Talk:Foucault pendulum

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It would be nice if a simpler intuition about the pendulum was provided. --128.135.82.146 02:42, 21 March 2007 (UTC)

[edit] The first "Foucault" pendulum

A couple years ago I was fascinated to discover that an Italian scientist shortly after Galileo had already experimented with the rotation of pendula, without making the connection to the rotation of the Earth. I would like to include this information in the article, but I can't remember the name. Can anyone help? In this context, it would be good to include a note on the fact that it is not trivial to build an accurate Foucault pendulum. See e.g. South Pole Foucault Pendulum. --Art Carlson 09:38, 16 October 2005 (UTC)

This is curious. A reliable reference should be provided. Stammer 08:16, 2 February 2007 (UTC)

There is no reference to Viviani's work in William Tobin's "The Life and Science of Léon Foucault", which provides historical background for the problem. Unless an authoritative reference can be provided, I surmise that the Viviani story is a canard. Stammer 08:50, 23 February 2007 (UTC)

I have commented out the mention of Viviani, pending a reference. Rracecarr 14:48, 23 February 2007 (UTC)

In the book "Pendulum: Leon Foucault and the Triumph of Science" by Amir D. Aczel, 2003, a paragraph is included at the end of chapter 2 commenting on the difficulties that Galileo faced, as follows;

"There is a bizarre footnote to this story. In an obscure paper of 1660 or 1661, cited in a manuscript of 1841 found at the library of the Grand Duke of Tuscany, Vincenzo Viviani (1622-1703). who was a student of Galileo, wrote cryptically: "We observe that all the pendulums hanging on one thread deviate from their initial vertical plane, and always in the same direction." Viviani did not elaborate, and the work itself disappeared. It was rediscovered only after Foucault's pendulum experiments became public in 1851." (8). Note 8 provides the reference as, Academie des Sciences, "Comptes rendus hebdomedaires de l'Academie des Sciences, April 4, 1851.

I do not know if the 1851 reference is verifiable. Perhaps both references and a brief note could be included with the article because of the historical significance of the Foucault pendulum. David Harty 05:13, 24 March 2007 (UTC)

I checked the journal (name: "Comptes rendus hebdomadaires des séances de l'Académie des sciences" is probably the one that is meant here), there is an article published on March 31 1851 and the next is on April 7 1851. If there were more information, like a page range, then I could have another look. --ShanRen 15:01, 24 March 2007 (UTC)

This is all quite interesting. The Univerity Library in Bologna, not far from where I live, has the complete "Comptes Rendus" collection from 1835 onwards. When I'll have time, I'll have a look at the March 31,1851 and April 7,1851 issues and then, if I manage to identify and locate it, at the relevant(?) Viviani paper. Be patient (or faster than me). Stammer 21:40, 24 March 2007 (UTC)

The journal is available online and has free access I believe. The relevant sections are around page 460. Here is a link. Make sure you double check what I was looking at is the correct location, in case I missed something. --ShanRen 21:52, 24 March 2007 (UTC)

Thanks for the link. I checked both the March 31 (pp. 436-476) and the April 7 (pp. 477-520) issues. Foucault and his pendulum are mentioned on pp. 504-505 within some short communications by de Tassan, Franchot and Faye, concerning possible improvements. I could not find any reference to Viviani (please double-check). More in general, I wonder whether the alleged Viviani remark quoted by Aczel can be construed as describing a Foucault pendulum,i.e. "as an experiment to demonstrate the rotation of the Earth", even assuming its authenticity. Stammer 06:39, 25 March 2007 (UTC)

The reference does appear to be misleading, perhaps only referring to the pendulum experiment being made public, i.e., being moved to the Pantheon. I quoted the paragraph and note correctly, but one would have to ask the author what he found. Thank you for the link, as well. It was very interesting! Pages 135-138, 157-160, and 197-207 of the hebdomedaires are pertinent to the Foucault pendulum experiment in the Meridian room of the Observatory (part of the proceedings of Feb. 3, 10, & 17th respectively). The pendulum really did cause quite a stir!David Harty 12:44, 25 March 2007 (UTC)

In the "Dichiarazione d'alcuni altri strumenti adoprati per misuratori del tempo" (Description of some other instruments used as time meters), published in 1667 and available online at http://brunelleschi.imss.fi.it/cimentosite/Template.Asp?xmlfile=Immagini.xml&idx=17 a member of the Accademia del Cimento (probably Viviani) writes: "L'ordinario pendolo a un sol filo, in quella sua libertàdi vagare (qualunque ne sia la cagione) insensibilmente va traviando dalla prima sua gita, e verso il fine, secondo ch'ei s'avvicina alla quiete, il suo movimento non è più per un arco verticale, ma par fatto per un a spirale ovata, in cui più non posson distinguersi nè noverarsi le vibrazioni". I translate as follows: "The ordinary single thread pendulum, in its freedom to wander (for whatever reason) deviates imperceptibly from its first trip, and towards the end, as it approaches rest, its movement is no longer along a vertical arc, but looks made along an ovoidal spiral, where vibrations [oscillations] can neither be distinguished nor counted anymore" . Stammer 17:48, 2 April 2007 (UTC)

By the looks of it, what is described in that text is a problem that always comes up with pendulum setups. It's very hard to release the pendulum without any sideways motion. Inevitably, the motion of the pendulum is ellipse-shaped rather than planar. The natural swing frequency is slightly dependent on the amplitude of the swing, hence the swing along the long axis doesn't have exactly the same period as the swing along the short axis. It's like using two sinus-signals as perpendicular inputs on an oscilloscope. When the frequency of the two signals is slightly different, the line moves around over the screen. --Cleonis | Talk 18:07, 2 April 2007 (UTC)

I agree. Actually the Cimento paper discusses how to overcome the problem in order to build a better clock. Still, an excerpt from the passage above ("va traviando dalla prima sua gita", i.e. deviates from its first trip) is cited on the web in conjunction to the Foucault pendulum. Note also that also the phantomatic quote in Aczel's book refers explicitly to a single thread pendulum. To me the Viviani-Foucault connection looks like an urban legend. Stammer 05:23, 3 April 2007 (UTC)

There's another curious tidbit here. In 1841 the Cimento essays, which include the "Dichiarazione", were reprinted in Florence. That's the date of the "manuscript" mentioned by Aczel. Stammer 05:51, 3 April 2007 (UTC)

[edit] Precession wrt Fixed Stars

The discussion on this is faulty. For example, the sentence

"The blue line marks the precession of the plane of swing of a Foucault pendulum with respect to the fixed stars."

in combination with the adjacent graph suggests that the plane of swing of the pendulum precesses with respect to the fixed stars at a rate of 15 degrees per hour if the pendulum is at the equator. This is false if the pendulum starts out swinging parallel to the equator. In this case the pendulum continues to swing parallel to the equator forever, and the plane is fixed with respect to the fixed stars.

Of course the situation changes is the pendulum was swinging north-south, which shows that the question of how the plane moves with respect to the fixed stars does not only depend on the latitude. I don't see why this is important for understanding the pendulum anyway, so it is best removed imho. —The preceding unsigned comment was added by 70.224.71.98 (talk) 00:29, 11 December 2006 (UTC).

I agree with the above comment. What the blue line really shows is the precession of the line defined by the intersection of the plane of swing and the equatorial plane (unless they are parallel). It should probably be removed. I think, however, that it is worth mentioning in the caption that the maximum amount of precession actually corresponds to being stationary in the inertial frame. --PeR 08:24, 11 December 2006 (UTC)

I see. It seems to me that the focus on the inertial frame does not add much to the understanding beyond the north and the south pole. So how about removing the blue line from the graph and adding a comment as you suggest? —The preceding unsigned comment was added by 129.74.228.212 (talk) 15:00, 11 December 2006 (UTC).

[edit] The blue line in the diagram is relevant and correct

change of direction of the plane of swing of the pendulum in degrees per hour
Red line: change of direction with respect to the Earth. Blue line: change of direction with respect to the fixed stars.

In the above comment, the unsigned commentator 70.224.71.98 stated: "Of course the situation changes if the pendulum was swinging north-south, which shows that the question of how the plane moves with respect to the fixed stars does not only depend on the latitude."

That statement is faulty. The rate of precession of the Foucault pendulum depends only on the latitude. (There may be some directional dependency, but this is a far weaker effect than the main effect, which is the coriolis effect. The magnitude of the coriolis effect is independent of the direction of motion (in the case of motion parallel to the Earth's surface.) Hence, contrary to what the unsigned commentator 70.224.71.98 suggests, the initial direction of swing of the Foucault pendulum is not an operative factor.

Als long as a Foucault pendulum is not located exactly at the equator there is a non-zero angle between the plane of swing and the plane of the equator, and then the blue line in the diagram is mathematically well-defined.

For example: a Foucault pendulum located at 1 degree away from the equator will precess, it is just that its rate of precession will be extremely slow. Hence when a Foucault pendulum is very close to the equator the motion of the plane of swing with respect to the fixed stars will be dominated by the fact that the plane of swing is being pulled along with the 15 degrees per hour rotation of the Earth. (For any latitude away from the poles the change of direction of the plane of swing is a composite of precession due to coriolis effect and being pulled along with the rotation of the Earth. The closer to the equator, the more dominant the effect of being pulled along.)

The criticism by unknown commentator 70.224.71.98 and PeR on the diagram seems to be the following: that although the blue line is correct for all latitudes other than at the equator, the diagram does not explicitly acknowledge that in the theoretical case of a Foucault pendulum that is located exactly at the equator the plane of swing will not precess. One way of addressing that is to cap the end-point of the blue line with a small circle, indicating that mathematically the blue line does not extend to exactly zero degrees latitude. --Cleonis | Talk 11:07, 3 January 2007 (UTC)

The blue line is indeed well defined for non-zero latitudes, but it is still incorrect. For latitudes close to the equator the change in direction of the plane of swing is close to zero when the pendulum swings in an east-west direction. (The plane of swing stays nearly parallel to the equatorial plane).

As I stated above, the blue line is is only correct if the phrase "direction with respect to the fixed stars" is replaced by its projection on the equatorial plane.

The animation Foucault_pendulum_plane_of_swing.gif (good job, by the way) shows such a projection. Note that when the swing is east-west (around noon each day), the plane of swing stays nearly parallel to the equator for a couple of hours while continuing to rotate at the same rate in the 2D image.

--PeR 20:22, 3 January 2007 (UTC)

I think I see your point. It is closely related to the problem of projection that I had solved in manufacturing the plane of swing animation As you point out, in that 2D view, the trace of the pendulum bob is projected onto the plane of the equator. Projecting the trace gives a distortion that affects the apparent length and the apparent precession of the trace.

The trace is the line of intersection between the plane of swing and the section of Earth surface it is suspended above. The tricky question is then how to define the motion of that line of intersection with respect to the inertial frame. My approach was to ignore the projection issue, I just added the values. The Earth rotates at a rate of 15 degrees per hour, and in the diagram the distance between the red line and the blue line is a constant 15 on the vertical scale.

I need to figure this one out. For the time being, I truncated the image, removing the blue line and other elements that aren't necessary without the blue line. I have uploaded it to this temporary location. You can edit the wikipedia article and the caption, and then replace the current image with the reworked one. --Cleonis | Talk 11:17, 4 January 2007 (UTC)

Tried to repalce the image, but your link does not work. How about you make the change until you figure it out, Cleonis. I edited the image caption again to remove the erroneous assertion about the blue line, but got reverted without comment (vandalism?). Since everyone agreed that the caption is incorrect as stated, how come it is still in there?

—The preceding unsigned comment was added by 129.74.117.17 (talk) 14:40, 26 January 2007 (UTC).

[edit] IMHO, the blue line is NOT relevant/correct

Here is my understanding of the physics, and my opinion about what should be included.

Imagine a coordinate system defined by a Foucault pendulum (FPCS): the x-axis is horizontal, aligned with the swing of the pendulum, the y-axis is horizontal and perpendicular to the swing, and the z-axis is upward (from the equilibrium position of the bob toward the suspension point).

On Earth, the z-axis is fixed, so the rotation of the FPCS occurs in a horizontal plane; that is, the axis of rotation is the FPCS z-axis. The rate of rotation is ωsin(θ).

The FPCS also rotates with respect to the fixed stars. Rather than rotating about the z-axis, however, the instantaneous rotation of the FPCS relative to the fixed stars occurs about an axis which is defined by the intersection of the FPCS x-y plane and the plane defined by the FPCS z-axis and the axis of the Earth; thus in the frame of the fixed stars, the axis of rotation of the FPCS itself precesses at a rate of one cycle per day. The instantaneous rate of rotation of the FPCS is ωcos(θ).

The projection of angular velocity vector describing the instantaneous rotation of the FPCS relative to the fixed stars onto the axis of the Earth has a magnitude of ωcos²(θ). I'm not sure why one would care about this except that it's the 'constant' component of the rotation.

However, neither of these quantities is shown by the blue line. Instead, what is shown is ω(sin(θ) − 1). The most physical interpretation I can come up with for this quantity is that it is an average rate of precession of the projection of the swing of the FP onto the equatorial plane, as was I think pointed out by PeR.

So, I would be in favor of a figure in which the blue line is either removed, to save all the confusion, or replaced by ωcos(θ), which is the magnitude of the instantaneous angular velocity vector of the FPCS relative to the fixed stars. Another option is to leave the blue line as it is, but in that case I think it needs a better description. Rracecarr 22:32, 19 February 2007 (UTC)

A couple of days ago I made a reworked version of the diagram available again, a version without the blue line. It is (again) available at this temporary location. Since I personally prefer the version with the blue line, I won't put it in the Foucault Pendulum article myself. --Cleonis | Talk 18:14, 20 February 2007 (UTC)

If the image is to be replaced, then one should fix the other problems too. The graph should include the entire domain of the latitude function, not just one with the northern hemisphere. Moreover, one should plot -sin(lambda) instead of sin(lambda), since the rotation is negative (clockwise) for positive latitude. And while I am at it, I believe that the graph should give the rotation per day and not per hour. That way the important point that the pendulum does not return to its original orientation after one day gets emphasised. --129.74.87.3 20:20, 20 February 2007 (UTC)

And imho, the blue line should be removed. Any effort to make the description correct will be quite cumbersome and the "gained insight" to "spent effort" ratio too low to justify it.--129.74.87.3 20:24, 20 February 2007 (UTC)

For the southern hemisphere, the graph is symmetrical to the northern hemisphere graph. Including the southern hemisphere in the graph would lower the resolution (hence the clarity) of the diagram, without adding information. While it is customary to take clockwise rotation as a negative rotation value, I believe this is not mandatory. You have a point in stating that giving the precession in degrees per day is clearer in presenting that at 30 degrees latitude a full precession cycle takes two (sidereal) days. It would also resolve the mismatch that 24 terrestrial hours do not quite coincide with one sidereal day. --Cleonis | Talk 20:41, 20 February 2007 (UTC)

Version of the graph with precession in degrees per sidereal day. --Cleonis | Talk 21:09, 20 February 2007 (UTC)

I have no religious feelings regarding the positive/negative clockwise rotation, but since there is already a convention, why not use it. But if people feel that the other way around is better, I don't mind. About the southern hemisphere, I don't really see the problem including them. There is not all that much information in the puicture, and one can always magnify it. It just seems to me that this would prevent any "northern bias" type arguments, and would be polite toward people that want a quick answer to what the precession is but that live on the southern hemisphere. But I am not religious on this one either. --129.74.87.3 21:03, 20 February 2007 (UTC)

I like the new version of the graph posted by Cleonis, with y-axis in degrees per day. Rracecarr 21:26, 20 February 2007 (UTC)

[edit] General Cleanup Suggestions

Some suggestions for cleaning up the article:

1) The introductions seems good to me, except for the last paragraph, which is just some trivia that is not important to the Foucault Pendulum

2) The section on the dynamics needs to be cleaned up. It currently contains incorrect information regarding the blue line.

3) The discussion on the dynamics of the pendulum should mention that the pendulum merely undergoes parallel transport. In other words, it turns with respect to the coordinate system on earth because paths of fixed latitudes are not straight lines on the sphere (except the equator). From this point of view, it is not the pendulum that turns, but the path of fixed latitude that bends underneath the pendulum. This way the pendulum fits into the larger picture of a geometric phase. The exposition should mention the geometric result that the difference in initial and final angles is just given by the enclosed solid angle of the path (no matter if the path is fixed latitude as in the classical foucault pendulum scenario, or a more general path that the pendulum traverses). I remember a nice and simple argument by Frank Wilczek that shows this, I will try to find it.

4) The references need to be cleaned up.

- The dead links need to be fixed or removed.

- The articles that just deal with the coriolis force should be removed from the foucault pendulum article. If need be they can be included in the article on the coriolis force, but imho a standard textbook will serve much better.

- The article "What Makes the Foucault Pendulum Move among the Stars?" by Norm Phillips is incorrect and should not be used as a reference. In that article it is claimed that the centrifugal force is essential for understanding the foucault pendulum, which is false. If need be, a subject expert should verify this.

5) The link to the vector diagram derivation should be removed or the vector diagram page needs to be seriously reworked. There are many ways to draw vector diagrams and come up with a sin(lambda) in the end. The only way to see if a particular derivation is rigorous is to essentially write down the differential equation. This is done in some of the linked websites (e.g. the one by Wolfe). Including an elementary explanation of the pendulum is nice, but what good is it if it is not rigorous. If one wants an intuitive understanding, I suggest using a geometric argument as mentioned in 3) above.

--129.74.87.3 20:56, 20 February 2007 (UTC)

The article by Norman Phillips is fundamentally correct. His terminology may not be well chosen, making his point hard to follow, but his identification of the physical mechanism is correct.

The only case that can be understood purely in terms of geometry is the case of a polar Foucault pendulum; that case is fully described in terms of coordinate transformation. At all other latitudes, there is physics taking place. For comparison, in the case of gyroscopic precession, the precession occurs if and only if a torque is being exerted. This torque is not doing work, but without its presence the gyroscopic precession does not occur. Likewise, in the case of a non-polar pendulum the precession deviates from the 360 degrees per sidereal day because of the presence of a force, as described by Norman Phillips.

Just to be on the safe side: I take it for granted that the wikipedia article about the Foucault pendulum should discuss the dynamics only in terms of newtonian dynamics. (The termininololy 'parallel transport' has a relativistic flavor to it.) --Cleonis | Talk 21:24, 20 February 2007 (UTC)

In 1851, a paper by Wheatstone was read to the Royal Academy, in which he described a device he had constructed that displayed the same dynamical behaviour as the Foucault pendulum, but much faster. The device works with a helical spring.

Click here for an image that represents Wheatstone's device. Click here for a transcript of wheatstone's original paper. --Cleonis | Talk 21:33, 20 February 2007 (UTC)

Will take a look at the Wheatstone article, sounds interesting. But probably not of central relevance to this wikipedia entry. Also, I agree with you that relativistic effects should not be included. Parallel transport simply describes what it means to be an "inertial direction" (no external forces) tangent to earth along a path.

We can have a more detailed discussion about Phillips article if necessary, but the underlying problem is that his force G is in fact not responsible for the motion of the pendulum -- contrary to what he claims. In the standard textbook derivations, G is ignored. Including G leads merely to a small correction of the sin(lambda) law, and is by no means responsible for it.

The pendulum can (and imho should) be understood geometrically -- at any latitude. I dug up the quote from Wilczek mentioned above that outlines this:

How does the pendulum precess when it is taken around a general path C? For transport along the equator, the pendulum will not precess. [...] Now if C is made up of geodesic segments, the precession will all come from the angles where the segments of the geodesics meet; the total precession is equal to the net deficit angle which in turn equals the solid angle enclosed by C modulo 2pi. Finally, we can approximate any loop by a sequence of geodesic segments, so the most general result (on or off the surface of the sphere) is that the net precession is equal to the enclosed solid angle.

This argument can now be found in modern physics textbooks (in particular ones with the words "geometry" and "physics" in the title). For a similar argument that only looks at paths of fixed latitude, you may look at [1]

--129.74.87.3 22:14, 20 February 2007 (UTC)

[reset indent]
OK, as you meant it, the expression 'parallel transport' was in terms of inertial direction.

I need to emphasize here that the physical force that is affecting the (non-polar) pendulum bob is not G alone. The force that is affecting the pendulum bob is the resultant force of newtonian gravity (without correction for centrifugal effect) and the normal force. In the case of the Foucault pendulum the normal force is provided by the suspension wire. The normal force is perpendicular to the local surface, the newtonian gravity (without correction for centrifugal effect) does not act perpendicular to the local surface. (Of course, the poles and the equator are specific regions where newtonian gravity does act perpendicular to the local surface) This resultant force, referred to as the poleward force must be incorporated in derivations (and correct derivation do incorporate it). The poleward force is a major factor in determining the deviation from the one cycle per sidereal day rate.
The way the poleward force is incorporated is usually a quite indirect one. The fact that the poleward force is always taken into account is easily overlooked. Specifically, when gravity is "corrected" for centrifugal effect, understanding the dynamics is hampered.

In terms of newtonian dynamics: in the case of a non-polar pendulum, the pendulum bob is not in inertial motion, the pendulum bob is deviated from inertial motion by the poleward force. (Interestingly, an accelerometer inside a pendulum bob would at each point in time only register acceleration in the direction parallel to the suspension wire.)

A pendulum bob during a swing from west to east is circumnavigating the earth's axis slightly faster than the earth's angular velocity, hence the pendulum bob will swing wide during a west to east swing. On the return swing, from east to west, the pendulum bob is circumnavigating the earth's axis slightly slower than the earth's angular velocity, hence the pendulum bob will be pulled towards the nearest pole by the poleward force. (of course, the poleward force also affects the pendulum bob when it swings north-south) This is illlustrated with the diagram on the right.
In my opinion, it is most unfortunate that Norman Phillips phrased his considerations in a way that invokes 'centrifugal force'. In his article, he uses the expression 'centrifugal force' inconsistently, making it quite difficult to follow his essentially correct line of thought.
The reason that the metorologist Norman Phillips was in a good position to appreciate the dynamics is the following: in the absence of a pressure gradient, air mass is in the same situation as the the pendulum bob: two forces are acting upon it: newtonian gravity (without correction for centrifugal effect) and the normal force. In meteorology and oceanography, a phenomenon that is called. inertial oscilations is recognized; motion with respect to the Earth, of mass that is subject to just two forces: newtonian gravity and the normal force. --Cleonis | Talk 23:45, 20 February 2007 (UTC)

I think what anon/129.74.87.3 means is that the equation of motion for the Foucault pendulum would be almost the same if the centrifugal term is dropped from the equations, and hence it is incorrect to say that the centrifugal force is "essential".

I haven't read the article by N. Phillips, but my interpretation of WP:NOR is that a peer-reviewed publication should not be removed from references because an editor thinks it's wrong (unless another peer-reviewed publication supports that editor). We could add more references, however, and possibly point out inconsistencies between them.

Interestingly, the statement that the rate of precession is 1 rev/s.day times the sine of the latitude is incorrect in reality, but becomes correct if the centrifugal force term is dropped. The formula describing the rate of precession works in both cases, if the term "latitude" is interpreted as geodetic latitude in the real case and geocentric latitude when the centrifugal force is dropped. A statement that is true in both cases is "The rate of precession is proportional to the cosine of the angle between the earth's axis of rotation and the direction of the wire at the equilibrium position."

If the centrifugal force is disregarded, then the equilibrium position will point towards the center of the earth, while in reality it points "down", which is slightly to the south of that in the northern hemisphere.

--PeR 07:45, 21 February 2007 (UTC)

PeR, nice observation regarding the definition of latitude. I was previously not aware of this and I always treated latitude as geocentric latitude. Thanks for clarifying that the question of taking the horizontal component into consideration can simply be absorbed in the definition of latitude.

I understand the argument about peer reviewed publications. However, I would like to point out that the article in question was not published in a physics journal, but in an education journal. It is likely to have received less scruteny in that environment, and it is not likely that any physicist will take the effort of contradicting the article directly. This is especially true since this is a basic physics topic that is covered well in many standard textbooks -- that clearly contradict the notion that the "horizontal component" is essential. So maybe we can build consensus amoung the editors that the reference should be removed, even if it is on the grounds of lack of clarity which Cleonis pointed out, rather than fundamental problems.

Cleonis, to distinguish an intuitive argument about the forces from a derivation of the motion, one needs to quantify the forces and be able to use these to write down a differential equation to show that it leads to the correct motion. I am unaware that this has been done by people that claim that the "horizontal force" is essential. To the contrary, most derivations neglect the effect (treating earth as a sphere and using the geocentric latitude and neglecting the centrifugal force as in e.g. Joe Wolfe's derivation between equations (8) and (9)). But maybe I am beating a dead horse here, after PeR clarified things so nicely...

--ShanRen 15:04, 21 February 2007 (UTC) (finally got myself an account, i am the anon from above)

I THINK (not sure) that the fundamental disconnect between Cleonis and PeR/ShanRen is the result of thinking about the problem from the Earth frame vs. an inertial frame. In the Earth frame, there are plenty of derivations available, and it makes sense to lump centrifugal force and Newtonian gravity together and call it "gravity". I think the article by Norm Philips (wasn't able to find it just now, but I looked at it about a year ago) is an analysis of the motion in an inertial frame. As Cleonis says, inertial motions (in the atmosphere/ocean), analyzed from an inertial frame, are the result of a poleward force. In the Earth frame, using the "effective" gravity, the surface of the Earth is a geopotential surface, and inertial motions are subject only to the Coriolis force, which makes them circular. In an inertial frame, geopotential surfaces are spherical, and so the surface of the Earth, which is oblate, crosses geopotential surfaces. Imagine a frictionless puck at the north pole. If you give it a kick, essentially what you have is a Foucault pendulum with a period of 1 day. In the earth frame, you see an inertial circle with a period of 12 hours, but in the stars frame, you see linear oscillitory motion with a period of 24 hours--the surface of the Earth slopes "upward" away from the pole (crosses geopotential surfaces), and there is a poleward component of the gravitational force, pulling the puck back toward the pole and producing the oscillatory motion. Of course, in a real Foucault pendulum, the restoring force is provided by gravity and the wire, rather than by a "poleward force." I believe that the argument made by N. Philips, and by Cleonis, is that, even so, this force is instrumental in the analysis of the precession of the pendulum in an inertial frame. I don't know if they're right or not, not having done the analysis myself, but when I was thinking about it last year, while arguing with Cleonis over content in this article which has now been removed, I remember becoming convinced that they were. Anyway, perhaps you all understood each other already, in which case this post is superfluous. Otherwise, hope this helps get everyone on the same page. Rracecarr 17:55, 21 February 2007 (UTC)

To be on the safe side: one must distinguish sharply between the restoring force that is a central force that acts in the direction of the equilibrium point of the pendulum's local swing, and the poleward force, that is a central force that acts towards the nearest pole. The poleward force is due to the oblateness of the Earth. The total force that acts upon the pendulum bob is the vector sum of the restoring force and the poleward force. --Cleonis | Talk 01:23, 24 February 2007 (UTC)

Cleonis, I read Wheatstone's article you posted. It's a neat design that simulates the foucault pendulum. In fact, one derivation linked in the article actually derives the motion of this model instead of the foucault pendulum. There are many similar devices that act the same way. For example a perfectly balanced bicycle wheel that is mounted with axes perpendicular to the surface. After one day, the wheel will have undergone a rotation of angle . Of course one can speed up the process by changing the orientation of the axis to simulate this motion. Another device that acts the same way is the South Pointing Chariot (my personal favorite), when one runs this machine on a ball along a path of fixed latitude. The difference of the initial and final directions of the pointer is again . I could go on with examples from optics and quantum mechanics and other areas of physics. The fact that these systems behave the same way is no coincidence, in general these phenomena are referred to as geometric phases. Mathematically they have in common that they perform some kind of parallel transport and exhibit some holonomy after traversing a closed path in some parameter space.--ShanRen 15:29, 22 February 2007 (UTC)

An essential aspect is the follwing counter-intuitive situation: On one hand, in the specific case of a polar Foucault pendulum, what is taking place is to be understood in terms of parallel transport; in the case of a polar pendulum the motion of the pendulum bob projected onto the Earth's surface is a section of a great circle. (A great circle that is stationary with respect to the inertial frame.) In the case of a non-polar Foucault pendulum however, it is not a case of parallel transport; each successive swing of the pendulum bob with respect to the inertial frame does not follow a (section of a) great circle. Attempting to understand the motion of a non-polar pendulum in terms of parallel transport goes awry. For example this geometric approach contains two errors, that happen to cancel each other. The attempt at derivation in that article does end up with the desired result, but the motion of the pendulum bob is misrepresented.

Mathematically, one has the choice of representing motion either with respect to a rotating coordinate system or with respect to an inertial coordinate system. Representing the motion with respect to a rotating coordinate system introduces terms that do not correspond to forces; the centrifugal term and the coriolis term. Those terms are there to facilitate the coordinate transformation, they do not stand for forces. Representing the motion with respect to an inertial coordinate system is a clean way of representing motion; all terms represent forces. Example: if you want to teach to a novice the dynamics of the solar system then you use a heliocentric model, because that is a clean model. Using the Heliocentric model, you assert that the motion of the planets is due to the inverse square law of gravitation. It would be silly to use a geocentric model in teaching the dynamics of the solar system to a novice (Animation that compares the heliocentric and the geocentric view). Likewise, the only clear way to present the dynamics of the pendulum bob is in terms of the motion with respect to an inertial coordinate system (that is co-moving with the Earth's center of mass.) --Cleonis | Talk 23:33, 23 February 2007 (UTC)

To clarify my language:

1) "Parallel transport". The plane of oscillation of the pendulum defines a direction tangent to earth (e.g. its normal). It is this direction that the foucault effect is concerned with. From the perspective of an inertial frame, we see that this direction travels along a path of fixed latitude during the course of a day. One may ask what mathematical concept governs this orientation. The answer is parallel transport along the path of fixed latitude.

2) The "derivation" you link is not one that I consider as "geometric". Maybe it could be called "trigonometric", they just draw some vectors and write down some angles, it is pretty much impossible to check if this is correct or not unless one interprets the pictures and writes down the "infinitesimal forces" and the resulting differential equation. I generally don't buy into these kind of derivations. What I consider "geometric" are ones like the argument outlined by Wilczek or the one I linked above. Or the argument I tried to give below. Essentially all use (spherical) geometry to derive the motion.

3) I also prefer the description from an inertial frame. To do this I believe you have to say how the orientation of the pendulum evolves in 3-space along the path of fixed latitude. To simplify things we see that the orientation is tangent to earth. But then you are describing parallel transport.

--ShanRen 02:37, 24 February 2007 (UTC)

[edit] Allais effect

Should the Allais effect be mentioned in the article? I think it's basically bogus science, but there has been a fair bit of noise about it over the years, so maybe it's appropriate to mention. See here: [2]. Rracecarr 15:17, 23 February 2007 (UTC)

Bogus or not, this has little to do with the Foucault Pendulum. --ShanRen 16:05, 23 February 2007 (UTC)

Anomalies in the precession of a Foucault pendulum have little to do with the Foucault pendulum? Rracecarr 16:33, 23 February 2007 (UTC)

The Allais effect is concerning a possible problem with the current theory of gravity, not with the Foucault Pendulum in particular. Given that for more than 30 years now, nobody has observed an anomaly with the Foucault pendulum, I would say that we leave this to the gravitation related articles to deal with this. That's my take on this.

Could not help to notice that the linked NASA website has a note implying that the plane of oscillation remains fixed in 3-space. (When tlking about the pendulum at the Smithsonian: "As the Earth rotates under the pendulum....") Apparently not a very reliable source for info on the Foucault Pendulum.... :-)

--ShanRen 17:59, 23 February 2007 (UTC)

Ahhhh, I wouldn't be too hard on them. It's a matter of semantics. Geophysical type people get very used to thinking of "rotation" as the compontent of the Earth's rotation that takes place about the local vertical--half the local planetary vorticity. In this sense, the Earth does just rotate under the pendulum. As for the Allais BS, I'm more than happy to leave it out. Thanks for the input. Rracecarr 18:13, 23 February 2007 (UTC)

[edit] Rewriting the "dynamics" section

Here is a suggestion of how to rewrite the dynamics section. It is quite different from what is there right now, so I would like some feedback before changing the article. I think the article should give an explanation of the effect. I propose that two explanations should be given, one from an intertial frame and one form an earthbound frame.

There are essentially two perspectives when describing the physics of the pendulum. One from the perspective of an observer in an inertial frame looking down onto earth, and one from the perspective of an earthbound observer.

In the latter case, the suspension point of the pendulum appears fixed, but the plane of oscillation of the pendulum apperas to be rotating in the earthbound frame during the course of the day. The reason for this is that the frame is not an inertial frame, giving rise to the Coriolis force which can be shown to be responsible for the precession.

When viewed from an inertial frame, the suspension point of the pendulum appears to traverse a path of fixed latitude in 3-space during the course of the day, but there are no external forces that could make the pendulum turn -- gravity acts perpendicular to the horizontal. Instead the plane of oscillation satisfies the constraint that it be perpendicular to the horizontal plane, as gravity exerts its pull on the pendulum. So the orientation of the pendulum undergoes free motion (no forces acting on it) while satisfying the constraint. Mathematically this means that the orientation undergoes parallel transport.

--Section deriving the motion using Coriolis force --

We describe the motion of a pendulum with natural frequency ω in an earthbound coordinate system with x-axis in east direction and y-axis pointing north using the small angle approximation. There are two forces acting on the pendulum, gravity and the Coriolis force. The Coriolis force at latitude φ is horizontal in the small angle approximation and given by

,

where Ω is the rotational frequency of earth. The force due to gravity, in the small angle approximation, is given by

.

Using Newton's laws of motion this leads to the system of equations

Switching to complex coordinates z = x + iy the equations read

To first order in Ω / ω this equation has the solution

If we measure time in days, then Ω = 2π and we see that the pendulum rotates by and angle of during one day.

--Section deriving the motion using parallel transport--

To quantify this, consider a path of fixed latitude φ on earth and cut out a small strip of width h around it. The strip has two boundaries, and the difference of the length of the northern boundary to the southern boundary is (to first order in h) − 2πhsin(φ). (Maybe need more detail here, use double angle formula...) Thus a vehicle with wheel distance h following the path of fixed latitude has the southern wheels travel a distance of 2πhsin(φ) more than the northern wheels, so it feels to the vehicle as if it made an anticlockwise turn of 2πsin(φ). The pendulum did not feel any forces, so it does not feel that it turned at all and will be at an angle of − 2πsin(φ) with respect to the vehicle after one day. Thus after one day, when the suspension point of the pendulum is at the same position as it was initially, and the vehicle points in the same position as it did initially, the orientation of the pendulum has undergone a net rotation of − 2πsin(φ).

This process can be nicely exemplified using a South Pointing Chariot.

The last paragraph is just some way to visualize parallel transport. If you have a better idea to describe this, let me know. --ShanRen 18:23, 23 February 2007 (UTC)

Definitely an improvement. The cart story is interesting--but I'm not sold (yet) on parallel transport. It seems to be an unnecessarily mathematically advanced concept. If you were explaining a Foucault pendulum to a professor of topology who had never heard of one, then sure, mention parallel transport, and he'll go "ah, of course". Most of us are not very familiar with the concept, and so it may not be a particularly useful tool in advancing understanding of the dynamics of the FP. Also, in my view, the parallel transport example you give is more of a mix of inertial and non-inertial analyses than it is an analysis from a purely inertial perspective. You start with an inertial frame (the Earth, imagined without rotation), and then define a non-inertial frame (the cart), and infer the precession by comparison with this non-inertial frame. A trully inertial approach would have to take into account the changing direction of gravity in an inertial frame. I honestly don't understand what's wrong with the general idea that "the earth rotates under the pendulum". As you say, the motion is constrained to the x-y plane. Hence rotation can only occur about the z-axis. So, only the component of the angular velocity vector of the earth in the direction of the z-axis counts. Therefore, the projection of the angular velocity vector of the Earth onto the local vertical gives you the (negative of the) rate of precession of the FP. To me, that seems to be a more direct way of finding a frame that doesn't have any rotation about the z-axis than the cart example. Of course, this is not an inertial analysis either, and would be more appropriate as a "physical interpretation" aside to the Coriolis analysis you suggest. Rracecarr 19:48, 23 February 2007 (UTC)

If "earth rotates underneath the pendulum", why doesn't the pendulum return to its original orientation after one day?

Your argument about projecting the angular velocity onto the normal is intuitive, but how do you show that it is correct? (As opposed to any other garden variety argument that comes out with sin(φ).) There may well be a way to explain precisely why this is a correct argument, but right now I can't think of an easy way that does not essentially repeat one of the two arguments I gave.

To your point of "mix of inertial and non-inertial analyses", I guess I did not work this point out as clearly as I wanted to. The point I am trying to make is that it's the cart (read path of fixed latitude) that turns, not the pendulum. Or in other words, the frame given by latitudes and longitudes is not an inertial frame along the path, but the one given by the pendulum is. I guess I should file on this a little. But if people feel that this explanation confuses more than it adds, then it should probably be dropped.

--ShanRen 20:15, 23 February 2007 (UTC)

The reason the pendulum doesn't come back to the same position after one day is that the angular displacement of the Earth reference frame about the local z-axis is not a multiple of 2 pi. The angular velocity vector of the earth can be expressed as two components: one vertical, and one horizontal. Together, they have a magnitude of one per day, and after one day, stuff stuck to the earth has come back to the same position it was in before. Because the FP is not stuck to the earth, it doesn't rotate with it. It is, however, constrained to move in the horizontal plane--its plane of oscillation is vertical, and it can only rotate about the vertical axis. Hence, only the vertical component of the earth's angular velocity vector comes into play. This is all basically a repeat of what I wrote before. What part of the argument strikes you as arbitrary (my interpretation of 'garden variety')? (As an aside, I'm very glad to know about the South pointing chariot--never heard of it before.) Rracecarr 20:54, 23 February 2007 (UTC)

You asked what the problem with the phrase "Earth rotates underneath the pendulum" is. The problem is that it implies that everything is back to normal after one day. Clearly there are other arguments that show that this is not the case, but that was not your question.

In your argument, it is not clear why the normal component of the angular velocity is exactly the amount by which the plane (and thus the pendulum) rotates. If the plane were to roate by that amount in an intertial frame, your argument would be a valid derivation. But it does not. From the perspective of an inertial frame, the plane is not invariant -- it does not make sense to talk about the plane "rotating" about the normal. To give a precise meaning to the argument, I belive you again need to talk about parallel transport or need to consider non-inertial forces. In short, I belive you need to explain better why it is that "only the vertical component of the earth's angular velocity vector comes into play" in this non-inertial frame you are using, and why does it have exactly the effect you say it does. --ShanRen 01:32, 24 February 2007 (UTC)

Well, I guess if it's not clear to you, than maybe it's not a very transparent argument. It's pretty clear to me. In an inertial frame, for motions confined to the surface of the earth, the vorticity of any object is defined as (du/dy - dv/dx), where u and v are the northward and eastward components of the velocity. The planetary vorticity is thus 2Ωsin(θ). That's just the vorticity of the ground, wherever you are. Anything with zero absolute vorticity, including a FP, will rotate with angular velocity Ωsin(θ) relative to the surface of the earth. Said another way, the Coriolis force is a result of the planetary vorticity. On a turntable rotating with angular velocity Ωsin(θ) opposite to the rotation of the earth, there is no vorticity, hence no Coriolis force, and so no precession, relative to the turntable. But really, that's all unnecessarily complicated. I disagree with you: it DOES make sense to talk about the plane "rotating" about the normal. The rate of rotation is just the vertical component of the angular velocity vector of the earth. Angular velocity vectors are just vectors--it wouldn't bother you to talk about the northward component of a wind vector, say, so I don't know why you don't accept the argument about the vertical component of the angular velocity vector. But regardless, I guess what we're looking for is an explanation that makes sense to everyone, without being unnecessarily technical. So I won't push this particular line of reasoning. Rracecarr 18:47, 24 February 2007 (UTC)

Yes, you can talk about the plane rotating about the normal. But then this is happening in a non-intertial frame. And your argument, that the plane is rotating underneath the pendulum, applies inertial laws of physics to this non-inertial frame. (The normal to the plane you are talking about is not fixed in any inertial frame). It is far from obvious why your argument holds. Of course this can be made precise, but it requires a lot more work. Here is an example of where the "rotating about the normal" argument has been made precise: [3]. --ShanRen 22:21, 24 February 2007 (UTC)

Username and password required to view that link. At any instant, the plane is indeed rotating at the stated rate about the normal in an inertial frame. At the same time, it is also rotating about an axis parallel to the surface of the earth. This part doesn't effect the motion of the pendulum because it is constrained to move horizontally (and has no angular momentum). It does, however, change the orientation of the "normal axis" a bit. But during each infinitesimal time period, the rotation about the normal is the important bit, and when you add up all the infinitesimal bits....

For free motion on a 2D surface, absolute vorticity (vorticity in the 2D plane, measured from an inertial frame) is conserved, due to conservation of angular momentum. It doesn't matter how you move the 2D surface through 3D space--you won't generate vorticity. But I concede the point. Things we come to think of as obvious when analyzing geophysical flows are maybe not that obvious. Rracecarr 17:26, 25 February 2007 (UTC)

You should be able to access the paper if you go through a univeristy (or any library that has subscrition to research journals). But this added explanation makes things more clear and I start to like the argument. The pendulum does have non-zero angular momentum (at almost all times), but we can neglect this since the time-average of the angular momentum is zero and thus all effect average out. There is still one issue that needs to be explained more carefully. You split the rotation into two components, the normal and one tangent. While this decomposition is unique, the order in which you apply the rotations matters. But as you are hinting this problem goes away at the infinitesimal level. But somewhat connected to this is the issue that the after a finite time step, the plane has changed in 3-space. So you are trying to say that a vector in that plane has rotated about the normal in that time. In order to make sense out of that statement, you need to be able to compare the two planes -- or you need to be able to say which vector in the second plane corresponds to the un-rotated vector in the first plane, so that you can rotate it. In my language that is called a connection, and there is an obvous choice in this setting. The way to identify vectors in the two planes is by means of the rotation axis that is in the plane, so the one about which the first rotation was carried out. (I hope what I am saying here makes sense, this is probably clear to you, but it needs to be addressed as part of the argument.) In the case of earth, the axis in the plane is in the direction of the latitude, so this gives a natural frame in which to measure the rotation of the pendulum. It happens to coinside with the earthbound frame. Thus the rotation of the pendulum in the earthbound frame is really just given by the rotation of the normal.

Btw, from my perspective this is simply a different way to describe parallel transport. Take a vector in the tangent plane and rotate the plane slightly (leaving the vector fixed in 3-space). Then the vector is not part of the plane any more. Take its orthoginal projection onto the plane and normalize (normalizing is only necessary after finite rotations, for infinitesimal rotations one does not need to normalize). The resulting vector is the same as the one obtained by your procedure (to first order, which is all that matters here). --ShanRen 18:17, 25 February 2007 (UTC)

[edit] In the case of the Foucault pendulum, the centrifugal term is a hundred times larger than the Coriolis term.

I calculate the centrifugal term and the Coriolis term at 45 degrees latitude.

I estimate the maximal velocity of the pendulum bob is, depending on the size of the rig, in the order between 1 and 5 m/s, I will use a velocity with respect to the Earth's surface of 3 m/s

For an object at rest with respect to the Earth, located at 45 degrees latitude:
Distance to the Earth's axis: 4500 km
Angular velocity (omega) of the Earth: 0.000072 radians per second
Velocity: 327 m/s
Magnitude of centrifugal term: v^2/r = 0.023 m/s^2
Magnitude of Coriolis term: 2 * omega * v = 2 * 0.000072 * 3 = 0.00044 m/s^2

In the velocity range of the Foucault pendulum, the centrifugal term is in the order of a 50 times larger than the Coriolis term. There is no way the centrifugal term can be ignored.

Yet, in correct-results-yielding-derivations (set up in terms of motion with respect to a rotating coordinate system) we see that the centrifugal term is not present while the Coriolis term is. The reason for the centrifugal term not being present in the equation is as follows: in an earlier stage of setting up the equation, a force was incorporated that acts in centripetal direction: the poleward force. The poleward force and the centrifugal term have exactly the same magnitude and they point in opposite directions: centripetal and centrifugal. Hence in the equation of motion (with respect to a rotating coordinate system) the poleward force and the centrifugal term drop away against each other. The act of dropping the centrifugal term is one and the same thing as incorporating the poleward force. That is why I have made the observation that all (mathematically) sound derivations do incorporate the poleward force. --Cleonis | Talk 00:46, 24 February 2007 (UTC)

Now I am confused. I believed the "polward force" you were referring to to be the component of the centrifugal/petal force that is perpendicular to gravity. Now you say it is the centripetal force. Please clarify.

Also, if you keep the centrifugal term in the calculation I made (i.e. using the geocentric latitude and not neglecting the centrifugal force), the solution will yield a small correction to the foucault formula (about half a degree at the middle latitudes). This corresponds exactly to the difference in geocentric and geographic latitude. Try it out, it gets a little messy though...

About your numbers, the magnitude of graviy is even bigger. So what. What's important is how these forces make the pendulum rotate.

--ShanRen 02:20, 24 February 2007 (UTC)

The blue arrow represents the poleward force. By definition, the poleward force is perpendicular to the normal force. See also the Image description page.

Definition of the poleward force: the resultant force of newtonian gravity and the normal force. (I use the expression 'newtonian gravity' to distinguish between the newtonian gravity on one hand, and effective gravity on the other hand. The effective gravity is a vector sum of newtonian gravity and centrifugal effect, the effective gravity acts perpendicular to the local surface of the oblate spheroid; exactly opposite in direction to the normal force.)

I have not considered difference between geographic latitude and geocentric latitude. I think it is pretty clear that if that needs to be corrected for, the correction will be a small correction to the Foucault formula. --Cleonis | Talk 11:03, 24 February 2007 (UTC)

If by "resultant force of newtonian gravity and the normal force" you mean "the difference of gravity and the normal (to the ellipsoidal earth) component of gravity" -- which is just the tangential component of gravity -- then we are on the same page. I still don't see it's importance for the precession of the pendulum. Taking it into conideration or ignoring it is exactly the difference of what definition of latitude is used. --ShanRen 16:02, 24 February 2007 (UTC)

Recapitulating: latitude is a way of assigning numbers to points in space. The magnitude of the poleward force is independent of the system of how latitude is notated. I suppose that one can define 45 degrees latitude either as exactly 45 degrees with respect to the plane of the equator, or one can define 45 degrees latitude as precisely half way the distance in kilometers from the equator to the pole, as measured along the surface of the Earth. Given the oblateness of the Earth, those two definitions of latitude will assign the number 45 to somewhat different locations. (I think the difference will be in the order of kilometers.)

I do appreciate the existence of that difference, but I think correcting for it, if necessary at all, will only involve a minor correction to the foucault formula, so I prefer to not discuss it in a first approximation.

The magnitude of the poleward force at 45 degrees latitude is pretty much the same as at 44 degrees latitude and at 46 degrees latitude. Therefore hopping from the one system of assigning latitude numbers to the other makes very little difference. (Conversely, if there would be a sharp gradient in the magnitude of the poleward force around 45 degrees, then latitude would be critical.)

The magnitude of the poleward force at 45 degrees latitude (give or take a couple of degrees of latitude) is 0.016 Newton of force for every kilogram of mass. --Cleonis | Talk 17:57, 24 February 2007 (UTC)

[edit] Properties of parallel transport

The ground track of a satellite orbit.
(Larger version (512x256 px)

Ground tracks of a constellation of satellites, all satellites orbiting in the same plane.
(Larger version (256x256 px)

I have uploaded some animations to wikipedia that I made a while ago. The animations illustrate the concept of parallel transport.

The first animation represents the ground track of a satellite that orbits the earth at a constant altitude, with a period of one sidereal day. In this animation the plane of the orbit is tilted with respect to the plane of the equator. In the case of very high altitude satellite orbit, the only force that is at play is newtonian gravity. The shape of the satelllite's ground track is a great circle. This great circle is stationary with respect to inertial space. (The expression 'with respect to inertial space' is shorthand for, 'with respect to the inertial frame of reference'. Here is an example of usage of this expression on an educational site.)

The second animation shows the ground tracks of a constellation of 4 satellites that take two sidereal days to circumnavigate the earth. All four satellites orbit in the same plane. What is shown in the animation is the motion of the ground tracks with respect to inertial space.

(Of course, since the Earth rotates underneath the satellite orbits, the ground tracks as mapped with respect to the Earth are not great circles.)

I take it that everybody grants the following properties of parallel transport along the surface of a sphere:

• A point that moves along the surface of a sphere in parallel transport will move along a great circle
• This great circle is stationary with respect to inertial space.
• An object that moves to and fro along a short distance in parallel transport, will likewise move along a section of great circle, with the great circle stationary with respect to inertial space.
  

--Cleonis | Talk 10:49, 24 February 2007 (UTC)

You seem to be confusing the concenpts of geodesic and parallel transport. --ShanRen 16:10, 24 February 2007 (UTC)

Yes, I did not fully appreciate the distinction between motion along a geodesic and parallel transport. Parallel transport is not necessarily inertial motion. For example, one can take a gimbal mounted, spinning gyroscope from one location to another. The relocation involves acceleration of the gyroscope's center of mass, so it is non-inertial motion, but during the relocation the gymbal mounted gyroscope retains its orientation with respect to inertial space. That is an example of flawless parallel transport. --Cleonis | Talk 17:18, 24 February 2007 (UTC)

[edit] The south pointing chariot and parallel transport

Recapitulating the operating principle of the south pointing chariot: let the wheelbase be 1 meter. Then the mechanism inside must be devised in such a way that when the outside wheel rolls 6.28 (2*pi) meters more distance than the inside wheel, the indicator must have turned a full rotation with respect to the chariot.

On a perfectly flat surface, this operating principle can perform flawlessly. For any trajectory that closes a loop, the outside wheel travels (2*pi*wheelbase) more meters than the inside wheel. This is independent of the area that is enclosed by the looping trajectory.

However, on the earth's surface, the chariot will not operate flawlessly, because the Earth surface is a curved space.
Let the chariot be located at 30 degrees latitude. Let the chariot go around the earth, always following the latitude of 30 degrees. Then as compared to the flat geometry case, the chariot's indicator will rotate half as much, because the difference in distance rolled is half as much as in the flat surface case.

This means that if you would actually use the south pointing chariot, and you would start with the chariot's indicator pointing due south, then after completing a circumnavigation at 30 degrees latitude, the chariot's indicator will point due north. (And for every latitude the relation is the same as in the foucault pendulum case)

The question is, does this way of using the chariot count as parallel transport? I seems to me it doesn't, for the chariot's operating principle works flawless if and only if the chariot moves over a geometrically flat surface. --Cleonis | Talk 15:34, 24 February 2007 (UTC)

The South Pointing Chariot only approximates parallel transport. It performes parallel transport in the limit where the distance between the wheels goes to zero (but of corse keeping the ratio of wheel distance to wheel size fixed). If the curvature is small compared to the wheel distance it gives a good approximation. Of course, if the curvature is zero, then it performs parallel transport exactly no matter what the wheelsize is. --ShanRen 16:07, 24 February 2007 (UTC)

As long as the wheelbase is sufficiently small, the operating principle of the South pointing chariot is independent of it's wheelbase. Taking the limit of wheelbase going to zero makes no difference. (Enlarging the wheelbase without any limit does eventually introduce flaws, of course.)

The deviation from flawless parallel transport is small if the circumference of the loop-closing trajectory is small relative to the radius of curvature of the planet's surface.

Importantly, it is precisely the flawed operation of the chariot in the case of travelling over the surface of the Earth that makes it a model for the Foucault formula! That is why I presented the case of the chariot circumnavigating the earth at 30 degrees latitude. If you start with the indicator pointing due South, then after circumnavigating twice (at 30 degrees latitude) the indicator points due South again, in perfect analogy to the case of the plane of swing of the foucault pendulum. --Cleonis | Talk 16:38, 24 February 2007 (UTC)

The wheelbase makes a difference, even when it is small. If you run the SPC centered on the 30 degree latitude circle starting pointing south, it will only approximately point north after one revolution. If it would perform flawless parallel transport, it would point exactly north. As the wheelbase gets smaller, the approximation will get better.

It seems to me that you are under the misconception that a direction undergoing parallel transport along a closed loop has to return to its original position. This only holds when the curvature is zero as shown by the Gauss-Bonnet Theorem. In general, the direction will pick up some holonomy, just like the pendulum does. In fact, on a sphere there it is impossible to transport tangent directions along paths (in a continuous fashion) so that it always (no matter what the path is) returns to the original position by the Hairy ball theorem. --ShanRen 17:34, 24 February 2007 (UTC)

I think I get your point. What I did was to label the behavior of an SPC on a flat surface as "flawless", and I labeled the behavior of the SPC on a curved surface as "flawed". The idea is, I gather, to take the actual behavior of the SPC on any surface as the very definition of parallel transport within the plane of that particular surface, be it flat or curved, and proceed from there.

I am glad this issue appears to be cleared up. I suppose the parallel transport approach can be made to work in the context of relativistic theory of gravitation, which is a quite exciting thought. On the other hand, I don't think there is a way of applying the parallel transport concept in the context of newtonian dynamics. --Cleonis | Talk 18:13, 24 February 2007 (UTC)

Except that the Foucault Pendulum, when viewed from an inertial frame, clearly undergoes parallel transport. No forces that can make it rotate + constraint to remain tangent = parallel transport. --ShanRen 21:35, 24 February 2007 (UTC)

In terms of newtonian dynamics, the Foucault effect is accounted for by the presence of the poleward force.

I'd like to formulate the equation of motion for a more general pendulum case, that is also more symmetrical: a conical pendulum. For example, a centrifugal governor that rotates with constant amplitude is an example of a conical pendulum. As is well known, the motion of a conical pendulum can be thought of as a linear combination of two perpendicular harmonic oscillations that are 90 degrees out of phase. I want to formulate the equation of motion for motion with respect to inertial space. First, I want to set up the equation for the case where the conical pendulum completes just a few revolutions during each revolution of the rotating platform (planet) it is located on. --Cleonis | Talk 23:02, 24 February 2007 (UTC)

Cleonis, you are wrong about the poleward force. It has little to do with the precession of the pendulum. This can easily be seen by the fact that the answer does not change by much if you ignore it or keep it in the equations. It is time that we settle this issue. If you wish, I can write down these computations for you to look at. Or you may want to do them yourself. Or maybe you have another idea on how to settle this?

You have to be very careful with the conical pendulum or the centrifugal governor. Unlike in the case of the planar pendulum, the average of the angular momentum is not zero for these systems. In that case the gyroscopic forces do not average out to zero and must be considered. These systems do not behave like a Foucault pendulum. If you are looking at the case where the period is very small the angular momentum is also very small and things will still work. But then you might as well consider the (bicycle) wheel that is not spinning as I suggested above. It behaves exactly like a Foucault pendulum. --ShanRen 03:15, 25 February 2007 (UTC)

Unfortunately, I have insufficient clues to reconstruct where we disconnect. Our points of view could not be more opposite.

There is a geophysical phenomenon, recognized in oceanography, that is more elementary than motion of the Foucault pendulum bob. This geophysical phenomenon involves only a single force: the poleward force. (Strictly speaking two forces, newtonian gravity and the normal force, by definition the poleward force is the resultant force of those two, resolved in the direction tangent to the local earth surface)

In this webcourse http://oceanworld.tamu.edu/resources/ocng_textbook/chapter09/chapter09_01.htm the phenomenon of inertial oscillations is described in terms of motion with respect to a rotating coordinate system (co-rotating with the earth) I maintain a website of my own, with articles about physics subjects. In my own article about inertial oscillations, I present analysis of the motion as mapped with respect to inertial space.

In order to account for the phenomenon of inertial oscillations, the poleward force is indispensible; it is the only force that is present. I am very curious now: do we also disconnect in the analysis of inertial oscillations, or can we establish common ground?

Thank you for pointing out that in the case of a conical pendulum, one has to take into account the tendency of any gyroscope to conserve angular momentum. I hadn't taken that into consideration. Since the conical pendulum I want to analyse is not gimbal mounted, but has a fixed orientation with respect to the rotating platform, it may not be an issue, but that remains to be seen. --Cleonis | Talk 12:23, 25 February 2007 (UTC)

I will take a look at the inertial oszillations some other time, right now the Foucault pendulum is taking enough of my time.  :-) --ShanRen 16:51, 25 February 2007 (UTC)

[edit] The gravitational geophysics of the Earth's oblateness

I'd like to address a potential source of confusion.

In the case of an oblate spheroid, the center of gravitational attraction and the geometric center are very close to each other, but they do not quite coincide. If all of the earth's mass would be concentrated in a single point, where would that point have to be in order to exert the same gravitational pull as the earth does in its current form? For an object located on the equator, the corresponding center of gravitational attraction is located at a distance of about 10 kilometers away from the geometric center of the earth. (This off-centeredness is called 'the quadrupole moment' of the Earth's gravitational field.) Also, we have that the equator is about 21 kilometers further away from the geometric center than the poles.

If one were to assume - erroneously - that the center of gravitational attraction is always the geometrical center then one would expect that in travelling from the equator to the pole one loses 21 kilometers of altitude. Difference in altitude corresponds to difference in gravitational potential energy. E = m*g*h and in this case h would be 21 kilometers.

However, one needs to take a smaller altitude difference into account. An object located on the equator has around a 107 KiloJoule per kilogram of mass gravitational potential energy more than the same object located on the pole. This difference in gravitational potential energy corresponds to a height difference of about 10 kilometers. Another example, moving from the equator to 45 degrees latitude corresponds to a drop in geopotential altitude of about 5 kilometers, hence around 53 KiloJoules per kilogram of mass of gravitational potential energy will convert to kinetic energy. --Cleonis | Talk 13:20, 25 February 2007 (UTC)

ShanRen, my hunch is that you are overcompensating for the quadrupole moment. My hunch is that you are - erroneously - assuming that the quadrupole moment is so large that in fact at every latitude the newtonian gravity acts perpendicular to the local surface. That would explain your peculiar assertion that "the tangent to the surface component of newtonian gravity can be absorbed into the definition of latitude." --Cleonis | Talk 13:44, 25 February 2007 (UTC)

Let's keep the discussion on the pendulum. Essentially there are two models.

• Spherical earth model. We neglect the centrifugal force, assume earth is spherical and use geocentric latitude. Gravity acts radially, that is perpendicular to the surface. From the perspective of an inertial frame there are no external forces that can make the pendulum turn, so the pendulum undergoes parallel transport and we end up with the sin law.
• Ellipsoidal earth model. We consider the combination of gravity and the centrifugal force and assume earth is an ellipsoid that forms an equipotential surface. We use the geographic latitude. The combination of forces acts perpendicualr to the surface. From the perspective of an inertial frame there are no external forces that can make the pendulum turn, so the pendulum undergoes parallel transport and we end up with the sin law.

When looking form the perspective of an intertial frame, the pendulum is travelling around earth along a circular path. This is how the "centrifugal force" comes into play from the perspecitve of the inertial frame. Note that the magnitude of the normal force is different in the two cases. This influences the period of oszillation of the pendulum, but not the Foucault precession. Thus essentially, the only difference between the two models is the horizontal component of the centrigugal force, that is the "poleward force". This difference has been absorbed in the definition of latitude and is quite small.

If you prefer an argument explaining this in an earthbound frame, I can give you one on your talk page. I think this discussion has caused enough clutter on this page.

--ShanRen 17:19, 25 February 2007 (UTC)

When the motion of the pendulum is mapped in inertial space, then there is no centrifugal term in the equation of motion.

I notice that you approach the analysis from an earthbound point of view. You combine gravity and the centrifugal term, and you assume the reference ellipsoid is an equipotential surface; those decisions are consistent only with the earthbound point of view.

For the purpose of explanation, I approach the problem from an inertial point of view. Aproaching from the inertial point of view, there's no combining gravity and centrifugal term, for in the inertial view there is no centrifugal term. Approaching from the inertial point of view, the equator is at a higher gravitational potential than the poles.

In my opinion only an explanation in terms of motion with respect to inertial space can be illuminating.

Modeling the Earth as a perfect sphere is not an option of course, as a first approximation the reference ellipsoid must be assumed as the Earth's shape. --Cleonis | Talk 23:05, 25 February 2007 (UTC)

I guess the sentence

When looking form the perspective of an intertial frame, the pendulum is travelling around earth along a circular path. This is how the "centrifugal force" comes into play from the perspecitve of the inertial frame.

is not clear enough, so let me explain in more detail. From the perspective of an inertial frame, the suspension point is travelling on a circular path. The suspension point of the pendulum is accelerated, and the force corresponding to this acceleration is exactly the centripetal force. In the case the pendulum is not oscillating at all, it would hang exactly in the direction of the plum line, that is perpendicular to the surface of the elliposiodal earth. The force at work here is exactly the centrifugal force. Since it has a name already, I just keep referring to that force that the pendulum feels on its path as the centrifugal force. I don't believe that this should be too confusing.

Of course you could ignore the acceleration of the suspension point. It will make the computations more cumbersome, but it will not change the result by much. Again, this can be absorbed in the definition of latitude (although I have not taken the time to see if one of the wikipedia definitions fits this purpose).

Also, please specify in what variable the ellipsoidal earth differs from the sphere in "first order" in your opinion. For pretty much any reasonable variable I can come up with, both are the same to first order. --ShanRen 23:46, 25 February 2007 (UTC)

We agree on the following: a Foucault pendulum in non-swinging state is a plumb line. A plumb line shows the direction of effective gravity. The suspension point circumnavigates the Earth's axis. In the case of a non-swinging pendulum the bob circumnavigates the Earth's axis along a circular trajectory.

The next question is: when the analysis is approached from an inertial point of view, what forces are being exerted upon the bob of a non-swinging pendulum? Answer: two forces: newtonian gravity and the tension of the suspension wire. Question: is a centrifugal force being exerted upon the bob? The answer is a definite no.

Conversely, when the analysis is approached from an earthbound point of view, then some people opt to regard the centrifugal term as a force in centrifugal direction that is being exerted upon the bob. As I mentioned before, I rigorously maintain an inertial point of view. In terms of the inertial point of view, no centrifugal term enters the equations, no centrifugal term figures in any consideration. I believe this level of rigor is necessary here.

I use the expression 'first approximation' not in a technical way here, I'm not referring to a hierarchy of 'first order', 'second order' etc. To me, first approximation is the a level of approximation that captures the bare essentials, while neglecting factors that are, say, 1/20th or less of the magnitude of the bare essentials. --Cleonis | Talk 01:14, 26 February 2007 (UTC)

[reset indent]Cleonis, as you say, there are only two forces acting on the hanging (non-oszlillating) bob. And their difference is .... a force that is equal to the centripetal force. Let's call this force F for now, since continuing to use the familiar name of the equal "centripetal" force seems too confusing. So please replace the term "centrifugal force" in my discussion above by "the force that causes the acceleration of a hanging bob". If you also still object to the term "equipotential surface" then replace this with the "normal in the direction of the plum line". Maybe then you are able to comment on the substance? --ShanRen 01:43, 26 February 2007 (UTC)

All I can do right now is to select the part of your statements that I agree with: the poleward force provides the required centripetal force for maintaining the trajectory of the pendulum bob, as it circumnavigates the earth's axis.

Other than that it's not working out, I find myself disregarding statements by you that contain the expression 'centrifugal force' for they don't seem to add up to some coherent thought.

In meteorology, the expression 'geopotential surfaces' is in wide use. A geopotential surface is defined as a continuous surface that is at all points perpendicular to the local plumb line (which is what you described). The best known example of a geopotential surface is the worldwide sea-level. Thinking in terms of geopotential surfaces implies commitment to the earthbound point of view.

There is no point in commenting on a representation that is cast in terms of an earthbound point of view. I am trying to persuade you to shift to an approach from an inertial point of view. So far I'm unsuccesfull in that. The conceptual tools that you use, such as concept of geopotential surface as an equipotential surface, are tools that slot in only with the earthbound point of view. If you shift to the inertial point of view, then there will be substance to comment on. --Cleonis | Talk 03:44, 26 February 2007 (UTC)

One more try. I will explicitely do the replacements that I asked you to do, but you are apparently you are not able or willing to carry out mentally. And just to make sure you don't miss it, I will copy the text down here.

• Spherical earth model. We neglect the force accelerating the suspension point, assume earth is spherical and use geocentric latitude. Gravity acts radially, that is perpendicular to the surface. From the perspective of an inertial frame there are no external forces that can make the pendulum turn, so the pendulum undergoes parallel transport and we end up with the sin law.

• Ellipsoidal earth model. We consider the force F that is the combination of gravity and the (negative of the) force accelerating the suspension point and assume earth is an ellipsoid whose surface is perpendicular to F. We use the geographic latitude. The combination of forces (that is F) acts perpendicualr to the surface. From the perspective of an inertial frame there are no external forces that can make the pendulum turn, so the pendulum undergoes parallel transport and we end up with the sin law.

Please keep your comments focused on the Foucault pendulum. --ShanRen 12:58, 26 February 2007 (UTC)

To ShanRen: You keep voicing your intention to cast the representation in terms of the inertial point of view, but despite this intention, you remain immersed in earthbound thinking. I demand mathematical rigor here; you are faulting me for demanding mathematical rigor. My attempts at explaining the physics to you are unsuccesful, so after this comment I will withdraw for a significant amount of time.

The comments I wrote the previous weekend were aimed at persuading you to actually do what you announce: shift to the inertial point of view.

We have that according to the laws of newtonian physics, a centripetal force is required to sustain motion along a circular trajectory. I will refer to the circular-motion-sustaining-force as "the required centripetal force".

It is only when the motion is cast in terms of the earthbound point of view, that there is the centrifugal term, which - as you point out - coincides with the negative of the required centripetal force.

When the motion is cast in terms of the inertial point of view, there is no centrifugal term. I claim that in this particular situation, we cannot afford any relaxation of mathematical rigor.

How the presence of the poleward force is significant:

Set up a Foucault pendulum on one of the poles. The swinging pendulum has an equilibrium point: the line from the suspension point to that equilibrium point coincides with the earth's axis. Now the case of a Foucault pendulum located somewhere between 44 and 46 degrees latitude (or any other non-polar/non-equatorial region for that matter). The line from the suspension point to the locally meassured equilibrium point coincides with the local plumb line. If that local (mid-latitude) plumbline is extended into the earth, it does not pass through the geometric center of the Earth. Also, and this is the significant feature, the local plumb line, when extended into the earth, does not point in the direction of newtonian gravity. In the case of a rotating planet, such as the earth, with an oblateness that matches its angular velocity, the direction of newtonian gravity is not perpendicular to the local Earth surface, but the plumb line is. (NB The direction of newtonian gravity is neither towards the geometric center of the oblate spheroid, nor perpendicular to the local surface. In fact, it is to a good approximation halfway between those two directions.) This particular angle corresponds to the fact that the required centripetal force must be provided.

In physics, there is the work-energy theorem: when an object that is subject to a force moves in the direction of that force, then that force is doing Mechanical work, altering the kinetic energy of the object. On a pendulum swing in south-to-north-direction (equator-to-pole) the poleward force is pulling the pendulum bob closer to the center of rotation. (compare an ice-skater who increases her spin-rate by pulling here arms closer to her axis of rotation). One may be sorely tempted to underestimate the significance of the work done by the poleward force. The catch is: while the effect is small on each separate swing, it is a cumulative effect. No matter the direction of the swing, swing after swing it just adds and adds and adds. (NB when the pendulum swings away from the pole the poleward force is doing negative work.) This cumulative effect accounts for the precession of the plane of swing when the motion is analysed from the inertial perspective.

For discussing the bare essentials, the different in geometric latitude and geographic latitude is not significant. More importantly: there is no such thing as "absorbing the poleward force into the definition of latitide". Compare the following example: at the equator it's hotter than at the poles: there is a latitudinal gradient of avarage temperature. But there is no such thing as "absorbing the temperature gradient into the definition of latitude". The idea of "aborbing the poleward force into the definition of latitude" is entirely flawed.

I am pleased to see that the article by Wheatstone has been added to the references. My intention is to make some animations that illustrate the operating principle of the Wheatstone-Foucault device. When those animations are finished to my satisfaction, I will return to the Foucault pendulum article. --Cleonis | Talk 23:13, 2 March 2007 (UTC)

[edit] blue line (again still)

I don't understand the value of the blue line. Why is the projection of the orientation of the pendulum onto the equatorial plane important? What is it's physical significance? The only value that I can see is that it points out that (a) the plane of osciallation of the pendulum changes and (b) that it does not return to its original orientation after one day. But this can be said a lot clearer without the blue line (and it's contrived definition). There is already a lengthy discussion on this and related issues above. Now at least the text has been fixed to correctly reflect the nature of the blue line, but maybe people can quickly indicate how they feel about removing it alltogether? --ShanRen 22:06, 27 February 2007 (UTC)

I'm in favor of removing it. Rracecarr 22:47, 27 February 2007 (UTC)

Ok, made the change -- and some more.... Please question, change or revert as you see fit. --ShanRen 21:07, 1 March 2007 (UTC)

To Rracecarr and ShanRen; Thank you both for updating the Foucault Pendulum article. You have made the article enjoyable to read again for a novice and tourist who is interested in the Foucault pendulum. 15 months ago there was no mathematical discussion of the motion of the pendulum. I tried to write a separate article regarding a vector analysis of the pendulum relative to the surface of the Earth but it certainly got bogged down. I am happy that it is no longer referenced. Thank you both for restoring clarity to operation of the Foucault pendulum and my view is that the blue line should not be included. David Harty 06:43, 8 March 2007 (UTC)

[edit] Alternate Caption for First Diagram

The caption of the first diagram says;

"Change of direction of the plane of swing of the pendulum in angle per sidereal day as a function of latitude. The pendulum rotates in the anticlockwise (negative) direction on the southern hemisphere and in the clockwise (negative direction) on the northern hemisphere. The only points where the pendulum returns to its original orientation after one are the poles and the equator."

It appears that this caption reflects a certain perspective. An alternative perspective in a note could be provided or further explanation could be added. When the pendulum is mounted at the poles the maximum rotation of the plane of the swing is observed. When one starts the pendulum in the direction towards a star while located at the poles then the plane of the swing is always directed towards that star. What is observed is the Earth rotating underneath the plane of the swing of the pendulum and this effect is a maximum at the poles. There is no force which causes the change of direction of the plane of swing of the pendulum. The caption is written for a certain perspective.

Anticlockwise and clockwise occur in the caption but this only occurs because of a change in perspective in viewing pendulum. The "directions" are not clear because the "point of view" is not clear and apparently changes. It also may be that the "point of view" is rotating with the Earth for the caption. Anticlockwise is observed for a pendulum in the southern hemisphere from a viewpoint on the southern hemisphere. Clockwise is observed for a pendulum in the northern hemisphere from a viewpoint on the northern hemisphere. The Earth only rotates in one direction. The caption is changing the "point of view" which is inconsistent. As viewed from the "point of view" of very high above the North Pole then clockwise motion would be observed for the southern pendulum plane of swing (and the point of view would have to be rotating). The Earth only rotates in one direction. However it is not the plane of swing that is moving it is the Earth turning underneath the plane of swing. The observer on Earth at the respective pole observes the anticlockwise or clockwise direction. If this discussion is correct then the diagram should be a mirror image about the equator rather than a sine function if plotted from a consistent viewpoint.

Since the plane of swing of the pendulum at the equator doesn't change with respect to the surface of the Earth then the orientation with respect to the Earth doesn't have to return to its original orientation with respect to the Earth. Only with respect to the stars does the pendulum return to its original orientation at the equator. At the poles the pendulum doesn't change with respect to the stars through the entire day. There is a mixed-up basis in this last sentence which is misleading. The subject of the article is the Foucault pendulum which by definition is the motion of plane of swing of the pendulum in relation to the surface of the Earth, not in relation to the stars. However, a consistent point of view can be helpful to the discussion.

An alternative caption or explanation for the first diagram might read;

Rotation of the Earth with respect to the plane of swing of the pendulum in angle per sidereal day as a function of latitude. The pendulum rotates in the anticlockwise (positive) direction on the southern hemisphere as observed from the South Pole and in the clockwise (negative) direction on the northern hemisphere as observed from the North Pole. The southern hemisphere pendulum if it could be observed from the North Pole would rotate in the clockwise direction. When observed from one fixed "point of view" high above the North Pole both of the polar pendulums would be observed to have a stationary plane of swing and the Earth would be observed to move counter-clockwise because the Earth is actually turning underneath the pendulum in the opposite direction then the Earth-based observed motion. The plane of swing of the polar pendulums do not rotate with respect to the fixed stars and return to their original orientation with respect to the Earth in one day. The plane of swing of the pendulum at the equator does not change with respect to the Earth for the entire day and returns to the original orientation with respect to the stars after one day (as does the Earth).

Four or five gif images for the polar pendulums would be needed to adequately identify the "point of view" perspectives that are described above. It would be excellent to have these images so that it would be clear in the article what is being discussed for the lay reader and the tourist. This would also clarify any comments made in the discussion. For the polar pendulums it would be necessary to have a dot on the circumference of the Earth to show that the Earth is rotating and the plane of swing of the pendulum is not when observed from high above the North Pole. A pendulum located at the equator could also be represented with a gif image. David Harty 08:42, 9 March 2007 (UTC)

[edit] Explanation for Second diagram

The caption of the second diagram says;

"The precession of the plane of swing at a latitude of 30 N. The view is from high above the north pole. The oval represents a circle that appears distorted because it is viewed at a 60 degree angle. The line inside the oval is the trace of the plane of swing over the surface that the pendulum is suspended above. On the left the view as seen from a non-rotating point of view, on the right the co-rotating-with-the-Earth point of view."

It is difficult for me to understand how the images relate to the dynamics section. The images show that the plane of swing of the pendulum is rotating. Therefore a conclusion is identified that there is a force acting on the plane of swing of the pendulum in order to make it rotate. However, it is determined from the polar pendulum discussion that the angular velocity with respect to the Earth is a maximum for the polar pendulum and this angular velocity is observed because the Earth is turning under the plane of swing of the pendulum, not a force acting on the pendulum. To be consistent, then one has to say that the Earth-bound frame of reference must be used to describe the relative motion of the Foucault pendulum to the surface of the Earth. This relative motion is equal to the "magnitude of the projection of the angular velocity of earth on the normal direction to Earth". Maybe this quoted statement in the article needs to specifically refer to this diagram so that it is clear that there is less angular velocity observed for the Earth turning underneath the pendulum, or maybe the implied conclusion needs to be identified that there is a force acting on the plane of swing causing it to precess, or maybe what is meant is that there is a Coriolis force acting opposite to the observed angular velocity of the turning of the Earth underneath the pendulum. An explanation is warranted to explain how the dynamics section applies to this diagram so that the article is consistent. David Harty 08:42, 9 March 2007 (UTC)

An alternative caption or explanation for the second diagram might read;

For the pendulum located at the equator the turning of the Earth is not observable as the plumb line of the pendulum is located perpendicular to the axis of rotation of the Earth and thus, turns with the Earth. As the latitude for the location of the pendulum increases the Earth turns underneath the pendulum to a greater extent corresponding to the decrease of the angle of the plumb line of the pendulum to the Earth's axis of rotation. The angular velocity observed increases according to the sine of the latitude, or increases according to the cosine of the angle in relation to the axis of rotation of the Earth (90 degrees minus the longitude). For the pendulum located at the poles the Earth is observed to fully turn underneath the pendulum as the plumb line of the pendulum is located on the axis of rotation of the Earth.

The above explanation ascribes the mechanism for the observed motion to the rotation of the Earth rather than a force acting on the pendulum. Any forces that are applied to the pendulum swing occur at the point of release of the pendulum. David Harty 12:34, 17 March 2007 (UTC)

It seems to me that a 3D demo might be more useful here. I quickly put together a little demo that lets you explore the pendulum at different latitudes. It probably needs some polishing. Also, I am not sure if there are any copyright issues, the java code for the demo was written by some student as part of a bachelor's thesis, I just grabbed it off the internet since it seemed useful.

I don't know how to put code for an applet into wikipedia, but just paste the following code under "import data" in the demo applet (sorry about the mess).

filever 100.0 env {function $S(theta,phi)=cos(theta)*sin(phi),sin(theta)*sin(phi),cos(phi)$ interval $v=-0.3,0.3,2$ interval $u=-0.3,0.3,2$ interval $phi=0,pi,30$ interval $theta=0,2*pi,60$ variable $lambda=-pi/2+0.0001,pi/2-0.0001,30$ 0.3141592653589793 variable $x=0,2*pi,60$ 0.0 function $lat(theta)=S(theta,-lambda+pi/2)$ function $E1(x)=lat'(x)/length(lat'(x))$ function $E2(x)=cross(lat(x),E1(x))$ function $V(x)=cos(sin(lambda)*x)*E1(x)-sin(sin(lambda)*x)*E2(x)$ function $P(u,v)=lat(x)+u*E1(x)+v*E2(x)$} objects [{controls obj0 {constants [] intervals [] objects [] readouts [] functions [] variables [x lambda]}} {frame.3d obj1 {canvas {canvas.3d obj9 {tool 100.0 points true graph {graph.3d obj10 {light true graph {plots [{plot.vector obj8 {base $lat(x)$ endstyle 1.0 dir $E1(x)$ color {coloring.constant obj11 {color [1.0 1.0 1.0 1.0]}} dim 3.0 thick false basestyle 0.0 subdivs 1.0 len 0.3 basesize 10.0 super {transpt $0$ visible true addedz 0.0} endsize 10.0}} {plot.surface obj7 {intrpl true super {super {transpt $0$ visible true addedz 0.0} color {coloring.group obj12 {colors [{coloring.constant obj13 {color [1.0 0.8 0.0 1.0]}}] opacities [0.33] mixings [0.0]}} expr $P(u,v)$} vrtnml true}} {plot.curve obj4 {super {super {transpt $0$ visible true addedz 0.0} color {coloring.constant obj14 {color [0.0 1.0 0.0 1.0]}} expr $lat(theta)$} thick false}} {plot.vector obj3 {base $lat(x)$ endstyle 1.0 dir $E2(x)$ color {coloring.constant obj15 {color [1.0 1.0 1.0 1.0]}} dim 3.0 thick false basestyle 0.0 subdivs 1.0 len 0.3 basesize 10.0 super {transpt $0$ visible true addedz 0.0} endsize 10.0}} {plot.point obj5 {super {transpt $0$ visible true addedz 0.0} color {coloring.constant obj16 {color [1.0 1.0 1.0 1.0]}} label $$ size 10.0 style 0.0 showlabel false dim 3.0 expr $lat(x)$}} {plot.surface obj6 {intrpl true super {super {transpt $0$ visible true addedz 0.0} color {coloring.group obj17 {colors [{coloring.constant obj18 {color [0.75 0.75 0.75 1.0]}}] opacities [0.23] mixings [0.0]}} expr $S(theta,phi)$} vrtnml true}} {plot.vector obj2 {base $lat(x)$ endstyle 1.0 dir $V(x)$ color {coloring.group obj19 {colors [{coloring.constant obj20 {color [1.0 0.0 0.0 1.0]}}] opacities [1.0] mixings [0.0]}} dim 3.0 thick true basestyle 0.0 subdivs 1.0 len 0.3 basesize 10.0 super {transpt $0$ visible true addedz 0.0} endsize 10.0}} {plot.axes obj21 {super {transpt $0$ visible true addedz 0.0} labels [x y z]}}]} alpha true scale 225.0286081679876 surfacemode 2.0 drawmode 1.0 transf [-0.6566753533157819 0.7541722120658989 -0.001324724584157749 0.0 -0.09168166812990104 -0.0780856249264418 0.9927220693168689 0.0 0.7485799570363302 0.6520175685727168 0.12042067179687313 0.0 0.0 0.0 0.0 1.0]}} group null hotspots [] drawmode 1.0}} graph {id obj10} frame {visible true plots [{id obj2} {id obj3} {id obj4} {id obj5} {id obj6} {id obj7} {id obj8}] title $Foucault Pendulum$ labels [$ Vector: V(x) starting at lat(x)$ $ Vector: E2(x) starting at lat(x)$ $ Curve: lat(theta)$ $ Point: lat(x)$ $ Surface: S(theta,phi)$ $ Surface: P(u,v)$ $ Vector: E1(x) starting at lat(x)$]}}}]

If anyone knows how to properly link to a java applet and pass on data from wikipedia, it would be great if this could get cleaned up.--ShanRen 14:46, 9 March 2007 (UTC)

Nice diagram but the image for the pendulum at one of the poles is missing. David Harty 12:34, 17 March 2007 (UTC)

Yes, that is a problem with the coordinate systems. The easisest way to fix this is to change the "lambda" range from -pi/2+0.001 to pi/2-0.001. That way the pendulum is actually never at the pole, but the difference is too small to tell, but the problem with the coordinate systems is avoided. With a little more work one could hide this adjustment so that the user does not see it right away. --ShanRen 15:13, 17 March 2007 (UTC)

When the view is changed to "up the z-axis" and the vector image is moved to the North Pole (pi/2) then the vector image disappears. The vector image is very helpful for defining the perspective. It might be helpful to add a second red vector to depict the full swing of the pendulum. Sorry I am not familiar with the code for an applet.David Harty 10:26, 19 March 2007 (UTC)

David, did you use the old code or the new one? (I changed it recently.) I simply change the range of lambda to only go to pi/2-0,001. I will find a better way around this problem when I have more time.

Not so sure about the second red vector. In that case it is difficult to distinguish a "half-turn" (180˚) from a "full turn" (360˚). But the second vector could have a slightly different color, if you think that it would be helpful to have two vectors.

In case you have not figured it out yet, you can use your mouse and drag the image to change the viewing angle to any angle you want. --ShanRen 11:43, 19 March 2007 (UTC) Thank you. The new version works well and provides a proper basis for discussion. I found it to be very helpful for visualizing the terminology and clearing up my own thoughts. The important thought for the Foucault pendulum seems to be the alignment of the plumb line with the axis of rotation of the Earth.David Harty 05:07, 20 March 2007 (UTC)

I have been looking into the possibility of generating a (semi3D)-animation with the help of the ray-tracing program POV-ray.
This is a preliminary result. The "triangle" in the animation that is suspended above the sphere represents the plane of swing. A full cycle of the animation represents one sidereal day. Things that I want to add to the animation: some latitude lines, an arrow on the North pole to show the rotation of the Earth, a circle on the surface just below the plane of swing, and I want to make the "Earth" almost entirely translucent. Any suggestions are welcome. --Cleonis | Talk 20:48, 11 March 2007 (UTC)

[edit] The dynamics of the Wheatstone-Foucault device

I have started a sandbox article about the dynamics of the Wheatstone-Foucault device. All that is necessesary to account for the Foucault effect is Newton's laws of motion and the theorem of conservation of angular momentum. Any theorems of higher abstraction than that are superfluous for explaining the Foucault precession.

Right now, the mathematical derivations in the sandbox article are still rather rough around the edges. When I have ironed out the wrinkles in the Wheatstone-Foucault device article I will merge its contents with the Foucault pendulum article.

If you have any comments, you can post them on the Talk page of the Wheatstone-Foucault device article --Cleonis | Talk 21:57, 11 March 2007 (UTC)

[edit] Stereographic animation

I have created a stereographic animation. To obtain the stereo-effect, the animation must be viewed with the eyes looking crosswise: the left eye looking at the image on the right, the right eye looking at the image on the left.

It is a bit uncomfortable that there is no indication that the sphere is turning. The current version of the animation is 26 KB. The animation is so small in Kilobytes because very little changes in the scene. Adding longitude lines to show the rotation of the sphere would result in a much larger file.

The transitions from frame to frame are very jerky. To get smooth motion, I think at least four times as many frames would be necessary.

Generally I prefer to keep the KB's as low as possible. I think there are still a lot of people who only have dial-in connection. --Cleonis | Talk 21:13, 18 March 2007 (UTC)

Looks a lot better than the current version. I don't really care for the stereoscopic version, one should be enough imo. I would prefer if the animation also included a vector showing the direction of the plane of swing, I like to keep things to the essentials. But that might just be me and I don't have a strong opinion on that. Personally, I don't see the need to make earth rotate (the pendulum does not care). I still believe that an applet that allows people to rotate earth and look at it from any angle, and that allows people to change the latititude interactively would be more useful. But I still don't know how to inculde java appletts into wikipedia pages... --ShanRen 21:43, 18 March 2007 (UTC)

Yeah, an applet would be best. But for the time being animated GIF will have to do. I'll make a non-stereoscopic version (The image caption can present a link to the stereograph.) It'll be a few days, maybe next weekend. --Cleonis | Talk 23:33, 18 March 2007 (UTC)

So there is no way to include java applets in wikipedia? Maybe we should just link it then? --ShanRen 23:37, 18 March 2007 (UTC)

I anticipate that in the future the wikipedia programmers will add the functionality to include java applets. Until then GIF-animations are the next best thing. By the way, the java applet doesn't seem to work on my PC. I get the sphere made out of wire mesh, and some vectors, but it doesn't move or anything, it just sits there. Probably I need some more software to run the applet, but I don't know which software. The advantage of GIF-animations is the reliability.

Hmm. Two windows should open. One with the graphics, that you can spin and rotate with your mouse, and another one with buttons that let you adjust the latitude and let you start and stop the animation (or just go through it step by step). --ShanRen 00:39, 21 March 2007 (UTC)

[edit] Version of animation with vector

Animation of pendulum plane of swing precession at 30° latitude.

The animation is smooth now. The number of frames is quite large, 196 frames, but since there is very little change from frame to frame the size in KB's is reasonable: 42 KB --Cleonis | Talk 21:16, 20 March 2007 (UTC)

Very nice. --ShanRen 00:07, 21 March 2007 (UTC)

I agree. Very nice. The applet downloaded from above is also extremely helpful. David Harty 16:34, 23 March 2007 (UTC)

Uploaded a final version of the animation. I increased the translucency, to make the vector more constant. Smoothness is increased by increasing the number of frames, plus changing the frame rate from 10 per second to 16 per second. For comparison: the previous smooth version.
--Cleonis | Talk 21:30, 23 March 2007 (UTC)

[edit] Foucault Pendulum Thought Experiment

I agree with the previous commenter that a more intuitive example could be provided. The example would be provided for showing how to visualize the forces at work and how the effect is observed. In response to this request an attempt is made through the following thought experiment which could be proposed as a separate caption to the article. However, I think the explanation falls short in trying to explain the observed effect of the Foucault pendulum. I would appreciate any comments, especially related to items 14 and 15. I have replaced the steps below with an updated version but am sure that the description still falls short.David Harty 13:32, 24 March 2007 (UTC)

The following thought experiment is provided to visualize the forces at work and how the effect is observed.

1. Construct a flat turntable in a non-gravity environment such as the space station.

2. Construct a Foucault pendulum on the flat turntable such that vertical support tower is located at the center of the turntable on the axis of rotation of the turntable and has a armature that can be extended horizontally from the top of the tower.

3. Construct a pendulum bob hanging from the end of the armature by a wire. The pendulum bob has magnetic properties.

4. Construct a strong magnet located at the center point of the turntable. The bob suspended from the armature is attracted to the center point of the central magnet creating an angle of the plumb line of the pendulum with the vertical support tower. With the addition of the rotating turntable a new plumb line and angle is established for the rotating system (compare geocentric latitude with geographic latitude). This angle is defined as "phi".

5. If it is desired to observe the pendulum hanging directly over the axis of rotation of the turntable, then it is necessary to provide a curved support away from the armature such that the pendulum swing is not interfered by the support structure. However, it is noted that there is an absence of an effect that is observed when the pendulum hangs directly over the axis of rotation.

6. What is important to the construct is the point of attachment of the wire which suspends the pendulum. This point defines the vertical length "T" of the tower above the turntable (and the same as the distance above the center point of the magnet) and the horizontal displacement "A" from the axis of rotation provided by the armature.

7. The attractive force between the magnets represents the gravitational force provided by the Earth.

8. The rotation of the turntable represents the turning of the Earth.

9. The extension of the armature represents the changing of the latitude from the poles to the equator. When A is zero the pendulum is located directly on the axis of rotation such as the location of one of the poles of the Earth (cosine phi equals one). When A is infinite in relation to T the pendulum is located directly perpendicular to the axis of rotation such as the location on the equator of the Earth (cosine phi equals zero). When A is equal to T then phi is equal to 45 degrees such as the location of 45 latitude (cosine phi equals .707). When A is equal to 1.732 times the value of T then phi is equal to 60 degrees such as the location of 30 degrees latitude (cosine phi equals .5). The cosine of phi is equivalent to the sine of the latitude.

10. There are two forces at work on the pendulum. The force of attraction of the magnets (gravity) and the Coriolis Force generated by the rotation of the turntable (the turning of the Earth). The resultant force acting on the pendulum bob at all times is defined by the cross product of these two forces.

10a. The Coriolis Force is a force appearing in the equation of motion in a rotating reference and causes the Coriolis Effect. For further information also see the Visualization of the Coriolis Effect. The presence of the Coriolis Force results in a motion described as inertial circles.

10b. There are two manifestations that could be observed as a result of the rotation of the turntable, the angular velocity and the inertial circle velocity.

11. One more construct is needed. A support stand needs to be constructed on the turntable underneath the armature with a flat, rigid surface located underneath the pendulum bob. The support stand needs to be able to extend outwards on the turntable similar to how the armature of the pendulum extends. The plane of the flat surface needs to be angled such that the surface is perpendicular to the plumb line of the pendulum which is directed towards the center of the central magnet located at the center of the turntable. The plane of the surface represents the surface of the Earth at the location of the pendulum bob.

12. When the pendulum bob is not swinging the two forces of gravity and Coriolis are still acting on the pendulum bob, however, there is no observed effect in relation to the plane of the flat surface (the flat surface is not affected any differently than the center point of the pendulum bob).

13. When the swinging of the pendulum bob is initiated at the polar location then the plane of swing of the pendulum is established over the rigid, flat surface. The plane of swing of the pendulum bob is observed to rotate in relationship to the surface and this is the effect observed by the Foucault pendulum.

14. The rotation is observed because the resultant combination of the two forces acting on the bob changes with the change of position caused by the swing of the pendulum. The plane of the swinging bob is constrained differently than the plane of the rigid, flat surface so that all or part of the rotational velocity can be observed by the change in relationship of one plane to the other. (Would it be incorrect to say that change that is observed is related to the change in centrifugal force that occurs moving inwards and outwards in relation to the center of the turntable?)

15. For a given location of the Foucault pendulum the magnitude and direction of the gravity term doesn't change significantly over the range of arc of the pendulum swing (though gravity does decrease by the square of the distance between the two objects). What does change with location (latitude) is the direction of the attractive force supplied by the magnetics (the gravitational attraction of the Earth) in relation to the Coriolis Force suppled by turntable (the rotation of the Earth). (Would it be correct to say that the change that is observed is related to the increase in the horizontal component of the gravitational force that is acting in opposition to the centrifugal force? Is it correct to say that the two horizontal forces are 90 degrees out of phase but over the course of a full rotation of the turntable (Earth) the forces cancel out?)

16. The magnitude of the change that is observed between the two planes established is determined by the rotation of the Earth divided by the cosine of phi (the sine of the latitude).

16a. If the central magnet were moved outward along the turntable to the point directly underneath the connection of the pendulum then the pendulum bob would hang perpendicular to the turntable and the full rotation of the Earth could be observed again in relation to the pendulum swing.

16b. When the armature of the pendulum is extended the location of the equator is approached. At the equatorial location the angular velocity can't be observed by the motion of the pendulum swing because the forces acting on the pendulum cancel out on opposite sides of the pendulum swing.

16c. The Coriolis Force cancels out on opposite sides of the pendulum swing.

17. If there was no rotation of the turntable than the only force acting on the pendulum would be the constant force of gravity and no change would be observed by the swinging pendulum in relation to the surface of the Earth.

18. If there was no attractive force then the only force acting on the pendulum would be the Coriolis force and the pendulum bob would be observed outward and upward(?) to the armature. It is the action of gravity that is always attracting the pendulum bob to obtain the maximum closeness, i.e., align the connection wire with the plumb line of the pendulum.

What is needed to observe the effect of the Foucault pendulum is an attractive force supplied by gravity, a Coriolis force suppled by the rotation of the Earth, a rigid surface for comparison supplied by the Earth's surface and a plane of swing of the pendulum. David Harty 13:32, 24 March 2007 (UTC)

For the case of a reduced attractive force between the magnets, the case of diminishing gravity, one has to be more precise in defining the construct.

The angle "phi" is more precisely defined as the angle between two lines; one line created between the center point of the central magnet and the center point of the bob and the other line created by the axis of rotation of the turntable. The values of T and A are more precisely defined in diminished gravity using the center point of the bob in relation to the center point of the central magnet. With the above construct one can increase or reduce either of the two forces to observe the effect. The attractive force component can be changed by changing the strength of the magnetic field or changing the distance between the two magnets. The centrifugal force component can be changed by changing the rotation rate of the turntable. Other construction parameters can be changed to observe the effect such as changing the center point of the central magnet, changing the location of the vertical support tower in relation to the axis of rotation, or changing the plane of the support stand. Differences from the Foucault pendulum will be observed.

Please provide comments below related to a numbered step. In particular steps number 14 and 15 where my explanation falls short. I have to give credit to the applet provided above which provided the visualization of the effect that is observed, then using the force diagram to construct a thought experiment to explain the change in the forces which could be observed. I would appreciate any comments and will let the subject digest for awhile while waiting and observing comments. Thanks in advance for all comments. I think an intuitive approach is needed because that is what Foucault was able to envision. David Harty 16:34, 23 March 2007 (UTC)

David, the more I think about it, the more I have to agree that the derivation in the article using the Coriaolis force is probably not very helpful. Anyone with the differential equations and physics background needed to understand the computation can easily do it by themselves. And anyone that does not have the background will not understand it. So maybe it should be removed.

So the question remains what other explanation could be used to substitute. A thought experiment is a good idea. In that case I would advocate the bicycle wheel as a model. It is a little abstract, but probably the easiest example to understand. The next complicated case would be the Wheatstone-Foucault device. I would favour that over your example. Maybe the simples explanation is that it is not the plane of oscillation that rotates, but the local surface of earth that rotates at a rate given by "the magnitude of the projection of the angular velocity of earth onto the normal direction to earth" (quote from article). This can be best understood (and visualized) using the bicycle wheel. Maybe this train of thought deserves more detail.

Some comments about your example:

4. The angle does not model the plumb line but the geocentric latitude. This will get adjusted to the plumb line (geographic latitude) by rotating the disk at the apporpriate speed, so that with the centrifugal force the rest position of the pendulum makes the same angle as the plumb line.

10. The combination of centrifugal force and gravity does not equal the coriolis force! The coriolis force is not directly connected to either one; what it has in common with the centrifugal force is that it arises because the rotating frame is not an inertial frame. Other than that it is not related to the centrifugal force (nor gravity).

14. Answer to you question: Yes, it would be incorrect. See also 10.

15. Partial answer (don't really understand the question) The magnitude of the gravitational force (of any force in the direction of the plumb line) has no effect on the precession of the plane of oscillation.

In general, I don't think an intuitive understanding can be established by looking at forces in the rotating frame. The coriolis force is velocity dependent, and that is very difficult to visualize. --ShanRen 17:57, 23 March 2007 (UTC)

Thank you for your comments. They have been very helpful. All I am trying to do is identify the why, then refer to other articles such as the Coriolis Effect article for more detailed development. It seems to me that there are two manifestations that have to be identified and that manipulating one manifestation provides elucidation to the other manifestation. That is all that I am trying to do and bring to the article. The two manifestations that need to be compared are the angular velocity of the Earth and the velocity of the inertial circles as caused by the Coriolis Force. That is why I chose the construct that I did, rather than have to convert other constructs into the Foucault pendulum reference frame. Identifying then changing the parameters in the construct allows one to compare how the observations change and what causes it. My intuition tells me that the inertial circle mechanism cannot be observed by the Foucault pendulum construction because the swings on opposite sides of the plumb cancel out the effect. If the pendulum were mounted on a fluid that would be a different construct... but a parameter which could be manipulated in the construct and the effect observed. I have replaced the section above and added comments and strikeouts as it is noted that this talk page is already large. Your comments are well received and it has occurred to me that it is not possible for me to provide a concise and accurate description of the interaction of the two manifestations. David Harty 13:24, 24 March 2007 (UTC)

In commenting on my own thought experiment it has occurred to me that I don't have a complete understanding of the forces at work and the resulting effects. Several of the statements above are actually intended to be questions, for example, 16a, 16b, 16c, 17, 18.

It is also noted that the surface velocity vector decreases as one moves inwards towards the axis of rotation. This surface velocity vector decreases from the equator to the pole with the cosine of the latitude since the projection of the latitude onto the equatorial plane is equivalent to the radius from the axis of rotation. The angular velocity remains the same so the ratio of the surface velocity to the angular velocity decreases with cosine of the latitude (or, the ratio of the angular velocity to the surface velocity increases with the sine of the latitude). So maybe 16a is incorrect in that it is the alignment of the angular motion to the axis of rotation that is important rather then with the force of gravity, and the pendulum swing would still be observable in 16a (basic question 1). I don't know the answer to this question posed by the experiment.

Related to this is a comment about the vector diagram provided with the applet. I think the white vectors represent the force of gravity (directed inwards) and the angular velocity along the rotating surface of the Earth, which is constant. Perhaps it may be helpful to depict the surface velocity vector, as well, along with the angular velocity, since the surface velocity changes with latitude. Thus, one will get the sense that the surface velocity is changing with latitude w.r.t the angular velocity, as well as, the direction of the force of gravity is changing with latitude w.r.t the angular velocity. This may provide a better representation of the basic premise.

One last fundamental question regarding cause and effect. Are the basic forces involved with gravity and angular momentum resulting in an effect that results in the observed motion of the pendulum swing, or do the basic forces give rise to an additional force, the Coriolis force, that than results in the observed motion of the pendulum swing, or is the Coriolis force just another effect of the basic forces (basic question 2). The article should be made clear in this regard. Your comments are well received. David Harty 09:10, 28 March 2007 (UTC)

David, some quick answers. The single important quantity for the Foucault pendulum and the related systems discussed here is the angle between the angular momentum of earth (or the disk) and the plumb line (equilibrium of the pendulum). The pendulum is assumed to oscillate in a plane, so it passes through the equilibrium point with every swing. (If not, other effects of comparable size may appear. For example, one can build a pendulum that traces out an ellipse, but the ellipse does not precess at all. This is because the period of a physical pendulum depends on the amplitude, and the mayor and minor axis precess just because of the difference in amplitudes. One can engeneer this to exactly counteract the Foucault effect (or double it if one wishes to do so.) From the perspective of an earthbound frame, the driving mechanism is the Coriolis force. It is difficult to visualize how it acts, because it depends on the velocity of the pendulum bob. From an inertial frame outside of earth the motion of the plane of swing can be visualized as follows. Model the plane of oscillation by its normal vector. Note that it is tangent to earth at all times. To understand its motion, take the vector and move it in 3-space along a path of fixed latitude for some distance. At it's new position it is not tangent to earth any more, but sticks out (or in) a little. Correct this by projecting it back onto the tangent plane. Repeat. In the limit of infinitesimal movements, this is exactly what the Foucault pendulum does. In the demo, the white vectors merely indicate the earth-bound coordinate system given by north and east directions. View this as axes drawn on the floor underneath the pendulum. The red vector gives the direction of swing of the pendulum. For 16a, this is (almost) the same as a Foucault pendulum just a meter or so away from the north pole. During the course of a day the suspension point traces out a circle, but the pendulum comes back to the same (almost) position after one day. --ShanRen 13:43, 28 March 2007 (UTC)

Thank you for your comments and the time it takes to prepare a response. The comments have been very helpful as has been the recent discussions. First off, I must apologize for my inaccurate portrayal of the white arrows on the applet. There really is no excuse for my inattention to that detail.

What I was thinking about was a force and velocity diagram to enhance the representation shown on the applet. The gravity force vector is always directed towards the center of the earth. The centripetal force vector is always directed inward and perpendicular to the axis of rotation in the plane of constant latitude (parallel to the equatorial plane). The centripetal force is reduced with increasing latitude because the radius of the rotating plane (defined by constant latitude) is reduced (decreases with the cosine of the latitude). Although the angular velocity remains the same with changing latitude, the surface velocity vector decreases with latitude. Thus, I thought it would be helpful to define the system of force and velocity vectors in which the pendulum is operating.

In regards to the bicycle analogy, I made another different interpretation in that I let the bicycle wheel represent the constant latitudinal plane (the armature indicated above) and then I constructed the magnets to represent gravity, then attached a pendulum to the rotating wheel. I did this only to visualize the construction. However, I see that angling the bicycle wheel to the disk (the equatorial plane) provides a similar effect by changing the angle between the centripetal force and the gravitational force. However, the plane of the bicycle wheel in this case represents the plane of the coordinate system that is shown in the applet diagram, not the plane of the pendulum swing. Since the Earth can't change like the coordinate system, it is necessary for us on Earth to have a pendulum which has the freedom of motion.

At this point pictures are worth a thousand words which is why I appreciate the representation in the applet diagram.

So I can visualize the gravitational force, the centripetal force, the angular velocity and the surface velocity. This system gives rise to the Coriolis force.

The solid Earth cannot show this change in relationship of the forces so a mechanism is needed which has freedom of motion to show this change as the latitude changes. The oceans have freedom of motion to show this change and gives rise to the inertial circles that are present in the ocean. Adding a Foucault pendulum provides an earth-bound reference for comparing the change in relationship of these forces which shows that the Earth is turning.

So from my simplified viewpoint it appears to me that the motion observed by the plane of swing of the pendulum is similar to how the Coriolis force is observed, in that they arise as a result of the forces and rotational velocity of the system. So maybe it would better to explain this system of forces as the cause, rather than saying that "the precession of the pendulum is explained by the Coriolis force", and also provide a link to the centripetal force article.

David Harty 12:43, 29 March 2007 (UTC)

David, the centripetal force has little to do with the Foucault precession. If you neglect it from the computation, the result changes only very slightly. I recommend you don't consider the centripetal force at all, since it seems to distract you from what is going on. If you would like to visualize this, consider earth (or your disk) rotating very slowly.

If you mount the bicycle wheel the way you said, it does actually represent the motion of the pendulum. Mount a bicycle wheel underneat the pendulum, with axis in the direction of the plumb line. Then start the swing of the pendulum, and mark on the bicycle wheel the initial plane of oscillation. Then over time, the marker on the bicycle wheel will always coincide with the plane of oscillation, the bicycle wheel turns exactly like the pendulum. --ShanRen 14:14, 29 March 2007 (UTC)

I thought I was referring to the bicycle wheel as the constant longitudinal plane rotating like the Earth, not angled like the surface of the Earth is to the axis of rotation. Unfortunately, the surface of the Earth (as shown by the coordinate system on the applet) can't rotate like the angled bicycle wheel. North is always North. So I agree with you, the angled bicycle wheel (axis in the direction of the plumb line) turns like the pendulum. I think that is what you mean and I appreciate your thoughts on the matter. 70.58.144.193 18:03, 29 March 2007 (UTC) Nevermind, I see what you mean. I am just adding extra steps that don't do anything and angling the bicycle wheel the wrong way. Thank you for your patience. 70.58.144.193 22:12, 29 March 2007 (UTC)

The Coriolis force is the cause of the precession of the pendulum swing, just like it is the cause of the inertial circles. It appears to me that it would be appropriate in the first diagram in the Foucault pendulum article to show the precession of the pendulum swing due to the Coriolis force. This would result in the curve being zero at the South and North Poles and -2pi at the equator. It is the pendulum that is precessing with respect to the surface of the Earth as caused by the Coriolis force, not the surface of the Earth that is turning underneath the plane of swing of the pendulum. Of course, the situation arises because the Earth is rotating and has a gravitational field. David Harty 08:14, 30 March 2007 (UTC)

Actually, the centripetal force becomes very important because it is the force that defines the angular velocity which is fundamental to the equations, even though the magnitude may be small. The centrifugal force at the outside of the wheel decreases with the radius from the axis of rotation. I think you missed the point and the force diagram is helpful in defining the motion of the Foucault pendulum.David Harty 11:26, 31 March 2007 (UTC)

If the Coriolis force were to be shown on the force vector diagram a lot could be explained from that one diagram.David Harty 14:58, 31 March 2007 (UTC)

[edit] Combination Diagram with Force Vectors

From Wikipedia the following statement is provided;

"The centripetal force is the external force required to make a body follow a circular path at constant speed. The force is directed inward, toward the center of the circle. Hence it is a force requirement, not a particular kind of force. Any force (gravitational, electromagnetic, etc.) can act as a centripetal force. The term centripetal force comes from the Latin words centrum ("center") and petere ("tend towards").

The centripetal force always acts perpendicular to the direction of motion of the body. In the case of an object that moves along a circular arc with a changing speed, the net force on the body may be decomposed into a perpendicular component that changes the direction of motion (the centripetal force), and a parallel, or tangential component, that changes the speed."

The Coriolis force acts like a centripetal force. I would like to propose a diagram that shows the forces acting on the surface of the Earth and the forces acting on the pendulum. I think this would be a combination of the applet provided above and the animated gif diagram provided in the article. The applet is excellent because it is able to move to a position on the entire globe and the animated gif diagram is excellent because it shows the pendulum and shows the change in the plane of swing of the pendulum. By showing the force vectors on the combined diagram many questions are answered, definitions provided and the Foucault pendulum becomes a basic demonstration of the Coriolis force.David Harty 16:01, 31 March 2007 (UTC)

Let me say it again. The coriolis force does not act like a centripetal force. Forget about the centripetal force, it is not important in this setting. Drawing force vectors for the coriolis force is difficult since the force depends on the velocity of the pendulum bob. It is not a force like gravity, or the centrifugal force in the case of earth for that matter, that are vector fields (as a function of space). One could include the actual movement of the bob in the diagrams, and then show how the coriolis force acts at each particular time. But this is quite cumbersome and not very illuminating. It takes many osciallations of the pendulum to see even the slightest effect. One could of course artificially slow down the period of the pendulum as has been done in the animations below, but there are other problems with this as these animations are generally unphysical. As I explained before, if the motion of the penudulum is not (to very good approximation) planar but elliptical, then other effects dominate the Foucault effect.

Also, the Foucault pendulum does not precess "because" of the coriolis force. The coriolis force is merely a tool to compute the precession rate in an earthbound frame. In an intertial frame, there is no coriolis force. --ShanRen 19:53, 31 March 2007 (UTC)

[edit] Visualizing the forces

Version (with a bob) of the Wheatstone-Foucault device.

In my opinion, the device described by Wheatstone, as shown in the current version of the Foucault pendulum article, is most suited for visualizing the forces. Actually, I prefer an adapted design, aimed specifically at being a tabletop model for the terrestrial mechanics of the Foucault effect. When the bob is displaced from the rest position the springs extend and/or contract, depending on the direction of displacement. Visualizing the extension of the springs gives a visceral impression of the forces that are acting. --Cleonis | Talk 22:13, 23 March 2007 (UTC)

[edit] Possible experiment

In the coriolis effect article it is described that the coriolis effect that is taken into account in meteorogy is the same as the effect that causes the Foucault pendulum to precess.

The following thought experiment is aimed at allowing for the connection with meteorology:
The most level surfaces on the Earth are salt flats that are flooded yearly. If memory serves me, the bonneville salt flats are flooded yearly with a thin layer of water, which then slowly evaporates, leaving a level surface that stretches for kilometers.

A hovercraft that hovers above a surface that is not perfectly level will tend to float down the incline. On the Bonnevill salt flats, a hovering hovercraft has no such tendency: perfect equilibrium.
What happens if a hovercraft driver starts going back and forth?
Let the hovercraft driver take on the role of providing the equivalent of the restoring force. A spot is marked, and the driver is orienting the thrust of the hovercraft in such a way that the restoring force is at all times directed towards that particular spot. The hovering of the hovercraft is analogous to the fact that air mass tends to remain at the same altitude (height above the Earth's surface) when it is set in motion.

It is also interesting to consider whether designers of vehicles for breaking land speed records need to be aware of the coriolis effect.
An object that is co-rotating with the Earth at the latitude of the Bonneville salt flats and Black rock desert is circumnavigating the Earth at about 300 meters per second, which is about the velocity of Thrust SSC. When Thrust SSC was going in west-to-east direction, it was circumnavigating the Earths axis at a velocity of about 600 meters per second. When Thrust SSC ran the course in the opposite, east-to-west direction, it was close to being at a standstill with respect to the Earth's axis. In both directions the car will have had a tendency to veer to the right. --Cleonis | Talk 18:53, 24 March 2007 (UTC)

[edit] Vincenzo Viviani's pendulum

It has been mentioned that Vincenzo Viviani observed a veering of a pendulum. It is unlikely that the crude pendulum of that demonstration actually showed the Foucault precession, for the following reasons.

Observing his pendulums on long runs, Léon Foucault became aware of perturbations from the theoretically expected precession.

In his book, William Tobin describes that Foucault tried to understand the perturbations by returning to the demonstration that led him to the pendulum in the first place. Foucault had clamped a metal rod in the chuck of a lathe, and he noticed that the direction of the (immaterial) plane of vibration remained the same when slowly turning the lathe.

However, when the lathe did not turn at all, the plane of swing would slowly veer, with a cycle time in the order of tens of seconds. William Tobin writes that Foucault reasoned as follows. Suppose that the metal rod is not perfectly uniform. Then the natural frequency of vibration in one direction will be different from the natural frequency in a direction perpendicular to it. Non-uniformity of the rod gives a directional bias in the rod. When a vibration of the rod is started, this induced vibration is unlikely to be perfectly aligned with the directional bias of the rod itself. Gradually, the energy of the vibration will transfer from one direction of swing to another.

Foucault reasoned that if the lathe rotates slowly, say one revolution per second, then any directional bias in the rod is averaged out. In fact Foucault contemplated incorporating this design feature in an actual pendulum, in order to average out bias in the suspension wire, but eventually he did not do so.

It is likely that what Vincenzo Viviani noticed was veering due to non-uniformity of the suspension wire. (or a non-uniformity of the upper attachment point of the wire. --Cleonis | Talk 08:39, 24 March 2007 (UTC)

Cleonis, you seem to have some more information on Viviani's experiment. I could not find anything, could you share your sources? --ShanRen 15:12, 24 March 2007 (UTC)

My comment was a general one. In Galileo/Viviani's time, no special effort was made to exclude effects other than the Foucault effect. On that ground it is reasonable to assume that in Viviani's time no Foucault effect was (accidentally) observed. You have pointed out that attempts to trace the origin of the Viviani story have been unsuccesful. So it's possible that the Viviani story is just a mystification. --Cleonis | Talk 18:22, 24 March 2007 (UTC)

As you have no additional information I have to conclude that your reasoning is bogus. You claim that Viviani's pendulum was "crude" without having any information to back this up. Contrary to what you claim, Galileo is known to have been a very capable experimentor. The reasons that you cite as a possibility for the observation, namely problems with the wire or the suspension point, are extremely unlikely to cause a veering if the experiment was set up somewhat carefully (which is quite safe to assume, looking at descriptions of experiments at the time). The problems that other people has with the wire and the suspension point are during long-term observations, they certainly don't come up in short-term observations. The fact that Viviani was not looking for the Foucault effect does not mean that he could not have done a careful experiment. From what we know so far, he paid attention to the most important factor in the experimental setup, namely that the pendulum was set into planar motion.

Of course, verifyability of the source is necessary for this to be included in the article. Once we have all the relevant information we may get a better idea what Viviani observed, and how relevant it is. And most importantly, we may be able to give an informed opinion. --ShanRen 19:21, 24 March 2007 (UTC)

[edit] Bicycle wheel mounted on a disk

The following passage from the section 'related physical systems' needs clarification:

a non-spinning perfectly balanced bicycle wheel mounted on a disk so that its axis of rotation makes an angle φ with the disk. When the disk undergoes a full clockwise revolution, the bicycle wheel will not return to its original orientation, but will have undergone a net rotation of

A rewording is necessary because the current formulation contains a selfcontradiction. When the bicycle wheel is non-spinning, then its angular velocity with respect to the disk is − Ω; the reverse of the angular velocity of the disk. After the bicycle wheel has been tilted, its angular velocity (with respect to the disk) is no longer the reverse of the disk's angular velocity.

I propose the following:

A perfectly balanced bicycle wheel is mounted on a disk in such a way that the direction of the bicycle wheel's axis is adjustable. In the initial state the axis of the bicycle wheel is parallel to the central axis (the axis of the disk). The disk is rotating with angular velocity Ω, and the bicycle wheel is initially non-spinning. Then, without touching the wheel, the wheel's axis is tilted so that it has an angle with respect to the disk. After that the bicycle wheel will have an angular velocity with respect to the disk: after each full revolution of the disk the bicycle wheel will not return to its original orientation, but will have undergone a net rotation of

The 'bicycle wheel mounted on a disk' is quite a neat example of the sine law in rotating systems with angled motion. I think I will make an animation that shows the above described process. --Cleonis | Talk 21:33, 26 March 2007 (UTC)

Let me clarify the system, it does not quite work the way you are describing it. The system you are descibing will not exhibit that exact phase shift. Here is a clearer version of what I meant to say:

a non-spinning perfectly balanced bicycle wheel mounted on a disk, which initially is also fixed, so that its axis of rotation makes an angle φ with the disk. When the disk is then rotated so that it undergoes a full clockwise revolution and is returned to its initial orientation, the bicycle wheel will not return to its original orientation, but will have undergone a net rotation of

--ShanRen 00:24, 27 March 2007 (UTC)

The detail of the disk not rotating initially introduces an element that is not present in the Foucault setup. Interestingly, it is not necessary for the disk to be non-rotating initially.

Let the disk be rotating with a constant angular velocity troughout. The angle of the wheel with respect to the disk, φ, is kept the same throughout. Initially, the wheel is driven (a torque is exerted) in such a way that the wheel counterrotates with respect to the disk, and after each full rotation of the disk the wheel is back in the same orientation. Then the wheel is released to rotate freely around its own axis. After the wheel has been released, it will undergo a net rotation of for each full circle of the disk.

The analogous procedure for a pendulum is as follows: a pendulum can be constructed in such a way that the swing is forced to go through a full circle (with respect to the Earth) in one sidereal day. That constraint of the pendulum requires a torque. When the pendulum is released to free swing, the plane of swing will undergo a net rotation of for each full circle of the disk.

In the two analogous cases the direction of the torque prior to release is the same. --Cleonis | Talk 15:19, 27 March 2007 (UTC)

Cleonis, having the bicygle wheel that is spinning initially just complicates the situation. It makes it difficult to read off the phase shift and it suggests the need for a relation between the initial frequency of the wheel and the frequency of the roation of the disk, which is totally artificial.

The setup you describe does not have the outcome you describe. If the wheel is initially "corotating" with the disk (i.e. returns to its original position after each turn of the wheel), then it will continue to do so -- if no external forces are applied.

As a side note, in your arguments you need to explain why the gyroscopic force (when you change the direction of the angular momentum of the wheel, or the conical pendulum) has no effect. This especially applies to your "analogy" using a conical pendulum (I assume that's what you mean). Anyway, there is something wrong with the pendulum you describe. You get into trouble if you have a period of 1 day. Just imagine it at the north pole!

Lastly, the initially non-rotating bicycle wheel relates to the Foucault pendulum in the following way. It simply defines an inertial frame along the path of fixed latitude, so it shows at what rate the ground rotates underneath the pendulum. --ShanRen 16:27, 27 March 2007 (UTC)

Shanren, do note that what I wrote is as follows:
initially, the wheel is driven (a torque is exerted) in such a way that the wheel counterrotates with respect to the disk, and after each full rotation of the disk the wheel is back in the same orientation.

You, on the other hand, are describing a setup in which the wheel is initially co-rotating with the disk, which obviously is an uninteresting initial condition.

When the wheel is at an angle with respect to the disk and the wheel counterrotates with respect to the disk, then the wheel is not motionless with respect to the inertial frame. Only if the wheel's counterrotation is an exact match for the disk's rotation, does the wheel return to the same orientation after every full turn of the disk. --Cleonis | Talk 21:33, 27 March 2007 (UTC)

Cleonis, ok you said "counterrotating". But what's your point? Either way, if your wheel is rotating (co or counter) in a way so that it initially comes back to its original orientation after each turn of the disk, it will continue to do so if there are no external forces. In short, the sytem that you describe does not exhibit the behavior you assert. So please focus on the issue instead of irrelevant co or counter discussions.

Also, what's with the "wheel is not motionless with respect to the inertial frame" business? The axis of the wheel is changin it's direction, so of course it is not motionless -- regardless of how it rotates. --ShanRen 01:14, 28 March 2007 (UTC)

[edit] How to define an inertial frame along a path of fixed latitude

I copy and paste from above:

Lastly, the initially non-rotating bicycle wheel relates to the Foucault pendulum in the following way. It simply defines an inertial frame along the path of fixed latitude, so it shows at what rate the ground rotates underneath the pendulum. --ShanRen 16:27, 27 March 2007 (UTC)

Shanren, you need to clarify what you mean by 'inertial frame' in this context

Recapitulating:
Inertial motion over the surface of a sphere follows a great circle. This is of course elementary spherical geometry. In euclidean 2-space inertial motion follows a straight line, and in spherical geometry this translates to motion along a great circle. It is appropriate to refer to motion along a great circle as 'inertial motion' for although there is a force perpendicular to the surface, confining the motion to that surface, there is no force in a direction tangent to the surface.

If it is assumed that the bob of the Foucault pendulum is at all times in inertial motion, then it follows that at all times the bob must be moving along a great circle.

We agree that the equation of motion for the Foucault pendulum is as follows:

In the image on the right, Great circle and coriolis motion, the red circle represents a great circle that is tangent to the latitude line of 30 degrees northern latitude, and the blue line represents the motion of the pendulum bob during the swing in which it swings in east-to-west direction.

The red circle represents a great circle that it tangent to the latitude line of 30 degrees north. The blue line represents the direction of motion of the pendulum bob when it swings from east-to-west, starting tangent to the latitude line.

representation of the precession of the direction of swing.

Motion along a great circle that is tangent to a latitude line has the following property: no matter the direction (west-to-east or east-to-west): the motion always proceeds towards the equator. It cannot move north of the tangent latitude line
On the other hand: motion of the pendulum bob as described by the coriolis term has the following property: motion in east-to-west direction that starts tangent to a latitude line proceeds towards the nearest pole.

It follows that the pendulum bob is not in inertial motion.
Hence a frame of reference that is co-rotating with the plane of swing of a pendulum is not an inertial frame of reference.

Shanren, that is why you must clarify what you mean by 'It simply defines an inertial frame along the path of fixed latitude'. --Cleonis | Talk 22:23, 28 March 2007 (UTC)

To answer your question. We are interested in observing a direction that is tangent to the sphere along a path of fixed latitude, namely the orientation of the plane of oscillation. To quantify what happens, we can define a moving frame along the path.

Let's look first at a simple example of a pendulum in the plane. In the plane the pendulum can act like a compass. If you take a planar pendulum on a car, and slowly drive along some path in the plane, the plane of oscillation does not change. (Lots of "plane"s here, hoepfully not too confsing.) The pendulum always swings in the same plane. Now suppose you drive along a circle and wish to understand the motion of the direction of the plane of oscillation. To record the motion of the pendulum, you can define a moving frame along then path. There are many choices to do so, and some are better or more natural than others. I want to draw your attention to two such choices. Firstly, in the plane we have an absolute notion of direction, say along the x and y axis (just make a choice). If we record the motion of the pendulum in these axis, the pendulum does not turn. The reason is that there are no forces that can make the direction of the pendulum turn, and the frame is an intertial frame. But we could also choose a different frame, namely one attached to the car, that is the tangent to the circle and the perpendicular direction. If we record the motion of the plane of oscillation in that frame, the pendulum rotates. Of course, there still aren't any external forces that make the pendulum turn, but the frame we chose was not an intertial frame.

Observe that while the path along which we took the pendulum was not a straight line, we could still define an inertial frame along the path. If we take a straigh-line path, then the two definitions would coinside.

Now for the Foucault pendulum, that is a pendulum on earth that moves on a circle of fixed latitude, there are also many choices to write down a moving frame along the path. One such choice is given by an earthbound frame, given by "east" and "north" (or by the tangent to the path and it's perpendicular). But there are many other choices. Some frames are more useful than others for doing physics. The earthbound frame has the problem that it is not an inertial frame along the path in the following way. (This is of course related to the fact that the circle of fixed latitude is not a straight line, except the equator.) If we observe a direction that does not feel any phsyical forces that could compel it to turn -- like the bicycle wheel -- then this direction will turn in the earthbound frame. However, just like in the plane, we can define an inertial frame along the path, that is a frame in which directions stay fixed, if there are no external forces that make it turn. The bicycle wheel defines such a frame. Of course we can give an intrinsic definition of such a frame. In physics this is called a parallel frame, and it is defined by the condition that the derivative of the frame in the direction of the path vanishes. (See parallel transport, although that wikipedia article is not very good.)

Hope that helps. --ShanRen 02:42, 29 March 2007 (UTC)

It appears that the definition of inertial frame that you are applying is different from the standard one. Standard in newtonian dynamics is the following: inertial motion is motion along a straight line.

Recapitulating:
In terms of newtonian dynamics, the motion of the planets around the Sun is considered to be non-inertial motion. In terms of newtonian dynamics the force exerted by the Sun is inferred from the fact that the planets do not move in straight lines. More generally, in newtonian dynamics, the equivalence class of inertial frames of reference can be singled out operationally in the following manner: observe a large number of test masses, and if they all are in inertial motion, then their velocity relative to each other can be represented as a linear relation. By contrast:

• If two of the testmasses exert a force on each other (attracting or repelling each other) then both of these test masses have motion that is not linear with respect to the other, inertially moving test masses.
• If the motion of the entire set of test masses is mapped in an accelerating frame, for example a rotating frame, then their relative motion appears to be non-linear. Choosing an inertial frame to map the motion in removes the appeareance of non-linear motion.

The very definition of the concept of inertial frame (in terms of newtonian dynamics) rules out the possibility of defining an "inertial frame" that follows a path of fixed latitude.

Great circle and coriolis motion. If the pendulum bob would be in inertial motion then it would follow a great circle all the time. But when swinging from east-to-west, the pendulum bob is deviated from the path of inertial motion by a force.

When the pendulum bob swings from east-to-west, it is circumnavigating the Earth's axis slower than the bob of a plumb line, so the east-to-west moving bob is subject to a surplus of centripetal force. Hence the east-to-west moving bob is pulled closer to the Earth's axis. The larger the east-to-west velocity (relative to the Earth) of the pendulum bob , the larger the surplus of centripetal force.

Let v_t,e be the total velocity, in tangential direction, of a plumb line, and v_t the total velocity of the east-to-west (relative-to-the-Earth) moving bob. Then the surplus of centripetal force (that causes acceleration in radial direction) is given by:

This can also be expressed as an equation that shows how the radial acceleration depends on the tangential velocity relative to the rotating system. Let v_t,r be tangential velocity, relative to the rotating system. We have: v_t = v_t,e + v_t,r

When the velocity relative to the rotating system is small the above expression simplifies to:

a_r = − 2ωv_t,r

This provides a straightforward explanation of why the coriolis term is proportional to the velocity relative to the Earth. --Cleonis | Talk 23:27, 31 March 2007 (UTC)

Cleonis, I understand that the concept of inertial frames is difficult and can be confusing. An inerital frame is simply a frame in which newtons second law holds. In euclidean 3-space, it is fairly straightforward to understand what is meant by that, since there is a global inertial frame. In our case we have a system that is constraint to tangent directions on the sphere, and there things are admittedly a little harder to understand. There is no global frame for tangent directions on the sphere, let alone a global inertial frame. There is not even a local inertial frame. But there is an inertial frame along any path (not just along geodesics), that is a frame in which Newton's second law holds for the constrained system. That is there is a frame along each path, so that a physical system that defines a direction in that frame will not change its direction in that frame as the system is moved along the path, as long as no external forces are present. If this is too abstract for you, look at a physics book and specifically look at the part where constrained motion is discussed.

While I am at it, you seem to confuse the notion of inertial frame and inertial motion. Paths of fixed latitude don't arise from inertial motion, but we can still study (inertial) motion of a tangent direction along that path. --ShanRen 00:34, 1 April 2007 (UTC)

More generally, in newtonian dynamics, an inertial frame is a frame in which all of newton's laws hold good, the three laws of motion and the law of gravity.

Indeed it is interesting to figure out how that translates to spherical geometry, for in the case of motion constrained to the surface of a perfect sphere, there is no global inertial frame.

In newtonian dynamics, gravitation is classified as an external force. You state the following condition: 'as long as no external forces are present'. The Foucault pendulum setup doesn't meet that condition, for in the case of a Foucault pendulum, there is an angle between the plumb line and the direction of newtonian gravity. For example, at 45 degrees latitude, that angle is 0.10 of a degree. The fastest way to obtain that angle is as follows:

Being at 45 degrees latitude corresponds to being at 4510.0 kilometers away from the Earth's axis, which, given the Earth's angular velocity, corresponds to having a tangential velocity of 328.9 meters per second. The amount of centripetal acceleration to sustain that circumnavigating motion is 0.024 m/s². Resolved into the direction tangent to the local surface that is 0.017 m/s². For the perpendicular to the surface component I take the average of polar and equatorial effective gravity: 9.81 m/s². The angle betweeen the plumb line and the direction of newtonian gravity is obtained from the ratio of 0.017 m/s² to 9.81 m/s². That yields 0.10 of a degree.

The angle between the plumb line and the direction of newtonian gravity is determined by the rotation rate of the rotating system, in this case the Earth, and by the the distance to the central axis of rotation. --Cleonis | Talk 07:11, 1 April 2007 (UTC)

Addendum: in the case of the Wheatstone-Foucault device, the motion is not rigidly constrained to motion along a spherical surface. The constraining force first needs to build up in the form of extension and/or contraction of (various parts of) the helical spring; which can be extension in any direction. A more general approach handles the Wheatstone-Foucault device just as well as the case of perfectly rigid constraint to motion along a spherical surface. --Cleonis | Talk 07:34, 1 April 2007 (UTC)

In the case of the Focuault pendulum, the condition "tangent to earth" means perpendicular to the plumb line. The question whether or not the direction given by the Foucault pendulum feels any forces is key of course, although that has nothing to do with the existence of an inertial frame. The point you raise about the angle between gravity and the plumb line is really a question about the definition of latitude but I guess I am repeating myself for the 10th time now. If you want to understand how this relates to spherical geometry, and how the Wheatstone device comes in, you need to understand the Gauss map. Geographic latitude has the Gauss map build in already.

And while I am at it, an inertial frame is really defined to be a frame where Newton's second law holds, it does not make much sense to sharpen the definition further. You should consult some physics books on this. --ShanRen 14:42, 1 April 2007 (UTC)

[edit] Traces of the motion

Thank you to Cleonis for his image which has been used in a discussion of the French page. In fact, the very nice image (B)

(B) Image of a Foucault pendulum launched from its center with an initial speed (South hemisphere)

from a German Wikipedian de:User:DemonDeLuxe (who died recently, I read)) was unrealistics in the sense that such a pendulum motion can't be easily obtained (see below). Foucault himself in order to launch the pendulum with a zero speed, bended the pendulum (on the East side for example), waited until the cable had no more oscillations and then burned a cord which retained the pendulum.

Thus, I started with the equations (A) above and tried to give an understandable solutions (see equation (2) of the French page). The nice German image corresponds to the initial conditions z₀ = 0 and which are hard to get!

(C) Standard view from the Panthéon (48° North, rotation in 110 seconds)

I tried to give two views of the pendulum: a standard view from the Panthéon and a second from the rotation plane (unfortunately the thumb of the second does not come). Would the addition of the speed in red, the projection on Earth (green) and of the trace in blue helps understanding the motion and the ellipse? That's my hope. Animations are done using Gnuplot, a freely available software on any operating system. The source code of the drawings are licensed under GPL and can be improved by anyone; the images are under GFDL and CC as usual. Images and sources are on the same address of Commons [4] .--Nbrouard 14:29, 30 March 2007 (UTC)

Hi Nbrouard.
The image with eight consecutive swings is a still image that represents the sucessive halfswings that are representes in the animation motion of the bob.

As you point out, the animation by DemonDeLuxe represents a pattern of motion that is difficult to obtain. Let an astronomical observatory be build smack on the South pole. Like all telescopes, the suspension of the mirror has motors that make the mirror remain stationary with respect to the stars. A polar pendulum that is attached to the star-stationary frame will on release follow a purely planar path.

On the other hand, when a polar pendulum is released from a point that is co-moving with the Earth, then the motion of the pendulum bob will be slighly ellipse-shaped. This deviation is imperceptible in the case of a real Foucault pendulum, but in schematic representations, with a ratio of swing frequency to rotation frequency in the order of 10 to 1, it makes a lot of difference. The animation by DemonDeLuxe assumed release from a star-stationary point, my animation assumes release from a point that is co-rotating with the rotating system.

If Foucault would have tried for the utmost, he would have made a rig to compensate for the rotation of the release point with respect to the suspension point. --Cleonis | Talk 12:15, 31 March 2007 (UTC)

I really like the animation (C), the one with the green trace and rotating blue ellipse. I have never seen one that so clearly demonstrates the motion in inertial and rotating frames simultaneously. I would definitely be in favor of including it in the article (perhaps with the minor change of English labels for the cardinal directions). Rracecarr 16:34, 2 April 2007 (UTC)

[edit] The concept of parallel transport

Shanren, you seem to be arguing the following:

You are discussing a setup with two constraining factors that play a separate, consecutive role: the overall constraining factor is that the motion is confined to motion along the surface of a perfect sphere. A subsequent (optional) constraint is that the motion is confined to motion along fixed latitude.

The path along fixed latitude is a non-inertial path. When a local physical system (such as a Foucault pendulum) is in motion along a fixed latitude, then the overall acceleration vector can be decomposed in a linear component and a rotational component. Being constrained to fixed latitude, the Foucault pendulum is barred from yielding to the linear component of the acceleration vector, but it is free to follow the rotational component.

This, I surmise, is what you are driving at by using the highly abstract expression 'parallel transport'.

Another way of putting it: if motion is confined to motion parallel to the surface of a perfect sphere, then the remaining two degrees of spatial freedom can meaningfully taken to represent inertial motion. So after constraining the motion even further (to the point that only rotation is left free), then that remaining freedom can still be taken to represent inertial motion of that very last degree of freedom.

If that is what you mean then I agree that that is a self-consistent way of representing the setup. Not suitable for novices however, it is necessary to understand the setup first in terms of basic principles before embarking on such abstractions. --Cleonis | Talk 08:49, 1 April 2007 (UTC)

No, that't not what I meant and what you say (in regard to your understanding of parallel transport) makes no sense. I agree that parallel transport is in general an abstract concept, that's why the article merely mentions it so that someone familiar with parallel transport can grasp immediately what goes on. For everyone else, the article tries to provide an explanation without direct reference to that term. In the end of course all explanations boil down to parallel transport. For example, the coriolis force shows up as the Christoffel symbols in the frame given by latitude and longitude. --ShanRen 14:56, 1 April 2007 (UTC)

[edit] The angle between newtonian gravity and the plumb line.

I copy and paste from above:

The point you raise about the angle between gravity and the plumb line is really a question about the definition of latitude. --ShanRen 14:42, 1 April 2007 (UTC)

Shanren, the question of latitude convention is where you are wrongfooting yourself really badly.
The angle between newtonian gravity and the plumb line is a function of rotation rate of the rotating system and distance to the central axis of rotation. Latitude comes in only indirectly; the total required centripetal force is resolved in the direction tangent to the local surface, and the closer to the Equator the smaller the component tangent to the local surface.

So let's do some calculations: I will compare two possible conventions of numbering latitude.

• Geometric latitude, the angle between the plane of the equator and the line from the Earth's geometrical center to a point on the Earth's surface.
• Geographically measured latitude, that I will count in kilometers. 5000 kilometer northern latitude is 5000 kilometers from the equator. (the distance from the Equator to the poles is 10000 kilometers)

(Another possibility is geodetic latitude, 45 degrees geodetic latitude is where the actual Earth's surface is at an angle of 45 degrees to the plane of the equator.)
The polar radius of the Earth is 6356.752 kilometers, the equatorial radius is 6378.137 kilometers. The cross section of the Earth can be approximated with the following parametric representation. Let the x-direction be the equatorial diameter and let the y-direction be the distance from pole to pole.

I used a computer program to calculate line integrals.
The geographical latitude of 5000 corresponds to the value 0.25022*π of the parameter t. Geometric latitude of 45 degrees corresponds to the value 0.25053*π of the parameter t

The difference between geometrical latitude and geographical latitude amounts to about 6 kilometers along the surface of the Earth, which corresponds to an angle of about 1/20th of a degree.

At 5000 kilometers latitude geographically, the distance to the center of the Earth is 6378.137*cos(0.25022*π) = 4506.9 kilometers. At 45 degrees geometrical latitude, the distance to the center of the Earth is 6378.137*cos(0.25053*π) = 4502.5 kilometers. The difference between those two is less than 1/1000th. Obviously, what latitude convention is used is insignificant compared to other factors. --Cleonis | Talk 16:15, 1 April 2007 (UTC)

Cleonis, please read before spamming this page with your irrelevant rambling. I am getting really tired of this. Nobody claimed that the choice of which latitude is being used makes a big difference, actually quite the opposite is true.

The defnition of latitude simply corresponds to the definition of "horizontal". And what counts for the Foucault pendulum is not the horzontal component of the centripetal force as you seem to suggest, but the horizontal component of the sum of gravity and the centrifugal force. And the horizontal component of this sum is zero in the case of the geocentric latitude. If one neglects the centripetal force, then "horizontal" means perpendicular to gravity, which pretty much corresponds to geocentric latitude. And yes, the difference for the Foucault pendulum is small. You claimed before that it is not, and this might be the source of your confusion here. So think about this. And by think I mean "think" and not ramble on about your ideas on this page. I don't think anyone is actually interested. If you actually have a calculation supporting your arguments (and by calculation I don't mean some pictures that come with some esoteric rambling), then please feel free to share these and we can actually have a discussion with substance. --ShanRen 17:00, 1 April 2007 (UTC)

Both the centrifugal term and the coriolis term are merely tools to compute the motion in an earthbound frame. It is meaningless to say that 'what counts for the Foucault pendulum is [...] the sum of gravity and the centrifugal force'. Centrifugal force is just a computational tool. --Cleonis | Talk 18:18, 1 April 2007 (UTC)

In choosing a calculation strategy for dealing with the angle between newtonian gravity and the plumb line, two approaches present themselves. To use a perfectly spherical planet, with the same size and the same angular velocity as the Earth, or to incorporate the oblateness of the Earth.

In either approach, the same angle between newtonian gravity and the plumb line is obtained: at 45 degrees latitude an angle of 0.1 of a degree.

• If a perfectly spherical model is used, then the model predicts that newtonian gravity is everywhere perpendicular to the surface, and a plumb line that is co-moving with the rotating sphere has at all latitudes (except at the poles and the equator) an angle with respect to the local surface.
• If the Earth's oblateness is incorporated, then the model predicts that there is at all latitudes an angle between newtonian gravity and the local surface (except at the poles and the equator), and a plumb line that is co-moving with the Earth is everywhere perpendicular to the local surface.

Either way, given the presence of rotation, there is in both models the same angle between newtonian gravity and the plumb line.

Only a model that incorporates the centripetal force will obtain the observed Foucault precession. Models that do not incorporate the centripetal force and yet obtain the Foucault precession are fudged. See http://en.wikipedia.org/wiki/User:Cleonis/Sandbox/Wheatstone-Foucault_device --Cleonis | Talk 18:18, 1 April 2007 (UTC)

Cleonis, the models you are talking about are not the models that I was talking about. I can't be bothered to do the calculation to check your assertion that in both of your models the angle is the same (and I doubt that you have done it), since this is irrelevant to the discussion of the pendulum. More importantly, you continue to push your assertion that the centrifugal force is central in understanding the pendulum, without providing adequate explanation. As I said above, a couple of pictures and some random assertions and formulas are not an argument. The computations in your sandbox article are not even finshed! While I appreciate your effort and some of your animations, I am tired of your bullshit. If you can't even write down the equations and solve them, how can you keep agressively making statements about what the equations are saying and how different terms in the equation affect the outcome? It's probably best if I withdraw from this discussion for a while, I just don't see how I can continue to respond to your posts in any meaningful way that moves things forward. --ShanRen 19:17, 1 April 2007 (UTC)

I have consistently been opposed to suggesting that there is a role for 'the centrifugal force'. All along,the problem has been that you have been atributing opinions to me that are miles away from what I actually write. Your last reaction is typical, I'm emphatically against attributing a role to "centrifugal force" and weirdly, you accuse me of the opposite; pushing for a role of "centrifugal force".

Both the centrifugal term and the coriolis term are merely tools to compute the motion in an earthbound frame. That has been my point all along.

I do need to correct myself on a particular matter. The significant feature is that there is an angle between newtonian gravity and the plumb line. Many people adopt the following erroneous reasoning: the plumb line that is co-moving with the rotating Earth is perpendicular to the local surface, ergo: the plumb line that is co-moving with the rotating Earth is not subject to any tangent-to-the surface force. To address that particular error, I emphasized the significance of the oblateness. Unfortunately, that emphasis was prone to being misunderstood. In the case of the Foucault pendulum, I now think it is better to emphasize the angle between newtonian gravity and the plumb line. --Cleonis | Talk 19:49, 1 April 2007 (UTC)