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NEWTON’S BUCKET (U-TUBE) EXPERIMENT

Датум креирања: понедељак, 28 април 2014

Vladan Panković, Darko Kapor, Miodrag Krmar, Stevica Djurović, Milan Pantić

Department of Physics, Faculty of Sciences, 21000 Novi Sad, Trg Dositeja Obradovića 4., Serbia , Ова адреса ел. поште је заштићена од спамботова. Омогућите JavaScript да бисте је видели.

Abstract

In this work we consider a simplified, "differential" form of remarkable Newton bucket paradox, simply called Newton (inverted) Π-tube paradox. Concretely we consider a tube (with relatively small radius) twisted in the form of a square (with side length much larger than tube radius) without one side. "Middle" part of this tube is placed in a plane parallel to Earth surface, while other two parts of the same tube are placed perpendicularly in respect to mentioned plane. (In this way mentioned tube has form of the inverted Π letter. Less strictly we can speak about U-tube.) Further, tube is initially filled with water till one half of the vertical sides height and then complete system become to rotate around symmetry axis of one, simply called "central" vertically placed tube part. By constant speed of the rotation, in final, equilibrium state, height of the water in the "central" tube part will be under initial height, while height of the water in the opposite vertical, simply called "peripheral" tube part will be over initial height. Dynamics of this system can be effectively very simply described by pressure equilibrium equation and Pascal law and it points out unambiguously and definitely that height differences (and, analogously, parabolic water surface in the original paradox) point neither existence of the absolute Newtonian space nor existence of the "relative" Machian space. It points out definitely real dynamical interaction between water (representing an incompressible fluid characterized not only by velocity but by pressure too) and tube (representing a rigid body).

PACS numbers: 47.10.-g
Key words: Newton bucket

1. Introduction

As it is well-known Newton [1] interpreted "paradoxically" simple experiment of the rotating bucket with water as an unambiguous and definite proof of the absolute space existence. He stated: "If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water; after, by the sudden action of another force, it is whirled about in the contrary way, and while the cord is untwisting itself, the vessel continues for some time this motion, the surface of the water will at first be plain, as before the vessel began to move; but the vessel by gradually communicating its motion to the water will make it begin sensibly to revolve, and recede by little and little, and ascend to the sides of the vessel, forming itself into a concave figure. ... This ascent of the water shows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour. ... And therefore, this endeavour does not depend upon any translation of the water in respect to ambient bodies, nor can true circular motion be defined by such translation. ... but relative motions ... are altogether destitute of any real effect. ... It is indeed matter of great difficulty to discover, and effectually to distinguish, true motion of particular bodies from the apparent; because the parts of that immovable space in which these motions are performed, do by no means come under observations of our senses." [1]
It is well-known too that Mach criticized Newton interpretation and suggest (without any explicit theoretical and mathematical formalism) that Newton absolute space must be changed by a "local" relative space determined by Sun and other relatively close stars. 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 bucket (vessel) rotates in relatively weak gravitational field of the Earth, 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 gravitational and mechanical arguments, prove failure in Newton interpretation of mentioned rotating bucket with water experiment.
In this work we shall consider a simplified, "differential" form of remarkable Newton bucket paradox, simply called Newton (inverted) Π-tube paradox. Main reason of the introduction of such variation of Newton original paradox is avoidance of the complex formalism of the fluid dynamics (unknown to Newton) and their reduction in simplest forms without any diminishing of the generality of the basic conclusions. Concretely we shall consider a tube (with relatively small radius r) twisted in the form of a square (with side length R much larger than tube radius) without one side. "Middle" part of this tube is placed in a plane parallel to Earth surface, while other two parts of the same tube are placed perpendicularly in respect to mentioned plane. (In this way mentioned tube has form of the inverted Π letter. Less strictly we can speak about U-tube.) Further, tube is initially filled with water till one half of the vertical sides height and then complete system become to rotate around symmetry axis of one, simply called "central" vertically placed tube part. By constant speed of the rotation, in final, equilibrium state, height of the water in the "central" tube part will be under initial height, while height of the water in the opposite vertical, simply called "peripheral" tube part will be over initial height. (As it is not hard to see in a simple change of the suggested experimental set-up with more vertical tube parts between "central" and "peripheral" final distribution of the water level heights using these parts will be approximately parabolic, i.e. "concave".) Dynamics of this system can be effectively very simply described by pressure equilibrium equation and Pascal law and it points out unambiguously and definitely that height differences (and, analogously, parabolic water surface in the original paradox) point neither existence of the absolute Newtonian space nor existence of the "relative" Machian space, but they point definitely real dynamical interaction (and pressure redirection) between water (representing an incompressible fluid characterized not only by velocity but by pressure too) and tube (representing a rigid body) walls.

2. Theory

In Newton bucket experiment really physically there are four principally important physical systems: 1. water, 2. bucket (tube), 3. gravitational field , and, 4. system (cord etc.) that rotates bucket (tube).
Firstly we shall consider gravitational field and its dynamical influence on the other systems. As it is well-known, water, without interaction with gravitational field, do not occupy internal space of the bucket (tube) but it leave bucket and forms one or more spherical bubbles in the bucket outer space. For this reason Newton bucket experiment cannot be realized at all without gravitational field and corresponding source of this field, e.g. Earth. However, dynamical influence of the gravitational field can be done effectively implicit. Simply speaking in case that in a relatively large space domain D much larger than the bucket linear dimensions d gravitational field is in a satisfactory approximation homogeneous with corresponding constant intensity, i.e. gravitational acceleration g representing effectively a parameter. Such situation is, for example, satisfied nearly Earth surface where gravitational fields of all other systems (Moon, Sun, Milky Way, other galaxies, etc.) can be effectively neglected in corresponding satisfactory approximation.
Secondly we shall consider system (cord etc.) that rotates bucket (tube) (in the discussed effectively constant gravitational field). This system dynamically interacts immediately with bucket only (not with the water!), so that it can be effectively treated as the integral part ("motor") of the bucket.
In this way, after discussed effective hiding of the gravitational field and system (cord etc.) that rotates bucket (tube) in Newton bucket experiment really physically stand only two principally important physical systems: 1. water and 2. bucket (tube).
Without real physical dynamical interaction between water and bucket (tube) water surface stands plane, what can see any biker (with photo camera) at the cylindrical dead wall that observe water in a glass placed in the center of the circular base of the wall.
So, for explanation of the Newton bucket (tube) experiment dynamical interaction between water and bucket (tube) must be analyzed relatively accurately. Roughly speaking, by this real dynamical interaction (that also includes adhesion forces and viscosity effects) there is energy-momentum exchange between water (representing effectively an incompressible fluid) and bucket (representing effectively a non-deformable body). All this for original Newton bucket experiment, i.e. in three dimensions, represents mathematically-technically a complex problem.
For reason of the technical-mathematical simplicity we can, instead of the original Newton bucket experiment, analyzed practically one-dimensional {"with degeneration of other two dimensions", we use pedagogical example from modern string theory) Newton tube experiment. Concretely we consider a tube (with relatively small radius r, so that two of three tube dimensions are effectively "compactified") twisted in the form of a square (with side length R much larger than tube radius) without one side. "Middle" part of this tube is placed in a plane parallel to Earth surface, while other two parts of the same tube are placed perpendicularly in respect to mentioned plane. (In this way mentioned tube has form of the inverted Π letter.) Further, tube is initially filled with water till one half of the vertical sides height and then complete system, tube with water, become to rotate by constant angular speed Ω around symmetry axis of one, simply called "central" vertically placed tube part. In final, equilibrium state, level of the water in the "central" tube part will be under initial height, while level of the water in the opposite vertical, simply called "peripheral" tube part will be over initial height. (As it is not hard to see in a simple change of the suggested experimental set-up with more vertical tube parts between "central" and "peripheral" final distribution of the water level heights using these parts will be approximately parabolic, i.e. "concave".)
In final situation an observer co-rotating with "peripheral" tube part cannot observe any mechanical motion of the water in respect to tube. But, of course it can observe decrease of the water height in "central" tube part and increase of the water height in the "peripheral" tube part what paradoxically, according to the Newton interpretation , could to point absolute space existence.
Now we shall definitely prove that suggested paradox, as well as original Newton bucket paradox is inconsistent.
Namely, according to introduced suppositions and simplifications on the water as incompressible fluid and tube as non-deformable one dimensional body any "point" of the water rotates with the same speed v=Ωx as corresponding "point" of the tube for 0≤x≤ R which yields a simplified but correct (in suggested approximation) description of the dynamical interaction between tube and water in form of immediate touch.
It implies that in the "central" tube part, where x=0, there is no rotation for any water "point". In the "middle" tube part there is a speed gradient 0≤Ωx≤ ΩR, while in the "peripheral' tube part, where x=R, there is rotation of any water "point" with speed ΩR.
Denote with hC and hP water height in "central" and "peripheral" tube part.
Initially, before rotation, both mentioned water heights are identical and equal h0 so that initial pressure equilibrium equation represents equilibrium between hydro-static pressures in the "central" and "peripheral" tube part equals
(1) ρghC = ρghP = ρgh0
where ρ represents the water mass density and g – Earth gravity acceleration. (Here, as well as in all further considerations, atmospheric pressure is approximately neglected.)
Finally, by stable rotation, pressure equilibrium equation, according to introduced suppositions, equals
(2) ρghC = ρghP – ρΩ2R2/2 .
First term at the left and right hand of (2) expresses hydro-static pressure in the "central" and "peripheral" tube part. Second term at the right hand of (2) represents the hydro-dynamical pressure in "peripheral" tube part done in any point of this tube part by centrifugal force, i.e. rotation realized using discussed dynamical interaction between the water and "peripheral" tube part. Redirection of this pressure toward up is the simple result of the Pascal law and supposition on the water as the effectively incompressible fluid and bucket as effectively a non-deformable body.
Equation (2) implies
(3) ρΩ2R2/2 = ρg(hP – hC)
so that, since left hand of (3) is positive, hP must be larger than h0 and it than hC. Also, equation (2) implies
(4) hP = hC + Ω2R2/(2g)
which, formally, expresses hP as a parabolic function of R identical to corresponding well-known functional dependence in original Newton bucket experiment.
In this way it is definitely proved that in the final, equilibrium state, increase of the height of water in the "central" tube part and decrease of the height of the water in the "peripheral" tube part will be over initial height is simple consequence of the real dynamical interaction between water and tube. Moreover, water represents a fluid and complete physical description of such system needs not only speed, precisely velocity vector field but also pressure tensor field. For this reason only absence of the relative motion, i.e. zero relative sped, is not sufficient for a consistent conclusion on the absence of dynamical interaction between water and bucket (tube). Or, an observer co-rotating with "peripheral" tube part, that cannot observe any mechanical motion of the water in respect to tube, cannot consequently conclude that here is no dynamical interaction between water and bucket (tube). Really, the same observer can simply observe decrease of the water height in "central" tube part and increase of the water height in the "peripheral" tube part and this heights changes or corresponding pressure changes are sufficient for existence of the dynamical interaction between the water (as a fluid) and bucket (as a rigid body).

3. Conclusion

In conclusion we can shortly repeat and point out the following. In this work we consider a simplified, "differential" form of remarkable Newton bucket paradox, simply called Newton (inverted) Π-tube paradox. Concretely we consider a tube (with relatively small radius) twisted in the form of a square (with side length much larger than tube radius) without one side. "Middle" part of this tube is placed in a plane parallel to Earth surface, while other two parts of the same tube are placed perpendicularly in respect to mentioned plane. (In this way mentioned tube has form of the inverted Π letter. Less strictly we can speak about U-tube.) Further, tube is initially filled with water till one half of the vertical sides height and then complete system become to rotate around symmetry axis of one, simply called "central" vertically placed tube part. By constant speed of the rotation, in final, equilibrium state, height of the water in the "central" tube part will be under initial height, while height of the water in the opposite vertical, simply called "peripheral" tube part will be over initial height. Dynamics of this system can be effectively very simply described by pressure equilibrium equation and Pascal law and it points out unambiguously and definitely that height differences (and, analogously, parabolic water surface in the original paradox) point neither existence of the absolute Newtonian space nor existence of the "relative" Machian space. It points out definitely real dynamical interaction between water (representing an incompressible fluid characterized not only by velocity but by pressure too) and tube (representing a rigid body).

References

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

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