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NEWTON’S ROTATING ANTI-MACE (WITH TWO SPHERES)

Created: Friday, 06 June 2014

Vladan Panković, Darko Kapor

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 present a simple classical mechanical system, simply called Newton's rotating anti-mace (with two spheres connected with a tiny, deformable string) and demonstrate that dynamics of this system (that includes precession effects) is time irreversible. All this can be very interesting for a better understanding of the basic dynamical principles of the classical mechanics (of rigid and deformable bodies) as well as for time reversal.

PACS numbers: 45.20.D-
Key words: Newton's rotating anti-mace, spheres, time reversal

As it is well-known all Newton's basic classical mechanical dynamical laws and gravity law (for rigid bodies) are symmetric in respect to time reversal transformation (according to which usual time direction or arrow is changed by inverse time direction or arrow, and vice versa) [1]. In other words there is no empirically strange phenomena for the observers by formal change of the usual time arrow (directed from past toward future) in opposite or inverse time arrow (directed from future toward past) by description of the classical mechanical motions. For example, as it has been pointed out by Feynman [1], we can record by a video camera (e.g. clockwise) rotation of a planet around Sun, as the consequence of the gravity interaction, and later we can move video material backward (formally simulating inverse time arrow) without any strange phenomena for observers. Namely, the planet rotates again around Sun (in opposite, anticlockwise direction) according to the same Newton's gravity law.
But, in the classical mechanics of the deformable bodies time reversal can be satisfied in some situations, while in some other situations time reversal cannot be satisfied. For example by an elastic spring, precisely a linear harmonic oscillator, time reversal is satisfied (backward motion of the video material does not imply any strange situation for observers). On the other hand by a phase transition from domain of the elastic deformation into domain of the plastic deformation and breaking phenomena of the spring time reversal becomes broken. Really, we can do a video record by usual time arrow of the event when a force can plastically deform spring coil in a linear wire without realistic inverse deformation corresponding to inverse time arrow.
In this work we shall present a simple classical mechanical system, simply called Newton's rotating anti-mace (with two spheres connected with a tiny, deformable string) and demonstrate that dynamics of this system (that includes precession effects) is time irreversible. All this can be very interesting for a better understanding of the basic dynamical principles of the classical mechanics (of rigid and deformable bodies) as well as for time reversal.
So, consider a simple classical mechanical system, simply called Newton's rotating anti-mace (with two spheres). It holds a cylindrical hollow tube with height H and relatively small (much smaller than H) radius r of the circular basis area. Through this tube a practically massless, tiny, one-dimensional string with deformable form and constant length L=H+l is dragged where l represents some length much larger than r but much smaller than H. At both string ends two spheres with the same mass m and radius R are fixed where R is larger than r and smaller than l.
Suppose that initially tube is placed vertically so that one tube end is placed up while other is placed down at this vertical axis. According to Earth gravity upper sphere touches the upper tube part, while down sphere and down part of string between down sphere and down tube end with length l do practically a resting mathematical pendulum with length l and mass m. (System, by which down sphere initially touches the down tube end while upper part of string between upper sphere and upper tube end with length l is switched over tube end toward down, is simply called Newton's mace.)
Suppose, further, that tube changes its previous vertical position in an inclined position so that upper end of the tube become deflected for some length D in respect to the vertical axis at which down end of the tube and down sphere as well as corresponding down part of the string remain.
Finally, suppose that using an external mechanism (e.g. manual action or similar) tube does a precession around vertical axis with some sufficiently large angular speed Ω. It implies that there is corresponding centrifugal force acting at the upper sphere (while, according to suppositions, there is no action of the centrifugal force at down sphere). It, further, implies action of the corresponding component of centrifugal force at the string so that upper part of the string (between upper sphere and upper tube part) becomes larger and larger tending finally toward l, while down part of the string (between down sphere and down part of the tube) becomes smaller and smaller tending finally toward zero.
It can be observed that the completely same effect appears if mentioned precession is clockwise or if it is anticlockwise oriented. It has an interesting consequence. Namely, according to remarkable Feynman suggestion [1], we can do a video record of the described dynamics of the Newton's anti-mace rotating, for example clockwise, in usual time direction (from the past toward future). Later we can move this video record backward that simulates inverse time direction (from the future toward past). In this case we can see how upper sphere by anticlockwise rotation tends closer and closer toward upper end of the inclined tube, while down sphere by an anti-vertical translation motion goes far away down tube end till final distance l. However, it represents extremely strange phenomena for the observer, since as it has been previously pointed out, real precession of the Newton's anti-mace, clockwise as well as anticlockwise, yield the completely same (but not opposite) final effect. It means in fact that by Newton's anti-mace dynamics time reversal transformation is not satisfied or that here time irreversibility appears caused by deformable character of the string.
Nevertheless, it can be observed that, roughly speaking, inversion of the time direction simultaneously with a discrete change of the Newton's rotating mace by corresponding Newton's rotating anti-mace (or vice versa) represents a symmetry transformation of the dynamics of mentioned systems.
In conclusion we can shortly repeat and point out the following. In this work we present a simple classical mechanical system, simply called Newton's rotating anti-mace (with two spheres connected with a tiny, deformable string) and demonstrate that dynamics of this system (that includes precession effects) is time irreversible. All this can be very interesting for a better understanding of the basic dynamical principles of the classical mechanics (of rigid and deformable bodies) as well as for time reversal.

Acknowledgements

Authors are very grateful to Prof. Dr. Tristan Hubsch, Branko Marčeta and Milan Mrdjen for illuminating discussions and observations as well as technical help.

References

[1] R. Feynman, The Character of Physical Law (Cox and Wyman LTD, London, 1965.)

 

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