by 2 Baron Loránd EÖTVÖS Member of Ac. Sci. Hung.
1. §. Introduction
It is a well-known postulate Of the Galilei-Newton mechanics that any body moving to the east on the Earth, should lose weight, while that moving to the west, should gain weight. This variation of gravitational acceleration in a static solar system:
1 Translated by dr. Éva Kilényi, linguistically corrected by Judy Elliott
2 The eminent author of this treatise completed the manuscript in German during a terminal illness which caused his death on the 8th of April, 1919. He had it sent to the editorial office of thejournal Annalen der Physik with the date 31st March, 1919. The paper appeared in Volume 59, pp. 743-752, towards the end of the year 1919. The proof-reading was done by the present editor of our Bulletin and Mr J. Fekete, who had been a collaborator of the deceased for many years. Simultaneously with the above date the author handed the manuscript over to the editor with the request that its Hungarian translation and the theoretical elucidation necessary in 3. §. be done by him and that he would be responsible for publishing it in the Bulletin. All these requests were carried out with the co-operation of Mr J. Fekete.
The unfortunate state of affairs in Hungary, the forced interruption of the activity of the Academy and printing difficulties caused the treatise to appear in Hungarian later than in German.
LORÁND EÖTVÖS
where Ω means the angular velocity of the Earth’s rotation, which is
Let us mark the geographical latitude on the Earth by φ, and by dy/dt the velocity of the body on the Earth in a coordinate system, whose X, Y, Z axes coincide with the celestial North, East and the vertical downwards direction, respectively.
3 Namely: 1. If the rotating Earth is regarded as a sphere in first approximation, at any point P of latitude φ, its peripheral speed caused by the rotation will be:
νΩ =O7?cOS(p, where Ω marks the angu
lar velocity and R the radius of the Earth (see Fig. 7). The centrifugal force exerted on mass m in point P is :
which is parallel to the plane of the equator and its direction is QP. Its component in the direction of CP is:
which acts against the gravitational force of the Earth.
2. If m has a velocity component dy/dt rela
tive to the Earth’s surface, acting in the direction of the prevailing peripheral veloc
ity vn, its mechanical influence on the cen- trifugal force can be formulated by substituting vq by dy/dt), where dyldt is significantly smaller than vq . Accordingly, the centrifugal force will be
In approximation:
t dy
mQ. flcosrp + 2mSl-γ-;
__ dt
Its component in the PC direction of the gravitational force is:
_ dy
-mClRcosq -2mQcos<p-f-.
at
EXPERIMENTAL DEMONSTRATION OF THE GRAVITY VARIATION ... 3
The experimental demonstration of this simple postulate, however, was not successful a quarter of a century ago. The present success is due to those mature considerations which have led us to the perception of gravity conditions at sea. And, oddly enough, it was a mistake which led us on the right path. (‘Ci tius enim emergit veritas e falsitate, quam e conftisione.’ Baco) Two noteworthy sea-trips made by professor Hecker of Berlin, the first on the Atlantic in 1901, the second on the Indian Ocean between March 23, 1904 and April 8, 1905, captured the interest of all specialists, including myself, who were studying terrestrial gravity4.
I soon recognised, that in the otherwise high-accuracy computations the influence of the ship's movement, which should have manifested itself with a value computable in advance, was not in accordance with these presumptions.
To avoid even a shadow of doubt it seemed desirable to go over the observations carefully and recalculate them. But professor Hecker, whom I requested to have these computations carried out, went far and beyond that.
Conquering all manner of difficulties he succeeded in persuading the Russian tsarist government to equip a new expedition; thus in May, 1908 he was able to make another sea-trip and carry out new measurements on the Black Sea, in some part on the same line but in opposite directions5. The difference between the ship’s speed towards the west and east was near to6 45 km/h, the
4 Bestimmung der Schwerkraft auf dem Atlantischen Ozcan, sowie in Rio de Janeiro, Lissabon und Madrid. Von O. Hecker. Veröffentlichung des königlich preussischen geodätischen Institutes. Neue Folge Nr. 11, Folio, pp. 1-137, mit neun Tafeln. Berlin,
1903.
Bestimmung der Schwerkraft auf dem Indischen und Grossen Ozean und deren Küsten, sowie crdmagnctische Messungen. Von Prof. Dr. O. Hecker. Zentralbureau der internationalen Erdmessung. Neue Folge der Veröffentlichungen. Nr. 16. Folio, pp. 1-233, mit zwölf Tafeln. Berlin, 1908.
5 Bestimmung der Schwerkraft auf dem Schwarzen Meere und an dessen Küste, sowie neue Ausgleichung der Schwerkraft auf dem Atlantischen, Indischen und Grossen Ozean. Von Prof. Dr. O. Hecker. Zcntralburcau der intemationalen Erdmessung.
Neue Folge der Veröffentlichungen. Nr. 20. Folio, pp. 1-160, mit vier Tafeln. Berlin, 1910.
6 See page 103. of the above-cited publication
4 LORAND EOTVOS
value of gravity difference according to Equation (1) approximately:
Ag = 0.707 · 0.000146 -4 500 000 = 0.129. (2)
3600 V
This gravity difference is high enough that it is possible to observe it even with the most simple experiments carried out with the methods to be described in the following section. Those apparent contradictions indicated by Hecker's observations at sea assisted us in carrying out the first experimental demonstration of the old theory.
2. §. The possibility of experimental demonstration at much smaller ve
locities in laboratory conditions. The method of resonance According to Equation (1) on latitude 45° each gram of an eastward moving body at a speed of 1 cm/sec suffers a gravitational decrease of Ag = - 0.000103, i.e. in the absolute c.g.s. system one tenth of the unit of acceleration, meaning the weight of the body changing by its one ten millionth part.
Thus if a well-fed man weighing 100 kg walks at leisure, at a speed of 1 m/sec on the surface of the Earth — supposed to be of spherical shape — towards the east, he will become lighter by approximately 2 100,000 100/10,000=2000 c.g.s. i.e. by 2 grams, which denotes two hun
dred thousandth part of his original weight.
However, the experiments, which need rectilinear motion of constant velocity are hardly realizable; we must therefore resort to the more easily and accurately realizable circular motion.
Let us rotate a beam-like body loaded on both ends round a vertical axis which passes through its centre of gravity in its state of balance; while during the swinging of the beam this centre of gravity stays near to the vertical axis of rotation. In this case the masses periodically move to the east and west, respectively, thereby initiating, due to the respective gravity variations, a periodic swinging motion whose amplitude increases by repetition before reaching a maximum value limited by attenuating forces. This forced swing
ing motion is a special case of forced oscillations, like that of acoustic
EXPERIMENTAL DEMONSTRATION OF THE GRAVITY VARIATION ... 5
resonance, whose respective theory of point motion is discussed by Helm
holtz in a masterly manner in his theoretical physics.
Here, however, the forced swinging of the beam has to be treated as the oscillation of a compact body, therefore necessitating a presentation of the theory of the beam’s motion in a simple but somewhat detailed descrip
tion.
3. §. Theory of a balance rotated and subjected to impulses. The maxi
mum amplitude due to resonance
In the following considerations we have restricted the procedure to a swinging body symmetric to three perpendicular planes, which can swing freely round one of its horizontal axes (e.g. on edges).
Let Oa, Ob, Oc axes of a Cartesian co-ordinate system in rigid connection with the body (see Fig. 2), where Ob is the constant horizontal swing axis. In the state of equilibrium the (aOb) plane is horizontal and the positive c is directed vertically downwards, coinciding with the (¾Oő^ plane and axis c0, respectively.
Let X, Y, Z be the co-ordinate axes of a celestial system in rigid connection with the Earth, whose X axis is directed to the north, Y to the east and Z vertically downwards, furthermore, let ρΛ and pcbe the radii of those circles or arcs which are described by an m mass-point of the swinging body round axes b and c0, respectively.
The planes of symmetry of the swinging body are: the first plane (bOc) and the second plane (cOa) plane. At the same time these two planes represent the two co-ordinate planes in rigid connection with the balance. The third plane of symmetry is not the (aOb) plane, but the parallel plane passing through O0. This point O0 is the centre of gravity of the beam.
The edge of the beam coincides with the constantly horizontaíOb axis which does not pass through the centre of gravity, O0, lying somewhat below the edge, similarly to a common pair of scales. Hence the position of balance is a constant position; if the whole instrument is not rotated round the vertical OZ axis, the beam swings as a common physical pendulum.
7 Vorlesungen über theoretische Physik. Band I. Abtheilung 2. Dynamik discrcter Massen-Punktc. Herausgegeben von O. Krigar-Menzel. Leipzig, J. A. Barth. 1898.
pg. 95, pg. 119.
6 LORÁND EÖTVÖS
Fig. 2.
Furthermore, let it be, according to Fig. 2, Oa^ Ob0, Oc0 the axes of a Cartesian co-ordinate system, rotating together with the whole instrument:
the stand and holder of the balance, round the vertical axis OZ in such a way that Oc0 and Ob0 always coincide with OZ and Ob, respectively, while Oa0 remains constantly in the horizontal plane.
According to this definition the (α0Οό0) plane always remains horizon
tal, while planes (ö0Oc0) and (c0Oo0) remain always vertical. The relation between the two co-ordinate systems of the common origin, O, can be recognized by rotating the (a000c0) system round the common Ob=ObQ axis by angle ε.
EXPERIMENTAL DEMONSTRATION OF THE GRAVITY VARIATION ... 7
/. Non-rotated beam. Simple harmonic oscillation
If attenuation is neglected, the torque exerted by the gravitational force on the beam can be expressed by:
where M represents the mass of the oscillating beam, g the acceleration of the gravitational force on the bodies resting on the Earth’s surface, s the distance between the centre of gravity of the beam and the edge of oscillation, i.e.
OOo=s, and ε the angle made by 5 and the direction of gravity. If Kb denotes the moment of inertia of the oscillating body referring to the edge 0b, the equation of motion of the beam can be described as:
where torque F(s) is negative, meaning it can decrease angle ε.
If the amplitude of the oscillation is small, the period of double oscillation, r0, in the first approximation, is:
2. Non-rotated beam. Attenuated simple harmonic oscillation
If attenuation is taken into consideration, its retarding torque can be described, at limited angular velocities, as:
the resulting equation of movement:
or at small amplitudes, in first approximation:
s LORÁND EÖTVÖS
where:
The common form of the well-known solution of the above equation is:
in which £ and δ are two integration constants, while the double oscillation period, T, from
The ratio of two consecutive maximum angular deviations:
meaning the attenuation ratio, its natural logarithm,
is the logarithmic decrement.
3. Rotated beam. The influence of gravitational variations and the centrifugal force on the oscillations of the beam
In this general case, defined in the title, the torques due to gravitational variations and to the centrifugal force are added to the torques and F(*>
described by Equations (3) and (6).
EXPERIMENTAL DEMONSTRATION OF THE GRAVITY VARIATION ... 9
3a. The rotation round axis OZ, at a period of 3 gives an angular velocity of
to the whole instrument, and with it, to the beam. The y co-ordinate of a mass-point m, in the (XYZ) co-ordinate system, in rigid connection with the Earth, can be described in the following way:
where pc represents the perpendicular distance of m from axis of rotation OZ, a the angle made by the rotating straight pc=O ’P and the rotating plane (aOc), while 2 π r/3 denoting the rotational angle made by the rotating plane (cOa) and the stable (XOZ) plane (the plane of the geographical meridian passing through O).
Here we have to take into consideration that angle a is independent of time, and that this angle has a constant value for each m point of the beam.
Accordingly, the prevailing gravity variation, i.e. the weight variation of the rotated mass-point m can be described by Equations (1) and (12) in the following way:
or, bearing in mind that angle a may have different values for different mass-points m:
According to Fig. 2, c¾, ό0, c0 are the current co-ordinates of point m in the Oűq, Ob^ Oc0 co-ordinate system, which — as mentioned in the beginning of 3. §. — has a common axis with the Oa, Ob, Oc co-ordinate system (Οό0=Οό), round which the latter is turned away by angle ε relative to the former one.
Hence the following relations hold between a and co-ordinates a^ bQ, c0 as well as co-ordinates a, b, c:
10 LORÁND EÖTVÖS
(15)
Using these relations, we get from Equation (13):
(16)
This gravitational variation is directed vertically upwards and creates the following torque round axis Ob (round the edge of the beam):
which is of opposite direction to torque expressed in Equation (3), as this tends to increase angle ε.
The torque on the whole oscillating body, caused by gravitational variation due to rotation, referred to the edge of the beam, accordingly:
The sums in these formulae: (Σ>α02) and (ZmaobQ) characterize the moments of inertia, and, according to transformation equation (15), they can be written yet another way:
experimentaldemonstrationofthegravityvariation ... 11
3b. Rotating the instrument round vertical axis OZ, the centrifugal force comes into being, also creating its torque which influences the period
of oscillation of the beam.
According to Fig. 2, pc marks — as in section 3a — the perpendicular distance of mass-point m in point P from axis of rotation OZ, i.e. O’P; due to the rotation the centrifugal force acting upon m in point O ’P, is
As the rotating beam is a rigid body, we may take this force as also acting in O This force can be broken up into two components: one acting along Pa the other along Pb their sizes being:
The first component creating a torque of
around edge Ob, the second component, however, being parallel to this edge, exerts no torque round Ob; thus it cannot influence the oscillation of the beam.
Thus the total torque, created by the centrifugal force, exerted on the rotated beam, referring to edge Ob:
This torque, especially in the case of a long beam and small OO0=s distance and small ε angles, is always negative, as it aims at decreasing ε.
This condition demands a positive sign here, as — according to Equations (22), (24) and (25) — the sum (LmaOcO) is negative.
This sum (Zwűoco) is also of torque character, by using the transfor
mation formulae (15):
12 LORAND EÖTVÖS
3c. The reduction of torques and
F
qj in the case of a symmetric beam.The sums in Equations (19) and (22) are reduced to the following sums:
At the beginning of 3. §. it was mentioned that the beam should be symmetric with planes of symmetry: (bOc), (cOa) and the third parallel to the plane (aOb) and passing through point O0.
It is easy to break up three of the above five equations, viz.
into sums of pairs of points, each being zero, thus making the whole sum zero.
In the sum mbc the pairs of points are providing products of +mbc and -mbc, because the co-ordinates of the point-pairs — being the plane of symmetry for this sum, plane (aOc) — the following:
+c and -b, +c.
Similarly, in the case of equation mca, the plane of symmetry being plane (bOc), the point-pairs to which this sum can be broken up are providing products of +mca and -mca, and the co-ordinates of the two m mass-points are:
and —a, +b, +c.
Finally for the equation Xmab the planes of symmetry are planes (bOc) and (aOc), the point-pairs, to which this sum can be broken up, provide, accordingly, the products of +mab and -mab. The co-ordinates of the two respective mass points may be either:
+a, +b, +c and -a, +b, +c;
or:
+
ű,
+b, +c and +a, -b, +c.Accordingly the sums in (23) are equal to zero.
Let Ka and Kc be the moments of inertia on axes Oa and Oc, respect
ively, for them the following holds:
Comparing this with Equation (22), we get:
Taking all these into consideration, the sums of Equations (19) and (21) are reduced to:
Accordingly, the torques of Equations (18) and (21), F^ and F^, are reduced to the following formulae respectively:
4. Equation of motion of the oscillating beam in first approximation.
The complete solution
From the above-defined torques — F^, F^, F(í^, F^, Equations (3), (6), (26) and (27), respectively — the equation of motion of the beam can be expressed as:
14 LORÁND EÖTVÖS
or:
It is supposed that the attenuation does not suffer any change during the rotation of the instrument, i.e. Γ remains the same for a rotated or non-rotated instrument. This presumption can be regarded as highly probable.
To prove the possibility of any deviation from this case special experiments would be required, aimed at the attenuation of both rotated and non-rotated instruments. Our experiments so far have not been decisive in this respect.
4a.
The equation of motion (28) was deduced without disregarding anything, thus the relation between ε and the current time,t
is rather complicated, making any further discussion difficult. We shall therefore discuss its first approximation.
Let an^le ε, Jn the following, be so small, that instead of sin ε, ysin(2e), cos“c, sin“a:
ε, ε, 1 and 0
can be written, respectively. Equation (28) thus becomes:
To make it shorter, we may use:
Thus the equation of motion becomes:
EXPERIMENTAL DEMONSTRATION OF THE GRAVITY VARIATION ... 15
The complete solution of this equation8:
where E and δ are the constants Of integration, and for Δ holds:
The general solution, in this form, shows that ε consists Of two terms Of different character.
Its first term
similarly to (8), represents a simple harmonic Oscillation, with the amplitude:
-1* (35)
decreasing in time, and becoming infinitesimal within a short time. The period of this oscillation is Τ’, and:
8 See also c.g. I. Fröhlich, Dynamics 198.§, pp. 456-459. Budapest, 1896. Espe
cially the second formula in (10).
16 LORÁND EÖTVÖS
(36) The logarithmic decrement of the oscillation, similarly to Equation (11)
thus
where we use the presumption of a constant k.
The second term of the complete solution in Equation (32) represents a simple harmonic, but forced oscillation, which remains stable in time:
(38)
This stable oscillation reaches maximum amplitude if
which means that a resonance of motion takes place if
where, according to Equation (30)
EXPERIMENTAL DEMONSTRATION OF THE GRAVITY VARIATION 17
Here 7θ means the double period of the beam valid for the fictitious case of a rotated instrument with acting gravitational and centrifugal forces but no attenuation and no variation of gravity.
Accordingly, the maximum amplitude — occurring at resonance, if 3 = 7q — of the forced oscillation, expressed in Equation (38), will be:
where, according to Equations (11) and (37):
and in it Tand 3 can be determined — according to Equations (9), (l 0) and (l l) — from the direct Observation Of the attenuated Oscillations Of a ηοη-το- tated instrument. The maximum amplitude, using Equation (30):
where, instead of the last term we may write
Let it be noted that the moment of inertia Kb can be determined by way of experiments or calculated in the case of a precisely designed and executed beam; the value of Amax is provided by direct observations (see 4. §.); and (Lma) means the moment of inertia of the beam referred to the plane of
Let it be noted that the moment of inertia Kb can be determined by way of experiments or calculated in the case of a precisely designed and executed beam; the value of Amax is provided by direct observations (see 4. §.); and (Lma) means the moment of inertia of the beam referred to the plane of