SELF

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S.B. Karavashkin and O.N. Karavashkina

6.4.2. The features that forces of inertia reveal under affection of distributed forces

In case of spatial forces, it is harder to reveal the accelerated pattern of motion within the frame. As opposite to lumped forces, spatial forces affect each elementary volume of a body and add, in case of homogeneous field, the same acceleration to all parts of it. Interior elasticity forces do not react to the exterior affection; consequently, deformations able to reveal the accelerated pattern of motion do not arise. This is why many authors think unrevealable the external revelations of centrifugal force. So, to make our study complete, it will be efficient to account, in revealing the features that differ the inertial and accelerated frames, the versatility of nuances of centrifugal forces revelation, dependently on the configuration of field and of pattern of body’s motion in this field.

6.4.2.1. The features of inertia forces revelation in the orbital motion in a central field

Let us first consider an orbital motion in a central field relative to which Galilee, as we saw above, proved impossibility to reveal the orbital motion within the frame involved in this motion. Einstein says the same: “We could, for example, expect that the height of sound of the organ tube would be other if its axis was parallel to the motion than in case if it was perpendicular to this direction. But our Earth, due to its orbiting about Sun, can be compared with a carriage moving with the speed of 30 km/s. So, in case if the principle of relativity was inapplicable, we should expect that the laws of nature would include the direction of Earth’s motion at each moment, i.e. the behaviour of physical laws has to depend on their spatial orientation about Earth. Actually, due to the seasonal change of direction of the speed of Earth’s orbital motion, this last cannot keep resting the year round relative to the hypothetical system. But with the utmost thorough of observation, we still were unable to reveal such anisotropy of the Earth’s space, i.e. the physical non-equivalence of different directions. This argument in favour of the principle of relativity is especially weighty” [50, p. 538].

Keeping the sequence of analysis, consider first a particular case of organ tube which Einstein took as a reason to prove the unchanged laws in the yearly motion of the Earth. First of all let us account for it that sound vibrations are impossible in absence of gas medium. While in gas medium, the constancy of speed of vibrations is determined not relative to an arbitrary point of the universe but only relative to the very medium in which the vibrations propagate. It is enough, the medium to drag with the Earth in its diurnal or yearly motion, the speed of sound relative to air to remain practically constant: “Simplicio. – The author goes on proving how, accepting the theory by Copernicus, we have to negate the sensible perception and strongest senses which would take place should we, feeling the breath of lightest wind, could not feel an impulse of permanent wind that strikes us with the speed of 2529 miles per hour, as such is the space that the Earth’s centre passes hourly in the large orbit, as the author thoroughly calculates. And, as far as, by his words, Copernicus’ opinion is “cum terra movetur circumpositus aer; motus tamen ejus, velocior licet ac rapidior celerrimo quocumque vento, a novis non sentiretur, sed summa turn tranquilitas reputaretur, nisi alius motus accederet. Quid est vero decipi sensum, nisi haec esset deceptio?” (“… With the Earth the surrounding air moves, and its motion, although faster than fastest wind, we do not feel but think it to be fully resting if not other motion. Well, what is it – a real illusion of feeling if this is not an illusion?”).

Salviati. – Seemingly, this philosopher thought as if the Earth, which Copernicus makes moving with the surrounding air in the circle of large orbit, is not that on which we live but some other and special, as this our Earth moves with itself, and with the same speed, with us and with the surrounding air. And which strikes can we feel if we fly with the speed equal to the speed of this which wants to strike us? This signor forgot that we, no less than the Earth and air, are involved in the circular motion and that therefore we always feel the touch of same part of the air which, thus, does not strike us” [39, p. 191].

Actually, if we consider the affection of yearly motion of the Earth onto the sounds propagating in the atmosphere, why not to account the diurnal motion? This also is the circular motion, and its affection is well and long ago known. “After the air started moving, due to the difference of pressures, there arise other forces able to considerably affect it. In the rectilinear motion this is ‘the deviation force of Earth’s revolution’ (‘Coriolis force’) and friction; in the curvilinear motion the centrifugal force is added to them. The deviating force is the consequence of fact that the motion of air, as well as all other bodies on the Earth’s surface, are relative motions that take place in the rotating coordinate system. It has a value 2v sin   where is the angular speed of Earth’s revolution (7,29 10^-5  per 1 second), is the geographic latitude, and v is the speed of air. Thus, the deviating force of Earth’s revolution grows from the equator (where it is zero) to the pole and, besides, it is the larger the more is speed of the moving air particle. It reveals itself on any motion of a body, irrespectively of direction, and moves always perpendicularly to the direction of motion: rightwards in the northern hemisphere and leftwards in the south” [66, p. 89]. And presence of the wind is known to affect the sound propagation also. Consequently, the Earth’s rotation and related orbital motion of all bodies and gases on the Earth’s surface causes additional forces whose regularities cannot be reduced to the gravitational interaction, as they relate to the motion of these bodies in the field of these forces and are directed perpendicularly to the motion, which makes impossible to model these forces through some potential centre of attraction.

From this, in the Earth’s orbiting about the Sun such forces that disallow us to identify the orbiting frame with that inertial will arise. And not only dynamically but kinematically, too, because, just as in case of lumped forces, under affection of Coriolis force the body’s trajectory will change, and we cannot model this change by the presence of additional centres. Actually, if we take several bodies with different directions of motion relative to the revolving frame that are affected by Coriolis forces different in their directions, we will encounter the difficulty: additional gravitating centres will affect some bodies and will not affect some others in this frame. Furthermore, the difference between the inertial and revolving frames will reveal itself in the complex motion of a body which, in particular, is the motion of any element of air on the Earth’s surface that participates in diurnal and yearly motion with the Earth.

Fig. 6.11. On determining the joint affection of Earth’s diurnal and yearly motion on the non-uniformity of the motion of a point on the Earth’s surface [39, p. 304]

If we account the yearly and diurnal affection, it was Galilee who has revealed their joint affection that causes the non-uniform motion of the point on the Earth’s surface: “In presence of such opposite motions of the parts of Earth’s surface, during rotation about its own centre, from the combination of diurnal motion with that yearly, we necessarily have to yield, for separate parts of terrestrial surface, the absolute motions, at some places much accelerated, at others – correspondingly decelerated. This is obvious if we consider a part about the point D whose absolute motion will be quite fast, as it is produced from two motions in one direction, namely leftwards. The first of them is a part of yearly motion common for all parts of globe, and second is the motion of the same point D moving leftwards with the diurnal rotation, so that in this case the diurnal motion increases and accelerates the yearly motion. It happens oppositely with the reciprocal part F which, being moved by the general yearly motion leftwards, moves with the diurnal motion rightwards, so that the diurnal motion is finally subtracted from that yearly; so the absolute motion yielded from addition of both motions will appear here much decelerated. Further, the absolute motion for points E and G appears to be equal to a simple yearly motion, because the diurnal motion at all or almost at all does not increase or decrease it, doing not deviating it leftwards or rightwards, neither upwards or downwards. So we come to conclusion that, as far as it is true that the motion of whole globe and of each its part would be uniform, should they move as only one motion, be it a simple yearly or only diurnal, with the same necessity it follows that when these two motions are composed for the parts of globe, it will produce two non-uniform motions, once accelerated, once decelerated, dependently on, whether the diurnal revolution is added to that yearly either subtracted from it. From this, if it is true (and this is true and experimentally proven) that the acceleration and deceleration of vessel’s motion make the water in it moving along it ahead and back, up and down near its brims, why not to assume that such phenomenon can and must occur with the sea waters contained in reservoirs susceptible to such changes, especially if these reservoirs are east-west stretched, i.e. in the same direction as their motion” [39, p. 304].

But if the joint diurnal and yearly motion of Earth affects the water cover of planet, it affects the air masses, too, moving them just as the water cover; this is the affection which we should expect in consideration of models related to the revolving frames, and affection which we can reveal in the very revolving system – of course, if we know the regularity. With it, such motion of air masses that arises due to the non-uniform motion of a body on the Earth’s surface will produce just the primary motions that will cause the Coriolis forces. So we see that all moveable masses, be they a water cover or a gas envelope of the Earth, come to motion due to the superposition of two revolutionary motions with the consequent affection of Coriolis forces onto the produced motion. And this total motion is so complicated and special that it is basically impossible and incorrect to substitute the causing forces by the forces of gravity.

It also follows from our consideration that the joint outwardly uniform motion in two or several revolving frames causes the linearly accelerated resulting motion of a body, and it does not occur if the body is involved in several inertial motions at the same time. It also basically differs inertial and revolving reference frames. The very fact of revolution – and here Newton was absolutely right – causes the acceleration of a body; consequently, it causes the effects related to this acceleration that curves the trajectory. And the fact that these effects to some extent are masked in application to small bodies on the Earth’s surface does not mean them absent. When considering large masses of water or air, we see effects caused by a complicated revolutionary motion and forces whose origin is impossible in inertial reference frames.