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 7.500 kg, fixed on a shaft of bronze,
suspended from a bronze bracket E, and hanging on a steel ball 6.5 mm 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 4.5 cm, 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 4.5 kg, the
total mass of the pendulum was 12 kg and the length of the equivalent simple
pendulum was about 83 cm.
The steel balls were high precision S.K.F. ball bearings, and their bearing
surfaces consisted of tungsten carbide and cobalt carbide.
The experiments took place in a basement, and the center of gravity of the
pendulum was located about 1.5 m 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 released from a position of rest
every 20 m, with an initial amplitude of about 0.11 radian. 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 9 m 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