable parallel bars B which could be displaced with respect to the sighting circle made it possible to measure the two axes of the ellipse, and also to determine the orientation of the central trihedral of inertia of the pendulum.
At the end of 14m the pendulum was stopped and was again released in the plane of the last observed azimuth. The observations in the series were accordingly chained, with the successive releases occurring every 20m, day and night. Each period of 24h accordingly included 72 series of chained observations.
5. In order to avoid any systematic influence, the steel ball on which the pendulum was supported was changed for each experiment, every 20m, and the surface S was changed at the beginning of each week of observations.
The support S was characterized by a very small difference of its own inertia in two perpendicular planes, such that under this influence the average position of the plane of oscillation tended to position itself parallel to the plane of least inertia of the support, as indicated by the vector PQ on the appended photographs, whose azimuth was about 171 grades, counting the azimuths from the South in the direct sense. When the pendulum was released in a plane different from that of PQ, there accordingly resulted a certain average tendency to form ellipses. These influences were determined precisely by experiments in which the pendulum was released in different azimuths, while eliminating the influence of time by a random choice of starting azimuth.
6. However, during a continuous series of observations, the tendency of the plane of oscillation was not to fix itself in the neighborhood of the direction PQ (as one might have expected, taking account of the Foucault effect); but the variation of its azimuth as a function of time appeared as an oscillation around the average direction PQ of very irregular appearance, at least at first glance. The observed departures were considerable. During the same period of 24h the variations of azimuth sometimes reached and passed 100 grades. The average observed azimuth P'Q' was in fact equal to 150 grades in June-July 1955, 22 grades less than the azimuth of PQ.
It is noticeable that the tangent at the beginning to the average curve of the different curves corresponding to the 2160 series of elementary observations corresponds exactly to the Foucault effect.
Analogous phenomena were observed in September-October 1955 with a symmetric pendulum consisting of a lead sphere.
Analogous results were obtained in 1953 in a small laboratory located in a brick apartment in Paris: 7 cité Fenelon. The mass of the pendulum was oscillating about 9m above ground level.
(*) Session of 4 November 1957.
(Extract of the Proceedings of the Sessions of the Academy of Sciences,
t. 245, pp. 1697-1700, session of 13 November 1957.)
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Correction — After the publication of the above Note, the author realized that he had made an error in the text which could be rectified as follows:
Page 4, §5, line 4:
instead of:
The support S was characterized by a very small difference of its own inertia in two perpendicular planes…
read:
The support S was characterized by a very small difference of its own elasticity and inertia in two perpendicular planes…
The four photographs of the experimental setup (shown elsewhere) were appended to this Note