ShanRen

The foucault pendulum

Hi ShanRen,

as you mentioned, the Foucault pendulum Talk page was getting cluttered with discussions between you and me with lots of repetition, so I'm moving to your Talk page.

Your proposal of trying to understand the motion pattern of the Foucault pendulum in terms of parallel transport.

Recapitulating:
In relativistic theory of motion and gravitation, two fundamental unifications are postulated, there is the unified concept of spacetime (which in the context of special relativity is called Minkowski spacetime), and the unification of the description of inertia and gravitation, a unification that is expressed in the form of the principle of equivalence.
In the 1920's the french mathematician Elie Cartan set out to find out the following: is it possible to formulate a theory of motion and gravitation that doesn't unify space and time into space-time, but that does unify the description of inertia and gravitation into a concept of motion in curved space. Cartan proceeded to show that indeed that is possible. In this exploratory theory by Cartan, gravitation is absorbed into a metric of three-dimensional space, with intrinsic curvature of this three-dimensional space.
This exploratory theory by Cartan is very interesting, since it deepens understanding of relativistic physics: the fact that the ramifications of this theory by Cartan can be worked out without any inconsistency arising shows that the concept of space-time and the principle of equivalence are independent physics postulates.
(Addendum: The wikipedia article about Elie Cartan mentions Einstein-Cartan theory which is another theory. Einstein-Cartan theory does incorporate the unification of space and time into spacetime.)

Your proposal for trying to understand the motion pattern of the Foucault pendulum exploits this independence, your proposal can be recognized as a variation on Cartan's theory. What you do is that you describe the motion of the Foucault pendulum in terms of parallel transport of a vector in a two-dimensional space with extrinsic curvature. Like Cartan's theory, you do not adopt the concept of space-time, but you do incorporate the principle of equivalence.

The differences with Cartan's theory are as follows: Cartan's theory involves curvature of three-dimensional space, and this curvature is intrinsic (no need to deal with some four-dimensional geometrical space that the three-dimensional space is embedded in)
Your proposal involves extrinsic curvature, the two-dimensional plane (the earth's surface) is embedded in a three-dimensional space.

Your proposal of a variation on Cartan's theory is interesting, and worth an article in some specialized journal, but for the purpose of teaching newtonian physics, it is not suitable. Postulating curvature of space is a non-newtonian concept.

In terms of newtonian theory, the precession of the pendulum (In the case of a non-polar pendulum) is accounted for in terms the poleward force oscillating between doing work and doing negative work, (the poleward force is doing work when the pendulum swings towards the pole, and it is doing negative work when the pendulum swings away from the pole. --Cleonis | Talk 08:08, 3 March 2007 (UTC)

• Explaining the Foucault pendulum through parallel transport is not my theory, but became popular in the mid-80s and is pretty much standard knowledge in physics by now. I learned about this from comments by Frank Wilczek in the book cited in the article.

• You do a lot of talking about how the poleward force is responsible for the pendulum. How about actually writing down a derivation for the motion of the pendulum? Or pointing me to a place where I can find one? All I have seen so far is a bunch of claims and some diagrams explaining the poleward force. Nowhere have I seen the corresponding differential equations been written down.

• The poleward force and the shape of earth have an influence on the direction of the plum line. This influence is not very big, and is commonly absorbed in the choice of which definition of latitude one works with. I claim (and my derivations show) that the direction of the plum line is really all that matters to the Foucault pendulum.

• You claimed that Wheatstone's device works like a Foucault pendulum. We agree on this. Do you believe that an analogue of the "poleward force" is also responsible for the precession in Wheatstone's device?
  --ShanRen 13:53, 3 March 2007 (UTC)

(About the way I respond: I copy your remarks so that the thread will be readable both on my own Talk page and yours.)

In the future, I will refer to the 'parallel transport approach' to understanding the Foucault pendulum. While the parallel transport approach is self-consistent, its assumption of curvature of space is not consistent with the postulates of newtonian physics. If the objective is to present a newtonian explanation, the parallel transport approach is out of bounds.

I will produce diagrams and animations, which I think are more illuminating than differential equations. See the animations in the mass flow meter article, no amount of equations can match the insight that is provided by that kind of animations. If I am able to produce animations with correct trajectories, then that is proof that I understand the motion. My purpose is to provide physics understanding to my readers, and I believe this understanding is best served with diagrams and animations. I will formulate equations, but first I want to represent the correct physics with animations.

1) There is no such thing as absorbing any physical feature into definition of latitude. I don't know why you persist in that error.
2) Of course, for a high-performance calculation, one must choose at the very start to either represent everything in terms of geometric latitude or in terms of geographic latitude. Shifting from one system to another within a calculation would introduce a systematic error. One can use either system, as long as one is consistent about it.
I don't have enough clues to reconstruct what you are thinking here. I suppose I should make a diagram.

I think we agree on the following:
The Earth is an oblate spheroid, with the oblateness matching the earth's angular velocity. With 'matching the angular velocity' I mean that it is clear that the earth's oblateness is determined by a particular equilibrium tendency. The oblateness gives rise to the poleward force, which provides the required centripetal force for any object that is resting frictionless on the earth's surface. Since the earth's shape is in a state of equilibrium, we have a lightning fast shortcut to calculating the magnitude of the poleward force. We simply take the centrifugal term, and take the negative of that.

My guess at differences between us:
I claim that the operative factor is the difference in direction between newtonian gravity and the plumb line. For any region of the earth the magnitude of this difference in direction can be calculated.
I claim that in and of itself the direction of the plumb line with respect to the geometric center of the earth is not the operative factor, because in the case of an oblate spheroid the direction of newtonian gravity is not towards the geometric center.

In the case of the Wheatstone-Foucault device, there is indeed an analogue of the poleward force. I am currently working on images in which that analogue is represented, the Wheatstone-foucault device suits that purpose very well. I intend to manufacture an animation for the case where the rotation rate and the vibration rate are relatively close in frequency, for example 6 vibration cycles to each rotation cycle. I think the differential equations that represent that case faithfully will be very interesting. --Cleonis | Talk 17:09, 3 March 2007 (UTC)

To simplify the analysis I use a concept version of the Wheatstone-Foucault device, with a spherical weight halfway the length of the helical spring representing the Foucault pendulum bob. I will use the common simplification of taking the weight of the spring as negligable. The image is in this temporary folder, the image is named 'wheatstone-foucault_bob.png' --Cleonis | Talk 17:53, 3 March 2007 (UTC)

You say that the "assumption of curvature of space is not consistent with the postulates of newtonian physics". You seem to be mixing up several concepts here. There is a difference between space and space, you seem to be mixing these up. Newtonian physics does not contradict that configuration space has curvature, as the example of a spherical pendulum shows (unrelated to foucault pendulum). You can find other implications of curvature in newtonian mechanics when considering nonholonomic constraints, the foucault pendulum being only one example of many.

You say that we can calculate the poleward force by "We simply take the centrifugal term, and take the negative of that." I thought we talked about this before, and agreed that this does not give the poleward force. In fact, I explicitly asked you about this before. Again, please clarify and stick with one definition of the poleward force so that the discussion can move forward, rather than getting stuck in circles.

I am all in favor of explaining things in basic terms and with illuminating animations. But right now we are arguing about what is a correct argument. To show that your argument is correct, I ask you to please write down the equations to prove it instead of just talking about it.

Please be advised that for Wheatstone's device to work properly (like a foucault pendulum), it is essential that the frequency of the vibrating string has to be large compared to the rotational speed of the disk, as Wheatstone points out in his article. In the same way, it is important for the pendulum's frequency to be much larger than the rotational frequency of earth. You may want to check through the computation to see what goes wrong to understand this better. Also, Wheatstone's device only involves a vibrating string and no "bob" in the middle of it. But of course you may also consider the system with the bob, and it will also behave like a foucault pendulum. But again, the frequency of the vibration of the bob needs to be much higher than the frequency of the rotation of the disk.

A thought experiment involving the poleward force. The poleward force is quadratic in the rotational frequency. This has the following implication: Suppose there were a series of planets with slower and slower rotational velocity, and suppose that their shapes are accordingly closer and closer to a sphere. Consider foucault pendula on all of them, at geographical latitude $\phi$. What do you expect to see? Since the poleward force is quadratic in the frequency, but the period is inverse proportional, the effect of the poleward force over one period will decline linearly in the frequency, so becomes less and less important. But I claim (and my calculation show) that the foucault formula will hold for all of these pendula. Of course we can't really carry out the experiment since we can't get our hands easily onto these planets. But Wheatstone's device gives us a way to test this.

--ShanRen 18:30, 3 March 2007 (UTC)

OK, you were referring all the time to representing the physics in a configuration space. I jumped to the assumption you were referring to curvature of physical space, because in the case of the Foucault pendulum it is actually possible to represent it that way. I stand corrected, you were referring to parallel transport of a highly abstract vector in configuration space.

question:
What is to be categorized as the Wheatstone-Foucault device behaving properly, and what is to be categoried as improper behaviour?

I endorse the following point of view: no matter the ratio of rotation rate to vibration rate, in all cases the motion pattern is to be regarded as proper, since in all cases the Wheatstone-Foucault device is operating according to the laws of physics.

Another matter is that some simplifying assumptions to speed up calculations are not valid in the range where the rotation rate is fast (rotation rate approaching the vibration rate). I surmise that that is in fact your demarcation criterium. In the range where certain simplifying ssumptions are not valid, you declare the Wheatstone-Foucault device to be "behaving improperly". I claim that you need to upgrade to more faithful equations: you need to set up the equations in such a way that they accommodate the full range of the device's degrees of freedom.

If you gradually increase the rotation rate, then you are certain to come along a point where there is a ratio of 1 to 6 of rotation and vibration. If the device can produce that ratio, then faithful equations ought to accommodate that ratio as well.
Addendum: I claim that the sine law will hold good when the rotation rate approches the vibration rate. I claim that the sine law arises exclusively because of the inclination between the vibration and the axis of rotation; no restriction of the sine law to a range where the rotation rate is much less than the vibration rate.

I claim that there is no discontinuity in the mechanics of the Wheatstone-Foucault device anywhere in the range of the device's degrees of freedom. If for slow rotation rate a different set of equations is used than for fast rotation rate, then the equations are not faithful. (That is my objection to relying on equations only. Usually it is possible to tweak equations into producing the desired final result, but that is in itself no warranty that the equations are actualy faithful.) --Cleonis | Talk 19:19, 3 March 2007 (UTC)

Well, that "highly abstract vector space" happens to be the tangent space to an ellipsoid in this case. You may also think of the underlying ellipsoid also as a subset of 3-space ("earth"), and as such it has curvature (earth is not flat!). And the vector that underges parallel transport also has a quite mondane interpretation in 3-space. The point here is that your argument that these systems are not Newtonian systems because their configuration space has curvature is absurd.

You can call proper whatever you want. But if you want the device to act like a Foucault pendulum, the wire needs to vibrate fast compared the the rotation of the disk, just as Wheatstone observed in his article. You may study the behavior of comparable frequencies if you wish. In the case of the Foucault pendulum, I already did the calculation for you. And yes, it also works if the frequencies are comparable. Only it gives a different answer. Just solve the differential equation in the Foucault article and don't make the approximation that the frequency of the pendulum is much larger than that of earth. So do the calculation.

--ShanRen 23:24, 3 March 2007 (UTC)

I do intend to perform the calculation. Also, I intend to doublecheck by setting up a numerical analysis. I intend to use a technique called leap-frog integration, which can be performed with a spreadsheet program. --Cleonis | Talk 00:39, 4 March 2007 (UTC)

How about doing the calculation before the talking? --ShanRen 03:35, 4 March 2007 (UTC)

The distinction between definition and calculational shortcut

The definition of the poleward force is as follows:
The resultant force (resolved in the direction parallel to the local surface) of newtonian gravity and the normal force.

The calculational shortcut that is available is not to be confused with the formal definition of the poleward force. It is common in physics to be pragmatic in calculational strategy. If a shortcut is available, one uses it. This has no bearing on the status of the formal definition. I recognize the availability of the shortcut because I understand the physics that is involved.

Are you indicating that you do not recognize that this calculational shortcut is perfectly reliable? --Cleonis | Talk 19:45, 3 March 2007 (UTC)

The problem is that your "calculational shortcut" actually does not give a result that matches your "defintion". According to your "definition" the poleward force is tangent to the surface of earth, but your calculational shortcut (the negative of the centrifugal force) gives a force that is not. In light of this, I do not recognize that this calculational shortcut is perfectly reliable. --ShanRen 23:08, 3 March 2007 (UTC)

The resultant force of newtonian gravity and normal force, resolved in the direction parallel to the plane of rotation. This provides the required centripetal force

The resultant force of newtonian gravity and normal force, resolved in the direction parallel to the local surface

In the definition of poleward force, I added the stipulation: resolved in the direction parallel to the local surface. Of course, the required centripetal force should act parallel to the plane of rotation, as shown in the first diagram. In the case of the Foucault pendulum (in the small angle approximation) the motion is constrained to motion parallel to the local surface. In the level of approximation we are dealing with here, the problem is cast entirely in terms of vector-components tangent to the local surface. Because of that it is worthwile to have a specific name for the tangent-to-the-local-surface component of the required centripetal force: the poleward force. This is shown in the second diagram.

Generally, my comments tend to be quite long, for whenever possible I want to try and avoid ambiguity. It appears that my description of the calculational shortcut was too terse. I gather that my descriptions should be more thorough. --Cleonis | Talk 00:13, 4 March 2007 (UTC)

A humble suggestion: The poleward force is the tangential (to earth) component of the centripetal force. Terse and correct. --ShanRen 02:48, 4 March 2007 (UTC)

The question what level of simplification is viable

I copy and paste from above:

if you want the device to act like a Foucault pendulum, the wire needs to vibrate fast compared the the rotation of the disk,
[...]

And yes, it also works if the frequencies are comparable. Only it gives a different answer.
 :--ShanRen 23:24, 3 March 2007 (UTC)

You state a different answer will be obtained, but you do not specify in which way it will be different

I expect that the answer will be different in the following way:
when the frequencies are comparable, the pendulum bob will quite clearly not follow a straight line on each separate swing. It will precess allright, it just will not move along a straight line on each successive swing. But your calculation hinges on the simplifying assumption that the pendulum bob moves along a straight line on each separate swing. (Of course, when the derivation is cast in terms of motion that is constrained to motion tangent to the Earth's surface, (motion in curved 2-space) the expression 'straight line' must be understood as 'straight line within that 2-space'.)

The challenge is to set up the equations for motion in geometrically flat 3-space. The limitation that you claim is not a limitation of the physical device, the limitation is merely an artifact of the level of simplification that you have opted to apply. --Cleonis | Talk 01:10, 4 March 2007 (UTC)

On the chance of repeating myself, the calculation I included in the Foucault article does not hinge on the simplifying assumption that the pendulum bob moves along a straight line on each separate swing. If you care to read it you see that the simplifying assumption enters at the last step. It's only purpose is to simplify the solution. Just go ahead and solve the equation yourself and you will see. Or if you don't know how to do it, then I can do it for you. What you will find is that the solution is different, and for comparable frequencies it does not make sense to talk about precession of the pendulum any more. (Alternatively check the derivations in the "external links" section, they are very similar.)

Question: Have you ever actually written down a solution for Foucault's pendulum?

--ShanRen 03:34, 4 March 2007 (UTC)

Yes, the simplifying assumption that I criticize enters at the last step, and its only purpose is to simplify the solution.
At every point in time of its swing, the pendulum has a particular direction. A natural point to sample the instantaneous direction is the equilibrium point. I can define as follows: the amount of precession from one swing to another is the difference in direction from one pass over the equilibirum point to the next pass. That gives a handle to assess/express the amount of precession from swing to swing.
I have not worked out an analytic solution that avoids the simplification that I criticise. (I'm moderately skilled in using differential equations. Example of me using differential equations to get at a result: the section 'the general case' in this article on my own website.) So for the time being we are finished discussing the issue.

Please give the analytic solution that you obtain without the simplifying assumption, I am curious how you approach that. As mentioned earlier, I will doublecheck with a numerical calculation. See Michael Fowlers description of solving dynamics problems numerically with Excel spreadsheets. --Cleonis | Talk 09:08, 4 March 2007 (UTC)

The only problem is that the pendulum does not pass through (or even close to) the "equilibrium point" on each swing if the frequencies are comparable. If you really want to extract some "precession" out of this experiment when the frequencies are comparable you can try and separate the "precession frequency" from the "swing frequency". Be careful though that the natural frequency of the bob will be shifted significantly. Anyway, let's stop this game, I will just tell you what the solution is (assuming that you understand complex variables). The differential equation in the article can easily be solved analytically. It is a seond order linear differential equation, and any basic textbook tells you how to solve it. The particular one is the same equation as for the damped harmonic oscillator, except that the damping factor is imaginary. In short, there are two fundamental solutions with frequencies $-i\Omega \sin(\phi )\pm i{\sqrt {\Omega ^{2}\sin ^{2}(\phi )+\omega ^{2}}}$. For large time frames, there is a problem with this solution because the small angle approximation has issues, but it will be fine for anything you can reach with your numeric methods.

--ShanRen 15:11, 4 March 2007 (UTC)

You are correct in pointing out that in swing the pendulum bob does not pass over the point the pendulum bob hangs above when it is not swinging. (Actually, the image on the right, that I manufactured and uploaded about a year ago, shows precisely that.) So the sampling point would have to be the point during the swing where the velocity of the pendulum bob relative to the Earth is greatest. (Alternative formulation: the point during the swing where the angle between the pendulum wire and the plumb line is smallest)

As you point out there is a range where separation between "precession frequency" and "swing frequency" is meaningless. So it is better to keep the explorations in the range where the swing frequency is much higher than the rotation frequency.

The rate of change of direction of the pendulum bob is not constant, it is proportional to the velocity of the pendulum bob with respect to the rotating system, and I want a solution to the equation of motion that incorporates that. In the image the successive swings aren't sections of a circle: close to the extremal points the curvature is close to zero, the lines curve strongest where the velocity is largest.

I know differential equations for damped harmonic oscillators, and how solving that kind of equations is facilitated by the relation e^it = cos(t) + isin(t) (I'm reproducing from memory here, so I may have a detail wrong.) In the differential equation for a damped harmonic oscillation the dampening force is parallel to the direction of the velocity, so I take it that a force perpendicular to the direction of the velocity is represented as an imaginary factor. --Cleonis | Talk 23:03, 12 March 2007 (UTC)

Cleonis, I am not really interested in writing on the Wheatstone-Foucault article. But here is a suggestion. If you assume that the rotation of the disk is slow, you can neglect the centrifugal force on the bob. It is quadratic, so that is a good approximation. Also, you don't need the small angle approximation in that case, since the restoring force of the spring is usually taken to be linear anyway (Hooke's Law). If you are having problems with the mathematical derivation, just use the one for the Foucault pendulum, it's essentially the same. The linked webpages in the Foucault article have more detail. --ShanRen 02:57, 14 March 2007 (UTC)

Yeah, I know you are not interested. We agree that the physical analysis of the Foucault pendulum and the Wheatstone-Foucault device are identical. The advantage of the Wheatstone device (for the purpose of education) is that it is much more visual. That is why I will add a discussion of the physics of the Wheatstone device in the Foucault pendulum article.

Since I discuss physics in terms of the inertial point of view, there is no centrifugal force.
As is well known, the foucault pendulum is very sensitive to even small influences. Any influence that is cumulative must be taken into account, and judging what influence will cancel out and what influence will be cumulative is not straigtforward.
I think the following comparison is interesting. Let a pendulum be inside a cabin and let that cabin be in linear uniform acceleration. In the case of Linear acceleration there won't be precession; linear acceleration will only result in a shift of the direction of effective gravitation, there is no cumulative effect. On the other hand: centripetal acceleration induces precession of the pendulum; the effect of the centripetal force is cumulative . This is correlated with the fact that the direction of the centripetalf roce rotates over time.)

In my sandbox article I present derivations (strictly from an inertial point of view) of how the centripetal force induces precession in the case of the Foucault effect. The formulas that I obtain are identical to what is obtained when the analysis is performed from the earthbound point of view. ( Of course they end up identical: it is two ways of looking at the same phenomenon.) http://en.wikipedia.org/wiki/User:Cleonis/Sandbox/Wheatstone-Foucault_device#Mathematical_derivations --Cleonis | Talk 17:32, 14 March 2007 (UTC)

The equation of motion in rotating coordinates

If you assume that the rotation of the disk is slow, you can neglect the centrifugal force on the bob. --ShanRen 02:57, 14 March 2007 (UTC)

You seem to suggest here that the centrifugal term can be neglected while retaining the coriolis term.

The centrifugal term and the coriolis term always go together. In the full equation of motion (full in the sense that terms that will drop away against each other are initially retained), in the full equation of motion for rotating coordinates there is always both a centrifugal term and a coriolis term. There is no such thing as neglecting the centrifugal term and retaining the Coriolis term. (In the case of the Foucault pendulum, the centrifugal term is a hundred times larger than the coriolis term)

I will discuss a simpler case than the Foucault pendulum, and then I will move on to the Foucault pendulum and the Wheatstone pendulum.

Two weights, connected by a helical spring. The force exerted by the spring provides the required centripetal force. A model like this can be used to represent rotation/vibration of a diatomic molecule.

The simplest case is one that involves planar motion and just a single force: a spring that is exerting a force on a weight. For the sake of simplicity, I assume that Hooke's law applies for the entire range of extension/contraction of the spring.

For the motion of each of the weights in the animation on the right, the equation of motion for inertial coordinates is (in vector notation):

${\vec {F}}=-k{\vec {r}}$

Of course, in cartesian cordinates the set of all solutions to that equation of motion can be represented as a linear combination of two perpendicular harmonic oscillations. The following parametric equation describes the set of solutions to the above equation of motion.

${\begin{array}{ll}x=acos(t)\\z=bsin(t)\end{array}}$

What is interesting about this set of solutions, of course, is that they all have the same period; in this case the period of a solution of the equation of motion is not a function of radial distance.

For rotating coordinates, the full equation of motion (in vector notation) is as follows:

${\vec {F}}=-k*{\vec {r}}+\omega ^{2}*{\vec {r}}+2*{\vec {\omega }}\times {\vec {v}}$

Where ω is the number of revolutions per unit of time of the rotating system.
In this equation of motion, the term for the centripetal force and the centrifugal term drop away against each other.
(Proof: the period of the rotation is determined exclusively by the properties of the spring. It is not possible to make the system rotate faster. While it is possible to increase the energy content of the system - appliying a torque will cause increase of energy of the rotating system - this torque will not induce a lasting change in angular velocity; the rotating system will absorb the influx of energy by expanding in radial size, and when the torque ceases, the angular velocity will be the same as in the beginning; all solutions to the equation of motion have the same period.)

The motion of the circling masses mapped in a coordinate system that is rotating at a constant angular velocity

The term for the centripetal force and the centrifugal term drop away against each other, and what remains is the coriolis term:

${\vec {F}}=2*{\vec {\omega }}\times {\vec {v}}$

The animation illustrates that in rotating coordinates the solution to the equation of motion transforms to motion along a socalled Coriolis circle.

The Foucault pendulum
In the case of the Foucault pendulum, there are two forces: the centripetal force that is required to sustain the circumnavigating motion, and the restoring force of the pendulum swing.

The full equation of motion (vectors resolved in the direction parallel to the earth's surface):

$\left\{{\begin{array}{ll}{\ddot {x}}=-{F_{x,centripetal}}sin(\phi )+\Omega ^{2}xsin(\phi )-\omega ^{2}x+2\Omega {\dot {y}}\sin(\phi )\\{\ddot {y}}=-{F_{y,centripetal}}sin(\phi )+\Omega ^{2}xsin(\phi )-\omega ^{2}x-\omega ^{2}y-2\Omega {\dot {x}}\sin(\phi ).\end{array}}\right.$

Since the Earth is an equilibrium shape (the Earth's shape matches its rotation rate; all parts of the solid earth have the same angular velocity) the term for the centripetal force and the centrifugal term drop away against each other, leaving the Coriolis term and the restoring force.

$\left\{{\begin{array}{ll}{\ddot {x}}=-\omega ^{2}x+2\Omega {\dot {y}}\sin(\phi )\\{\ddot {y}}=-\omega ^{2}y-2\Omega {\dot {x}}\sin(\phi ).\end{array}}\right.$

All authors who present the above relation as the correct equation of motion for rotating coordinates have implicitly incorporated the centripetal force into the full equation. If they would not have incorporated the centripetal force then in subsequent proceedings the centrifugal term would still be present.

(I may have a +-sign or a minus sign wrong here or there, but the structure of my exposition is clear enough.) --Cleonis | Talk 23:03, 16 March 2007 (UTC)

Cleonis, not really sure why you are writing all this. It seems that your main question is why you can neglect the centrifugal force over the coriolis force, at least if the rotation of the disk is slow. While your equations and arguments could use some improvement (I don't really want to comment on this), they keep the functional relations reasonably intact for me to explain this. in your equations of motion, the rotational frequency $\Omega$ enters in both, the coriolis and the centrifugal term. but in the centrifugal term it enters quadratically, while it only enters linearly in the coriolis term. you are looking for an effect that takes place after one turn, so that is a time of the order $1/\Omega$. So if you have the coriolis force act for that amount of time, you expect to see an effect that is independent of the rotational frequency. when you have the centrifugal term act that amount of time, you will see an effect of the order of $\Omega$. If the rotation of the disk is slow, that is $\Omega$ is small, the effect of the centrifugal term gets less and less, but the effect of the coriolis term is unaffected. Hope that helps. --ShanRen 02:46, 17 March 2007 (UTC)

Unfortunately, it doesn't help. The things you are writing do not add up to a coherent thought.

In the case of the Foucault pendulum, the centrifugal term is tens of times larger than the coriolis term. That is the only thing that counts. If a physicist wants to include the coriolis term, then he must also include the much larger centrifugal term.

Two weights, connected by a helical spring.

The science of physics is to investigate which forces are at work and how the sequences from cause to effect proceed. In the example I gave the analysis is as simple as it gets: one force: the force exerted by the spring, which is a harmonic force. When the spring contracts the spring is doing work, and potential energy is converted to kinetic energy. When the spring stretches again it is doing negative work: kinetic energy is converted to potential energy.

There is no such thing as a force acting in centrifugal direction that is being exerted upon the weights. The motion is accounted for in terms of Newton's laws of motion.

Of course, the centrifugal term and the coriolis term are eminently useful calculational tools.

The parametric equation

$\left\{{\begin{array}{ll}x=acos(\omega t)\\z=bsin(\omega t)\end{array}}\right.$

Motion due to a harmonic force described as circle + epi-circle motion

can be rearanged to the following form:

$\left\{{\begin{array}{ll}x=\left({\begin{matrix}{\frac {a+b}{2}}\end{matrix}}\right)\cos(\omega t)+\left({\begin{matrix}{\frac {a-b}{2}}\end{matrix}}\right)\cos(\omega t)\\y=\left({\begin{matrix}{\frac {a+b}{2}}\end{matrix}}\right)\sin(\omega t)-\left({\begin{matrix}{\frac {a-b}{2}}\end{matrix}}\right)\sin(\omega t)\end{array}}\right.$

And the animation on the right illustrates this rearrangement. The saliant feature is that the eccentricity in the trajectory, the deviation from symmetry, is itself a circular motion with respect to the co-rotating coordinate system.
Transforming to a coordinate system that is rotating with angular velocity ω :

$\left\{{\begin{array}{ll}x=\left({\begin{matrix}{\frac {a-b}{2}}\end{matrix}}\right)\cos(2\omega t)\\y=-\left({\begin{matrix}{\frac {a-b}{2}}\end{matrix}}\right)\sin(2\omega t)\end{array}}\right.$

The above parametric equation for the motion with respect to the rotating coordinate system corresponds to the coriolis term.

Metaphorically speaking, there is a wonderful "division of labour" between the centrifugal term and the coriolis term. In physics that involves rotation, the centripetal force is almost always to a good approximation a harmonic force (Nature tends to develop towards an equilibrium state). In many cases the centripetal force and the centrifugal term drop away against each other in the equation of motion. That explains why the centrifugal term and the coriolis term are so useful.

The reason that I write all of this is to show you the structure of how I approach the Foucault pendulum analysis. In physics analysis, only forces count. The physicist identifies the forces that are acting, and traces how the events proceed from cause to effect.

I will edit the Foucault pendulum article, and my edits will present straightforward application of newtonian principles. I have given it every effort to communicate to you that the approach that I favor is plain newtonian physics. --Cleonis | Talk 20:37, 18 March 2007 (UTC)

Cleonis, I was talking about the Wheatstone device in the case where the disk is rotating slowly. You respond by telling me that this does not match the scenario for the Foucault pendulum. So what? Please stick to one topic at a time. I welcome you to comment again -- but this time on the topic. The only reason why I suggest this approximation to you is that if you neglect the centrifugal force, the equation for Wheatstone's device are much easier to solve.

In the case of the Foucault pendulum you don't have the luxury of assuming that the rate of rotation is arbitrarily slow, it is given to you. If you actually would finally sit down and do the calculation (or at least copy it from a textbook!) you would see that the answers with or without the centrifugal force differ by less than a degree per day. To get you started on this, I recommend that you first settle for one definition of latitude, since there are several choices in the case you view earth as an ellipsoid. Without making that choice you can't write down any equations anyway, and you will see at the end that the choice is intimately connected to the question of how to deal with the coriolis force.

One last point, please spare me from your trivial physics triads about two masses on a spring. --ShanRen 21:21, 18 March 2007 (UTC)

Mathematical derivations

Image 2
Version of Wheatstone's device that underlines the similarity with the Foucault pendulum

In the following derivations, I call upon:

• Newton's laws of motion
• The work-energy theorem
• The principle of conservation of angular momentum

The following derivations apply for the mechanics of rotation in general. I take it as granted that the analysis of the Foucault pendulum is a problem in the mechanics of rotation

Motion towards and away from the central axis

At every point in time, the bob has a particular angular momentum with respect to the central axis of rotation. When the inside spring pulls the pendulum bob closer to the central axis of rotation, the centripetal force is doing mechanical work. As a consequence of the centripetal force doing work, the angular velocity of the bob changes. (Compare a spinning ice-skater who pulls her arms closer to her body to make herself spin faster.) The amount of angular acceleration can be calculated with the help of the principle of conservation of angular momentum.
When the pendulum bob is moving away from the central axis the centripetal force is doing negative work, and the bob's angular velocity decreases.

Overview of symbols

r	distance to the central rotation axis
ω	Angular velocity of the rotating system
vr	velocity in radial direction
vt	total velocity in tangential direction
at	acceleration in tangential direction
ar	acceleration in radial direction
vt,r	velocity in tangential direction, relative to the rotating system
vt,e	tangential velocity that corresponds to co-rotating with the rotating system, this is the equilibrium velocity

The formula for the angular momentum L is:

<math> L = mv_t^2 r </math>

What is sought is how acceleration in tangential direction (dv_t/dt) depends on the radial velocity (dr/dt). This can be found by first calculating the differential d(v_t)/dr.
The expression xy²=1 differentiates as follows:

<math> \frac{dx}{dy} = \frac{d(y^{-2})}{dx} = -2 y^{-3} </math>

Since x=y^-2, the expression -2y^-3 for the differential can also be written as:

<math> \frac{dx}{dy} = -2 \frac{x}{y} </math>

In the case of d(v_t)/dr this gives:

<math> \frac{d(v_t)}{dr} = -2 \frac{v_t}{r} </math>

The above equation can be transformed into a differential equation with time derivatives in the following way:

<math> \frac{d(v_t)}{dt} = -2 \frac{v_t}{r} * \frac{dr}{dt} </math>

When the tangential velocity relative to the rotating system is small then v_t/r is very close to ω. This gives the following relation between radial velocity and acceleration in tangential direction:

$a_{t}=-2\omega v_{r}$

Comment added march 27th:
I've stricken out passages that are wrong and/or irrelevant. Added nowiki tags around the <math></math>-tags, so the formulas won't be rendered anymore. --Cleonis | Talk

Motion tangent to the rotation

During a swing of the pendulum bob from west-to-east, the bob is circumnavigating the central axis faster than the equilibrium velocity, hence during an west-to-east swing the bob will swing wide. (Compare a car going round on a banked circuit, with the incline of the embankment designed for a particular speed. A car that goes faster than that will tend to climb up the embankment.) During a swing of the pendulum bob from east-to-west, the bob is circumnavigating the central axis slower than the equilibrium velocity, hence during an east-to-west swing the centripetal force will pull the bob closer to the central axis.

What is sought for is how acceleration in radial direction depends on tangential velocity v_t.

$a_{r}={\frac {v_{t,e}^{2}}{r}}-{\frac {v_{t}^{2}}{r}}$

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

$a_{r}={\frac {v_{t,e}^{2}}{r}}-{\frac {(v_{t,e}+v_{t,r}^{2})}{r}}$

$a_{r}={\frac {v_{t,e}^{2}-(v_{t,e}+v_{t,r})^{2})}{r}}$

$a_{r}={\frac {v_{t,e}^{2}-v_{t,e}^{2}-2v_{t,e}v_{t,r}-v_{t,r}^{2}}{r}}$

$a_{r}={\frac {-2v_{t,e}v_{t,r}-v_{t,r}^{2}}{r}}$

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

$a_{r}=-2\omega v_{t,r}$

Conclusions

I take it as generally granted that the analysis from the inertial point of view and from the earthbound point of view should confirm each other.

• When transforming back and forth between inertial coordinates and rotating coordinates, the amount of radial velocity is identical in both frames.
• When transforming back and forth between inertial coordinates and rotating coordinates, the amount of tangential acceleration is identical in both frames.

When there is a centripetal force that is a harmonic force then the coriolis term corresponds to compliance with the work-energy theorem.

The following relations apply when the motion is due to a centripetal force that is a harmonic force.

$\left\{{\begin{array}{ll}a_{t}=-2\omega v_{r}\\a_{r}=-2\omega v_{t,r}\end{array}}\right.$

--Cleonis | Talk 22:57, 20 March 2007 (UTC)

You sure know how to take something trivial and make it look complicated -- and get the answer wrong. I only read through the first part, and I am not sure if I want to wade through the rest. I am not sure why you are writing all of this, I thought you wanted to derive the motion for the Wheatstone device. Actually, I have no idea just what you are trying to derive here.

About your calculations, I don't really know what to say. I guess first you should consult the link you gave on the formula for angular momentum, your formula is wrong. But even given your wrong formula, your derivation of the derivative is not just convuluted, but also ends up with the wrong answer (no matter if your formula for angular momentum is used or the correct one). Not sure if this is useful, but here is a quick and correct way to take the derivative of your formula $L=mv_{t}^{2}r$, assuming that the left hand side is constant (it is not the angular momentum, but let's assume it is constant anyway) and the mass is constant. Taking time derivative on both sides gives $0=m\,2v_{t}{\frac {dv_{t}}{dt}}r+m\,v_{t}^{2}{\frac {dr}{dt}}$, which you can hopefully correctly solve for ${\frac {dv}{dt}}$. Also keep in mind that angular momentum is only preserved when there are no forces acting. --ShanRen 01:18, 21 March 2007 (UTC)

Good grief, the first half of the above derivations is a complete screw-up, starting with getting the formula for the angular momentum wrong. The correct formula for the angular momentum is of course:

$L=mv_{t}r$

Well, back to the drawing board. Any plans of mine to edit the Foucault pendulum article will have to be postponed.

I have finished two more animations, which can be used in the Foucault pendulum article, or in a separate article about the Wheatstone-Foucault device (or both) --Cleonis | Talk 23:27, 24 March 2007 (UTC)

Brian Fiedler's physics applets

I present as evidence two physlets by Brian Fiedler Professor in the school of meteorology at at the University of Oklahoma. Physlets are Java applets. See http://webphysics.davidson.edu/Applets/Applets.html

Brian Fiedler's Physlets for the Coriolis effect:
http://mensch.org/physlets/merry.html
http://mensch.org/physlets/inosc.html

I take it as granted that Coriolis effect as taken into account in meteorology and the Coriolis effect that is responsible for the precession of the Foucault pendulum are one and the same.

For example: in the Coriolis force article it is stated:

The Coriolis effect caused by the rotation of the Earth can be seen indirectly through the motion of a Foucault pendulum.

Therefore educational material that elucidates the Coriolis effect in general is relevant for understanding the precession of the Foucault pendulum.

I suppose I should have pointed out Brian Fiedler's physlets before. I came across them a year or so ago. --Cleonis | Talk 16:53, 27 March 2007 (UTC)

Evidence for what? Why are you writing this? --ShanRen 17:18, 27 March 2007 (UTC)