1078 lines
22 KiB
Plaintext
1078 lines
22 KiB
Plaintext
A confirmation of the Allais and Jeverdan-Rusu-Antonescu effects during the solar eclipse from 22 September 2006 , and the quantization of behaviour of pendulum
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V. A. Popescu 1*, D. Olenici 2*
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1Dept. of Physics, University "Politehnica "of Bucharest, Splaiul Independentei 313, 060042 Bucharest-Romania 2Planetariul Suceava, Str. Universităţii 13 A, 720229 Suceava-Romania
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In loving memory of our former Proffesors Gheorghe Jeverdan, Gheorghe Il.Rusu, Virgil Antonescu
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Abstract The experiments made with a paraconical pendulum at Suceava Planetarium
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(Romania) during annular solar eclipse from 22 September 2006 confirm once again the existence of the Allais effect (change of speed of rotation of plane of oscillation of a pendulum during an eclipse ) and Jeverdan-Rusu-Antonescu effect ( change of period of oscillation of a pendulum during an eclipse)
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Also is take in evidence the existence of the quantization of the azimuth of plane of oscillation of a pendulum which can be treated as a quantum oscillator.
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A large number of the excited states for a quantum Foucault pendulum are doubly degenerate in a similar way as the time dependence of the azimuths for a paraconical pendulum with a high sensitivity.
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The quantum eigenstates for a large energy of a Foucault pendulum predict that the probability density of finding the particle is largest near the classical trajectories.
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Although the annular solar eclipse from 22 September 2006 was not optical visible from Romania, a gravitational perturbation was detected with a sensitive paraconical pendulum and leds to ideia that gravitational perturbations who occur during an eclipse are similarly with tide when the Moon are at antimeridian .
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Keywords: Quantum mechanics, Foucault pendulum, Eclipse, Gravitation
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*Corresponding author: e-mail: cripoco@physics.pub.ro dimitrieolenici@hotmail.com
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2
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1. Introduction
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The Foucault effect It is well known that in the year 1851 J.B.L. Foucault has showed that the Earth rotation around his axis can be demonstrated by the rotation of the plane of oscillation of a long pendulum. This rotation is made constant after the law F = 15º sinφ/h, where F is the rate at which the azimutal angle change in time, φ is the latitude of the place of observation and h is the time in hours, and is named the Foucault effect. The Allais effect During solar eclipses from 30 June 1954 and 22 October 1959 Maurice Allais ( 1968 Nobel Prize in Economics ) utilized a short pendulum suspended on a ball (named paraconical pendulum) have discovered that during a solar eclipse, the speed of rotation of plane of oscillation of pendulum there are not constant as in the case of Foucault experiments, this is the eclipse effect or, the Allais effect[1]. The Jeverdan-Rusu-Antonescu effect During solar eclipse from 15 February 1961, Gheroghe Jeverdan, Gheorghe I. Rusu and Virgil Antonescu, have discovered that during a solar eclipse the period of oscillation of a Foucault pendulum is changed. In the same time this means that also the value of the gravitational acceleration g is changed, this is The Jeverdan-RusuAntonescu effect [2]. The pioneering effect In the years 1980 and 1989 the NASA's specialists have discovered that between the position of Pioneer 10, Pioneer 11 and Ulysses made by Doppler determinations and the positions calculated theoretical were a difference of around 400000 km. This anomaly is named now the Pioneering effect [3]. A similar gravity anomaly was measured during the line-up of Earth-Sun-JupiterSaturn in May 2001 [4]. During the total solar eclipse in 1977, the measurements with a high-precision gravimeter have detected a decrease in the earth’s gravity and the effect occurred immediately before and after eclipse but not at its height [5]. These anomalies have determined us to do pendulum experiments in distinct mode such: solare eclipses, lunar eclipses, planetary alignments etc. One of these results is presented in this papier.
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2.The confirmation of the Allais effect during solar eclipse from 22 September 2006
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In order to measure the azimuth of the plane of oscillation, the second author used a pendulum of 3.05m length and 8kg mass suspended with a ball (paraconical pendulum) with shaped like a horizontal biconvex lens and with distinctively rounded edges so as to reduce wind drag.
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For measure the azimuth is utilized a vernier with 0.1º precision.
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3
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Figs.1-3 show the experimental azimuth as a function of time for 6 series (successive chained observations each of 9 minutes) and the expected exact azimuth for Foucault effect (line named F), during an annular solar eclipse of 22 September 2006. Its maximum occurred at 14h 40.2 min Romanian Summer Time at Suceava, or 11h 40.2 min UT.
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From these graphs we see that in the days before and after eclipse on 21 and 23 September the curves have the tendency to have negative values whereas in the day of eclipse the tendency is to have positive values. This fact confirms the existence of the Allais effect.
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Another very interesting fact is that the position of plane of oscillation of pendulum has quantized values! The series 1, 2, 3 and 4, 5, 6 from 21 September 2006 (Fig.1) can be considered together because of similarities for the time dependence of the azimuth, analog with the degenerate states of quantum levels from atomic physics. Also, the series 3 and 4 from 22 September 2006 (Fig.2) can be grouped together. This leads us to attempt to give a quantum treates of behaviour of pendulum.
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3. Classical Foucault pendulum
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For a pendulum small compared with the earth, the vertical motion can be neglected and at the same time the term in ω2 where ω is the angular velocity of the
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earth. Using the relations between the coordinates q1, q2, q3 and the length l of the pendulum
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q12 + q2 2 + q3 2 = l 2 , q3 = −l
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1− q12 + q2 2 l2
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≅ −l + q12 + q2 2 + (q12 + q2 2 ) 2
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2l
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8l 3
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(1)
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and changing the origin in the center of the oscillating body, we have
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q3
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=
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q1 2
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+ q22 2l
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or
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q3
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=
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q12 + q2 2 2l
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+ (q12 + q2 2 ) 2 8l 3
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where
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(2a) (2b)
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q12 + q2 2 < 1
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(3)
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2l
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Using the approximations (2a), the Lagrange L and Hamiltonian H functions for a
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Foucault pendulum are given by the relations
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L
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=
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m(q&12 + 2
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q&
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2 2
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)
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+
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mω
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sin
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λ(q1q& 2
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−
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q2 q&1 )
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−
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mg 2l
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(q12
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+
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q
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2 2
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)
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(4)
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H
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=
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p12
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+
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p
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2 2
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2m
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+ ω sin λ( p1q2
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−
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p2 q1 ) +
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mg 2l
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(q12
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+
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q
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2 2
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)
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(5)
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4
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where m is the mass of the particle, l is the length of the pendulum, g is the earth
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acceleration, λ is the latitude, q1 and q2 are the Cartesian coordinates in the horizontal
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plane, q&1 and q&2 are the velocities and p1 , p2 are the momentum values
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p1 = mq&1 − mω sin λq2
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(6)
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p2 = mq& 2 + mω sin λq1
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(7)
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The corresponding motion equations are a system of coupled second ordinary
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differential equations that are the mathematical model for the Foucault pendulum within
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the approximation of little amplitude displacements
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q&&1
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− 2ω sin
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λq& 2
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+
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g l
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q1
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=
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0
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(8)
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q&&2
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+ 2ω sin λq&1
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+
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g l
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q2
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=
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0
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(9)
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Multiplying the equation (9) by i and adding it to the equation (8) gives the differential
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equation for motion in the horizontal plane
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ξ&&+ 2iω sin λξ& + ω02ξ = 0
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(10)
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where ξ = q1 + i q2 is a complex variable and ω02 = g/l .The solution of the equation (10)
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is (ω2 has been neglected in comparison withω02 )
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ξ (t) = [A exp(iω0t) + B exp(−iω0t)]exp(− iω3 t)
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(11)
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whereω3 = ω sin(λ) . If the pendulum is released from a point of maximum amplitude a
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= q1 (0) , q2 (0) = 0 , q&1 (0) = 0 , q&2 (0) = 0 (the initial conditions), it never passes through ξ
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= 0 and the solutions of the motion equations becomes
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q1 (t)
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=
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a cos(ω3t)cos(ω0t)
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+
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aω3 ω0
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sin (ω 3 t ) sin(ω 0 t )
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(12)
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q1 (t)
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=
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−a sin(ω3t)cos(ω0t) +
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aω3 ω0
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cos(ω 3 t ) sin(ω 0 t )
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(13)
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In the polar coordinates (ρ and θ) we obtain
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θ
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=
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arctan
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⎡ ⎢ ⎣
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q2 q1
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⎥⎤,θ& ⎦
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=
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−
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g
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2ω3 (−g + lω32 ) sin 2 (ω0t) + lω3 2 + (−g + lω3 2 ) cos(2ω0t)
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(14)
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The laws of conservation of energy E and angular momentum La are satisfied with
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these coordinates
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E r L
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=
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a
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ma 2l = rr
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2
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×
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(g − lω3 2 ) mvr = m(q1
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q&
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2
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− q2 q&1 ) + (q12
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+
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q2 2 )mω3
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=
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ma 2ω3
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(15) (16)
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Changing to a new variable
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⎜⎜⎝⎛
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q1'
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q
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' 2
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⎟⎟⎠⎞
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=
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⎜⎜⎝⎛
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cos(ω 3 t ) sin (ω 3 t )
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−csoisn(ω(ω33tt))⎟⎟⎠⎞⎜⎜⎝⎛
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q1 q2
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⎟⎟⎠⎞
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we obtain
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q1' = a cos(ω0t)
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(17) (18)
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5
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q
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' 2
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=
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aω 3 ω0
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sin(ω 0 t )
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(19)
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in a rotating system with the angular velocity ω sin(λ) . In this system, the Hamiltonian
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(5) is
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H'
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=
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p1' 2
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+
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p
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' 2
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2
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2m
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+
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m 2
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(ω
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2 3
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+
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ω
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2 0
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)(q1'
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2
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+
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q
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' 2
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2
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)
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The period on elliptical path is
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τ = 2π ω02 +ω32
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(20) (21)
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and the time for a complete rotation of the ellipse plan is
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T = 2π = 2π
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(22)
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ω3 ω sin(λ)
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The experiment on the Foucault pendulum is accompanied by the Airy precession
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(small ellipses appear) with an angular velocity
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ωA
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=
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−
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3 8
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αβω
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0
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(23)
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where α and β are the major and minor axes in radians of the elliptical trajectory of the
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pendulum
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α = a , β = aω3
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(24)
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l
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l ω02 +ω32
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In this case in the place of ω3 = ω sin(λ) we must to put
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ω
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c 3
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=
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ω
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3
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(1
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−
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3a 2 8l 2
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)
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(25)
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in the relations (12) and (13). Thus in place of the relations (14), (15), (16) and (21) we
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obtain
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( ) θ& c = (3a 2 − 8l 2 )ω3 − 64gl 3 + (3a 2 − 8l 2 ) 2 ω32 sin 2 (ω0t) ( ) 8l 2 64gl 3 cos 2 (ω0t) + (3a 2 − 8l 2 ) 2 ω32 sin 2 (ω0t)
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( ) E c = ma 2 128l 4
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64gl 3 − (3a 2 − 8l 2 ) 2 ω3 2
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Lr ca
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=
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ma
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2ω3
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(1
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−
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3a 2 8l 2
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)
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τc =
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2π
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ω02
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+ ω3 2 (1−
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3a 2 8l 2
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)2
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(26) (27) (28) (29)
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If we use the approximation (2b), then in place of the relation (20) we have
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H'
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=
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p1' 2
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+
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p
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' 2
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2
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2m
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+
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m 2
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(ω
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2 3
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+
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ω
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2 0
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)(q1'
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2
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+
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q
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' 2
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2
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)
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+
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mg 8l 3
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(q1' 4
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+
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q
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' 2
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4
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+
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2q1'
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2
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q
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' 2
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2
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)
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(30)
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For a classical Foucault pendulum with l = 3.05m, m = 8Kg, g = 9.81 m/s2, we
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obtain E = 0.68058884j, Ec = 0.68058885j, La = 0.0000228075Kgm2/s, Lac =
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6
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0.0000228076 Kgm2/s, τ = 3.5034448s and τc = 3.50523186s. For Suceava the astronomical latitude is λ = 47o39’ and ω3 = ω sin(λ) = 0.0000538929rad/s.
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Fig. 4 shows a simulation of the path of the Foucault pendulum (a), of the Allais pendulum (b), the differences between their coordinates (c), the temporal derivative of the azimuth for a Foucault pendulum (d), the differences between temporal derivatives for both pendulums (e) and the elliptical trajectory (the major and minor axes of the elliptical trajectory of the pendulum are a = 0.23m and b = 6.9×10-6m, respectively) in the rotating reference system of the Foucault pendulum (f).
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Fig. 5 shows the same graphs as in Fig.4 but for a longer pendulum (l =3.05×20m, τ = 15.66788s, a = 0.23m, b = 3×10-5m). θ& is zero for t = τ, 2τ, 3τ, 4τ, etc. An increase in the length of the pendulum leads to a larger of the Foucault effect in comparison with the Allais effect and the differences are visible in a perpendicular direction to that of the swing direction (Figs.4-5, (c)).
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4. Quantum Foucault pendulum
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The Hamiltonian operators for a quantum Foucault pendulum in a fixed system on
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the earth, without and with the Airy precession are, respectively
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( ) Hˆ
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=
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−
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h 2 2m
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⎜⎜⎝⎛
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∂2 ∂x 2
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+
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∂2 ∂y 2
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⎟⎟⎠⎞
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+
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m 2
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ω02
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x2 + y2
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− a 2 mω3 2
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(31)
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( ) Hˆ
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=
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−
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h 2 2m
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⎜⎜⎝⎛
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∂2 ∂x 2
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+
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∂2 ∂y 2
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⎟⎟⎠⎞
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+
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m 2
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ω02
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x2 + y2
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−
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a
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2
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mω3 2
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(1 −
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3a 2 8l 2
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)
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2
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(32)
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In this case the corresponding Schrödinger equations take the form
|
||
|
||
( ) −
|
||
|
||
h 2 2m
|
||
|
||
⎜⎜⎝⎛
|
||
|
||
∂ 2ψ ∂x 2
|
||
|
||
+ ∂ 2ψ ∂y 2
|
||
|
||
⎟⎟⎠⎞ +
|
||
|
||
m 2
|
||
|
||
ω
|
||
|
||
0
|
||
|
||
2
|
||
|
||
x2
|
||
|
||
+
|
||
|
||
y2 ψ
|
||
|
||
− a 2 mω3 2ψ
|
||
|
||
=
|
||
|
||
Eψ
|
||
|
||
(33)
|
||
|
||
( ) −
|
||
|
||
h 2 2m
|
||
|
||
⎜⎜⎝⎛
|
||
|
||
∂ 2ψ ∂x 2
|
||
|
||
+ ∂ 2ψ ∂y 2
|
||
|
||
⎟⎟⎠⎞ +
|
||
|
||
m 2
|
||
|
||
ω
|
||
|
||
0
|
||
|
||
2
|
||
|
||
x2
|
||
|
||
+
|
||
|
||
y2
|
||
|
||
ψ
|
||
|
||
−
|
||
|
||
a
|
||
|
||
2
|
||
|
||
mω
|
||
|
||
3
|
||
|
||
2
|
||
|
||
(1
|
||
|
||
−
|
||
|
||
3a 2 8l 2
|
||
|
||
)ψ
|
||
|
||
=
|
||
|
||
Eψ
|
||
|
||
(34)
|
||
|
||
The Hamiltonian operator and the Schrödinger equation for a quantum Foucault
|
||
|
||
pendulum in a rotating system are given by
|
||
|
||
( )( ) Hˆ
|
||
|
||
=−
|
||
|
||
h 2 2m
|
||
|
||
⎜⎜⎝⎛
|
||
|
||
∂2 ∂x 2
|
||
|
||
+
|
||
|
||
∂2 ∂y 2
|
||
|
||
⎟⎟⎠⎞ +
|
||
|
||
m 2
|
||
|
||
ω02
|
||
|
||
+ ω32
|
||
|
||
x2 + y2
|
||
|
||
(35)
|
||
|
||
( )( ) −
|
||
|
||
h 2 2m
|
||
|
||
⎜⎜⎝⎛
|
||
|
||
∂ 2ψ ∂x 2
|
||
|
||
+
|
||
|
||
∂ 2ψ ∂y 2
|
||
|
||
⎟⎟⎠⎞ +
|
||
|
||
m 2
|
||
|
||
ω02
|
||
|
||
+ ω32
|
||
|
||
x 2 + y 2 ψ = Eψ
|
||
|
||
(36)
|
||
|
||
If we use the approximation (2b), then in place of the relation (36), we have
|
||
|
||
( )( ) −
|
||
|
||
h 2 2m
|
||
|
||
⎜⎜⎝⎛
|
||
|
||
∂ 2ψ ∂x 2
|
||
|
||
+
|
||
|
||
∂ 2ψ ∂y 2
|
||
|
||
⎟⎟⎠⎞ +
|
||
|
||
m 2
|
||
|
||
ω02
|
||
|
||
+ ω3 2
|
||
|
||
x 2 + y 2 ψ + mg (x 4 + y 4 + 2x 2 y 2 )ψ = Eψ 8l 3
|
||
|
||
(37)
|
||
|
||
The values of the energy can be obtained from the solution of the Schrödinger
|
||
|
||
equations (33), (34), (36) and (37). A large number of the excited states for a quantum
|
||
|
||
Foucault pendulum are pair of the doubly degenerate.
|
||
|
||
7
|
||
|
||
We have solved the Schrödinger equations for the given boundary conditions (the
|
||
|
||
Dirichlet boundary condition at the ends of the interval where the wave function can be
|
||
|
||
approximated with 0) by using the Galerkin’s variant of the finite element method, with
|
||
|
||
triangular grid and variable step [7].
|
||
|
||
Fig. 6 show the calculated probability density, with the Schrödinger equation (37),
|
||
|
||
for the first pairs of excited doubly degenerate states of a quantum Foucault pendulum (l = 3.05m, a = 0.23m, b = 0.023m, m = 8Kg, g = 9.81 m/s2) with the energies
|
||
|
||
(in h = 1 units): e1 = e11 = 18.588058; e2 = e22 = 31.521995; e3 = e33= 48.414085;
|
||
|
||
e4 = e44 = 63.733378; e5 = e55 = 68.378136; e6 = e66 = 84.758123;
|
||
|
||
e7
|
||
|
||
= e77 = 91.269980; e8 = e88 = 112.904708; e9 = e99 = 117.030554;
|
||
|
||
e10 = e1010 = 136.740011; e11 = e1111= 144.932609; e12 = e1212 = 145.616593;
|
||
|
||
e13 = e1313 = 163.043895 (for two higher states e = ee = 291.502732;
|
||
|
||
E
|
||
|
||
= EE = 899.22807).
|
||
|
||
The quantum eigenstates for a large energy of a Foucault pendulum predict that
|
||
|
||
the probability density of finding the particle is largest near the classical trajectories
|
||
|
||
(compare the Figs.6 (E),(EE) with Fig.4 (a)).
|
||
|
||
A large number of the excited states for a quantum Foucault pendulum are doubly
|
||
|
||
degenerate in a similar way as the time dependence of the azimuths for a paraconical
|
||
|
||
Allais pendulum with a high sensitivity.
|
||
|
||
5. The variation of the gravitational acceleration - the JRA effect
|
||
|
||
First we use the method of Jeverdan-Rusu-Antonescu (applied for the first time to
|
||
|
||
the eclipse from 15 February 1961, developed and utilized by Olenici beginning with
|
||
|
||
solar eclipse from 11 August 1999) and we have determined the experimental value of
|
||
|
||
the period of oscillation of the pendulum before (Tb = 3.5060877s) and after
|
||
|
||
(Ta = 3.5061041s) the maximum of the eclipse with the average (Tm = 3.5060959s) assumed to correspond to the local (Suceava) gravitational acceleration gs = 9.81m/s2.
|
||
|
||
The experimental value of the period of oscillation of the pendulum during the
|
||
|
||
maximum of the eclipse is Te = 3.5060706s.
|
||
|
||
From the relations Tm = 2π l / g s and Te = 2π l / ge
|
||
|
||
we obtain
|
||
|
||
ge = gs (Tm / Te )2 = 9.8101412 m/s2. Thus, our measurements show that during the
|
||
|
||
maximum of the sun eclipse from 22 September 2006, the gravitational acceleration ge
|
||
|
||
is increased. This confirme the existence of the Jeverdan-Rusu-Antonescu effect.
|
||
|
||
For measure of period of oscillation was utilized a cronometer with 0.01s
|
||
|
||
precision, electrically ordered by point of pendulum, and where take in account 150
|
||
|
||
oscillation at every determination.
|
||
|
||
Also, in order to estimate the variation of the gravitational acceleration with a new
|
||
|
||
method, we put in the relation (14), the experimental azimuth value (in radians) in the
|
||
|
||
place of θ and solve a transcendental equation in g (we take only a value which is very
|
||
|
||
close to a given g).
|
||
|
||
Tables 1 – 3 and Fig. 7 show the calculated gravitational acceleration as a
|
||
|
||
function of 6 series in the day (22 September 2006) of the annular solar eclipse.
|
||
|
||
Fig. 8 shows the calculated differences between the gravitational accelerations
|
||
|
||
for successive series in the day (22 September 2006) of the annular solar eclipse. The
|
||
|
||
8
|
||
results are qualitatively similar with the measurements of Wang et al. [5] and the comments of Duif [8] (a decrease in the gravitational acceleration before and after eclipse and an increase in the time of eclipse).
|
||
6. Conclusions
|
||
Although the annular solar eclipse from 22 September 2006 was not optical visible from Romania, a gravitational perturbation was detected with a sensitive paraconical pendulum. This show us that gravitational perturbations who appear during an eclipse in a zone of antieclipse are similarly with tide produced by the Moon when are at antimeridian.
|
||
Thus we obtained a change in the velocity of the azimuth of plane of oscillation of a pendulum and a confirmation of the Allais effect for a measurement in a place where the eclipse is not optical visible.
|
||
The results obtained by using two methods show an increase of the gravitational acceleration at the maximum of the eclipse and a confirmation of the Jeverdan-RusuAntonescu effect (eclipses from 15 February 1961 , July 1991 , August 1999 etc).
|
||
A large number of the excited states for a quantum Foucault pendulum are doubly degenerate in a similar way as the time dependence of the azimuths for a lenticular aerodynamic paraconical pendulum with a high sensitivity in the time of the annular solar eclipse from 22 September 2006.
|
||
This result consolidate the ideia of quantization of gravity and the existence of a quantum mecanics at cosmical level.
|
||
References
|
||
[1] R. E. Matthew (ed.), Pushing Gravity: New perspectives on Le Sage’s theory of gravitation, Montreal, Quebec 219, 259 (2002) [2] M. Allais, L’Anisotropie de l’Espace, Paris 166 (1997) [3] G. T. Jeverdan, G. I. Rusu and V. I. Antonescu, An. Univ. Iasi 7, 457 (1961); G. T. Jeverdan, G. I. Rusu and V. I. Antonescu Science et Foi 2, 24 (1991); D. Olenici, Anuarul Complexului Muzeal Bucovina XXVI-XXVII-XXVIII, 659 (19992000-2001); XXIX-XXX, 403 (2002-2003) M. F. C. Allais Aero/Space Engineering 18 Sep. 46, Oct. 51 (1959) [4] http://www.allais.info/priorartdocs/olenici.htm; G. C. Vezzoli, Infinite Energy 9:53, 18 (2004) X .Amador , Jurnal of Physics,Conference Series, 24, 247-252, 2004
|
||
[5] Qian-Shen Wang et al. Physical Review D 62, 041101 (2000)
|
||
[6] J. –F. Pascal-Sánches, arXiv:gr-qc/0207122v1 30 Jul. 2002 [7] V. A. Popescu Phys. Lett. A 297, 338 (2002) [8] C. P. Duif, arXiv:gr-qc/0408023 v5 31 Dec 2004
|
||
|
||
9
|
||
Fig. 1. The experimental azimuth as a function of time for 6 series (1 from 11h 40min to 12h 40min, 2 from 12h 40min to 13h 40min, 3 from 13h 40min to 14h 40min, 4 from 14h 40min to 15h 40min, 5 from 15h 40min to16h 40min, 6 from 16h 40min to 17h 40min) and the expected exact azimuth for Foucault effect (line F) in a day (21 September 2006) before a partial solar eclipse.
|
||
|
||
10
|
||
Fig. 2. The experimental azimuth as a function of time for 6 series (1 from 11h 40min to 12h 40min, 2 from 12h 40min to 13h 40min, 3 from 13h 40min to 14h 40min, 4 from 14h 40min to 15h 40min, 5 from 15h 40min to 16h 40min, 6 from 16h 40min to 17h 40min) and the expected exact azimuth for Foucault effect (line F) in the day (22 September 2006) of the annular solar eclipse (its maximum occurred at 14h 40.2min Romanian Summer Time or 11h 40.2 min UT at Suceava).
|
||
|
||
11
|
||
Fig. 3. The experimental azimuth as a function of time for 6 series (1 from 11h 40min to 12h 40min, 2 from 12h 40min to 13h 40min, 3 from 13h 40min to 14h 40min, 4 from 14h 40min to 15h 40min, 5 from 15h 40min to 16h 40min, 6 from 16h 40min to 17h 40min) and the expected exact azimuth for Foucault effect (line F) in a day (23 September 2006) after eclipse.
|
||
|
||
12
|
||
Fig. 4. Simulation of the path of the Foucault pendulum (a), of the paraconical pendulum (b), the differences between them (c), the temporal derivative of the azimuth for a Foucault pendulum (d), the differences between derivatives for both pendulums (e) and the elliptical trajectory in the rotating reference system of the Foucault pendulum (f) for l = 3.05m, τ = 3.5034448s, a = 0.23m, b = 6.9×10-6m, b/l = 2.3×10-6.
|
||
|
||
13
|
||
|
||
Fig. 5. Simulation of the path of the Foucault pendulum (a), of the paraconical
|
||
|
||
pendulum (b), the differences between them (c), the temporal derivative of the azimuth
|
||
|
||
for a Foucault pendulum (d), the differences between derivatives for both pendulums (e)
|
||
|
||
with the same precision as in Fig. 1, and the elliptical trajectory in the rotating reference
|
||
|
||
system of the Foucault pendulum (f) for l =3.05×20m, τ = 15.66788s, a = 0.23m, b =
|
||
|
||
3×10-5m,
|
||
|
||
b/l = 5.1×10-7.
|
||
|
||
14
|
||
|
||
15
|
||
|
||
16
|
||
|
||
17
|
||
Fig. 6. The calculated probability density (with a contour plot) with the Schrödinger equation (37) in h = 1 units, for the first pairs of excited doubly degenerate states of a
|
||
|
||
18
|
||
quantum Foucault pendulum (l = 3.05m, a = 0.23m, b = 0.023m, m = 8Kg, g = 9.81 m/s2).
|
||
|
||
19
|
||
Fig. 7. The calculated gravitational acceleration as a function of 6 series in the day (22 September 2006) of a annular solar eclipse.
|
||
|
||
20
|
||
|
||
Fig. 8. The calculated differences between the gravitational accelerations for successive series (the differences between the successive columns in Tables 2-3) in the day (22 September 2006) of the annular solar eclipse.
|
||
|
||
Table 1. The calculated gravitational acceleration as a function of 6 series in the day before (21 September 2006) and after (23 September 2006) of a partial solar eclipse. For the
|
||
grouped series we have used a mean value.
|
||
|
||
t(min)
|
||
9 18 27 36 45 54
|
||
|
||
g (21 Sept.) (11h40m -14h39m) 9.8246212 9.8088554 9.8035510 9.8008999 9.8120359 9.8088538
|
||
|
||
g (21 Sept.) (14h40m -17h39m)
|
||
9.8247403 9.8086101 9.8035535 9.8009006 9.8120358 9.8088537
|
||
|
||
g (23 Sept) (11h40m – 17h39m)
|
||
9.8247576 9.8088481 9.8035456 9.8009027 9.8120361 9.8088549
|
||
|
||
Table2. The calculated gravitational acceleration as a function of 6 series in the day (22 September 2006) of the annular solar eclipse.
|
||
|
||
t(min)
|
||
|
||
g (22 Sept.) (11h40m -12h39m)
|
||
|
||
g (22 Sept.) (12h40m -13h39m)
|
||
|
||
g (22 Sept) (13h40m-14h39m)
|
||
|
||
21
|
||
|
||
9
|
||
|
||
9.8247375
|
||
|
||
9.8247572
|
||
|
||
18
|
||
|
||
9.8088794
|
||
|
||
9.8088480
|
||
|
||
27
|
||
|
||
9.8035533
|
||
|
||
9.8035470
|
||
|
||
36
|
||
|
||
9.8009006
|
||
|
||
9.8008955
|
||
|
||
45
|
||
|
||
9.8120361
|
||
|
||
9.8120295
|
||
|
||
54
|
||
|
||
9.8088536
|
||
|
||
9.8088569
|
||
|
||
9.8247626 9.8088513 9.8035495 9.8008990 9.8120352 9.8088529
|
||
|
||
Table3. The calculated gravitational acceleration as a function of 6 series in the day (22 September 2006) of the annular solar eclipse.
|
||
|
||
t(min)
|
||
9 18 27 36 45 54
|
||
|
||
g (22 Sept.) (14h40m -15h39m)
|
||
9.8247628 9.8088513 9.8035495 9.8008990 9.8120352 9.8088529
|
||
|
||
g (22 Sept.) (15h40m -16h39m)
|
||
9.8247592 9.8088504 9.8035489 9.8008986 9.8120348 9.8088525
|
||
|
||
g (22 Sept) (16h40m-17h39m)
|
||
9.8247592 9.8088498 9.8035483 9.8008979 9.8120343 9.8088523
|
||
|