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DE REVOLUTIONIBUS OF THE NEWTON’S SPHERES

Created: Wednesday, 21 May 2014

Vladan Panković, Stevica Djurović

Department of Physics, Faculty of Sciences, 21000 Novi Sad, Trg Dositeja Obradovića 4., Serbia , This email address is being protected from spambots. You need JavaScript enabled to view it.

Abstract

In this work we consider old classical problem of the Newton's rotating spheres experiment (including its opposite Newtonian (absolute space universe) and Machian (relative space universe) interpretations). We simply and definitely prove that in absence of all other physical systems except two rotating spheres connected with string (linear) tension or corresponding displacement in string, in full agreement with all basic principles of the classical Newtonian mechanics (and Hooke's elasticity law), determine infinitely many different - relative, but not unique - absolute, rest referential frames.

PACS numbers: 45.20.D-
Key words: Newton's rotating spheres

1. Introduction

As it is well-known Newton [1] interpreted "paradoxically" simple experiment of the two rotating spheres connected with a string as an unambiguous and definite proof of the absolute space existence. Newton said: "Indeed it is most difficult to known the true motion of bodies and actually to discriminate from apparent motion; therefore because the parts of the immobile space, in which bodies truly are moving, do not meet the senses. Yet the cause is not yet quite desperate. For arguments are able to be chosen, partially from apparent motions which are the differences of true motions, partially from forces which are the causes and effects of true motions. So that if two globes, to be connected in turn at a given distance from the intervening thread, may be revolving about the common centre of gravity: the exertion of the globes to recede from the axis of the motion might become known from the tension in the tread, and thence the quantity of the circular motion can be computed. Then if any forces acting equally likewise may be impressed mutually on the faces of the globes to increase or diminish the circular motion, the increase or diminish in the circular motion may become known from the increase or decrease in the tension of the thread; " [1].

It is well-known too that Mach criticized Newton interpretation and concept of the absolute space. He suggested (without any explicit theoretical and mathematical formalism) that Newton's laws and basic principles of the classical mechanics must be changed at least in the limit of the empty space. This idea represented main motivation factor for Einstein in formulation of his general theory of relativity. But in final form of the general theory of relativity "Mach principle" is definitely avoided, more precisely speaking rejected. Reason is fundamental and very simple. Namely, Mach principle implies unambiguously non-locality of the mass of a physical system as well as non-local (super-luminal) interaction between distant bodies. Moreover, in a small vicinity of any point of general relativistic space-time general theory of relativity can be locally approximated by corresponding classical Newtonian gravity and mechanics. In other words Newtonian but not hypothetical Machian classical gravitation and mechanics represent correct local approximation of the Einstein general theory of relativity.
All this unambiguously means that in such relatively small domains of the space in which two spheres rotate practically without any gravitational field, Einstein general theory of relativity and Newtonian classical gravitation and mechanics must yield effectively (approximately) identical predictions. In other words it is necessary, using exclusively Newtonian classical mechanical arguments, prove failure in Newton interpretation of mentioned two rotating spheres experiment.
In this work we shall problem of the Newton's rotating spheres experiment (including its opposite Newtonian (absolute space universe) and Machian (relative space universe) interpretations). We shall simply and definitely prove that in absence of all other physical systems except two rotating spheres connected with string tension or corresponding displacement in string, in full agreement with all basic principles of the classical Newtonian mechanics (and Hooke's elasticity law), determine infinitely many different - relative, but not unique - absolute, rest referential frames.

2. Formulation of the problem

Consider two identical spheres (with the same relatively small radius r and with the same mass m) that can be treated as the rigid bodies.
Suppose that between these spheres at mutual distance R (significantly larger than 2r) there is some attractive central force, as it has been originally supposed by Newton. Nevertheless, for reason of the technical simplicity (and without any diminishing of the generality of the basic conclusions) it can be supposed that between spheres there is no any force.
Suppose however that mentioned spheres are additionally connected with a practically massless, one-dimensional string, whose length in absence of any displacement force equals L (and whose surface of the cross area A is much smaller than L2 ). By displacement l of this string caused by some external force, in this string corresponding elastic or displacement force appears, that, according to Hooke displacement law, equals
(1) FE(L+l) = - EA l /L
where E represents corresponding Young's modulus of the elasticity of string.
Suppose that mentioned spheres connected with string are initially in rest in some appropriately chosen referential frame, and later that these spheres become to rotate about their mass center in the half of the string length with the same angular speed Ω in respect to rest referential frame. In this situation an equilibrium state appears described by expression representing in fact second Newton's law for single sphere
(2) - EA(l/2)(L/2) = - mΩ2(L+l)/2
where left hand of (2) represents the displacement elastic force for single sphere, while right hand of (2) represents the product of the single sphere mass and radial (centripetal) acceleration. It implies
(3) Ω = [2EA/(mL} l/(L+l) ]1/2
which means that angular speed is uniquely determined as the function of the displacement as the variable and A, m, E, L as the parameters.
On the basis of (3) Newton concluded the following. Consider the rotating referential frame that rotates with angular speed Ω in respect to mentioned rest referential frame. Observer in this rotating referential frame cannot immediately observe his rotation. However, it can conclude by immediate measurement of the displacement l and (1) that mentioned rotation and rest referential frame must exist. Newton supposed that the same conclusion stand completely correct even in case when except mentioned spheres and string there is no any other physical system what can be satisfied in an excellent approximation somewhere deeply in universe. It practically implies that by suggested experiment the absolute rest referential frame, i.e. absolute space can be detected.
It is well-known too that Mach criticized Newton interpretation and concept of the absolute space. He suggested (without any explicit theoretical and mathematical formalism) that Newton's laws and basic principles of the classical mechanics must be changed at least in the limit of the empty space. However, in such relatively small domains of the space in which two spheres rotate practically without any gravitational field, Einstein general theory of relativity and Newtonian classical gravitation and mechanics must yield effectively (approximately) identical predictions. In other words it is necessary, using exclusively Newtonian classical mechanical arguments, prove failure in Newton interpretation of mentioned two rotating spheres experiment. It will be done in the next section of this work.

2. Theory

Accurately speaking, (2) represents a square equation which holds two solution. First one solution is given by (3), but there is additional, second one solution of (2),
(4) Ω = - [2EA/(mL} l /(L+l) ]1/2 .
It practically means that there are at lest two different rotating referential frames, first one , simply called C, that rotates clockwise and second one, simply called AC, that rotates anti-clockwise in respect to supposed rest referential frame. Nevertheless, observer using measurement of the tension or corresponding displacement in string, can not differ C and AC at all.
Moreover, in absence of all other physical systems (except mentioned two spheres connected with string) and all other physical measurements (except mentioned tension or displacement measurement), impossibility of the experimental distinction of two mentioned rotating referential frames admits an additional option. Namely, without any contradiction experimental data it can be supposed that there are at lest two different rest referential frames, simply called R(C) and R(AC), so that C rotates clockwise in respect to R(C) and so that AC rotates clockwise in respect to R(AC). Simply speaking R(C) (corresponding to initially supposed unique rest referential frame) and R(AC) holds oppositely directed axes around which clockwise rotate respectively C and AC.
It can be very simply proved that mentioned and similar degeneration can infinitely increase. Concretely, it can be simply calculated and proved that the same classical elastic force as right hand of (2) appears in case when system spheres and string do a precession with angular speed Ωφ around an axis that turn out across middle of the string by angle φ from (0, π) angle interval so that the following condition is satisfied
(5) Ω2φ sin2 φ = Ω2 = [2EA/(mL} l/(L+l) ] .
This axis (with appropriately chosen two-dimensional basis of the vectors in the plane orthogonal to given axis) determines a rest referential frame. But since angle φ can have continuously many different values from (0, π) angle interval there is continuously many differential rest frames that all satisfies the same condition (2) or (3). Only in an especial case, for φ= π/2, we obtain situation, i.e. rest referential frame discussed by Newton, meanwhile all other continuously many values of the angle and corresponding continuously many rest referential frames exist too. All this unambiguously means that these rest referential frames can not be mutually distinguished using measurement of the tension or corresponding displacement in string.
In this way using exclusively Newtonian classical mechanical arguments (including Hooke's elasticity law), failure in Newton interpretation of two rotating spheres experiment as existence of the absolute space is definitely proved.

4. Conclusion

In conclusion we can shortly repeat and point out the following. In this work we consider old classical problem of the Newton's rotating spheres experiment (including its opposite Newtonian (absolute space universe) and Machian (relative space universe) interpretations). We simply and definitely prove that in absence of all other physical systems except two rotating spheres connected with string (linear) tension or corresponding displacement in string, in full agreement with all basic principles of the classical Newtonian mechanics (and Hooke's elasticity law), determine infinitely many different - relative, but not unique - absolute, rest referential frames.

References

[1] Isaac Newton, Principia Mathematica Philosophiae Naturalis, I, Scholium

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