User talk:David Harty

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[edit] Welcome

Welcome!

Hello, David Harty, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers:

I hope you enjoy editing here and being a Wikipedian! Please sign your name on talk pages using four tildes (~~~~); this will automatically produce your name and the date. If you need help, check out Wikipedia:Where to ask a question, ask me on my talk page, or place {{helpme}} on your talk page and someone will show up shortly to answer your questions. Again, welcome! – yes I wish you to see more edits from you. --Bhadani 12:48, 5 January 2006 (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 will be made through the the following thought experiment and also proposed as a separate caption to the article. latitude, Coriolis Effect, "Foucault pendulum", "Foucault Pendulum Vector Diagrams", centrifugal, centripetal, angular velocity, fictitious force

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.

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.

David Harty 14:13, 23 March 2007 (UTC) David Harty 11:57, 24 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 11:57, 24 March 2007 (UTC)

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. I don't know the answer to the question posed by the experiment. Related to this is 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, Coriolis, that than results in the observed motion of the pendulum swing. The article should be made clear in this regard. David Harty 08:48, 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 10:58, 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. Le terre, c'est tournez.David Harty 08:17, 30 March 2007 (UTC)

[edit] Combination Diagram

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 and the Foucault pendulum becomes a basic demonstration of the Coriolis force. David Harty 15:47, 31 March 2007 (UTC)

[edit] Diagrams For Foucault Pendulum

Why not include the information in the Foucault pendulum article? The articles are both short and could be merged easily.-localzuk 12:59, 5 January 2006 (UTC)

  • I agree with this comment Localzuk left on your talk page. It may not be small now anymore, but I think if you try to cut out how-to bits and leave those for Wikibooks you can considerably shorten your entry so it can be included in Foucault pendulum. They really belong together. - Mgm|(talk) 09:53, 9 January 2006 (UTC)

The comment was originally made when there was only one paragraph written for the article. The purpose of the first article is an overview of the Foucault pendulum and I would not like to clutter it up with details that are not pertinent to an overview. I think the first article is fine the way it is. I wrote the second article to explore details not fully explained in books and to attempt to provide diagrams rather than words. If the information gets moved to a Wikibook, that would be fine too, but this was as far as I intended to go at this point. David Harty 04:54, 10 January 2006 (UTC)

The article "Diagrams For Foucault Pendulum" was a draft page that was prepared while the article was being "wikified". The current article is entitled "Foucault Pendulum Vector Diagrams" and is linked at the end of the wikipedia article "Foucault pendulum". The article "Diagrams For Foucault Pendulum" needs to be deleted as originally recommended by Wikipedia. Sorry for any confusion this may have caused. David Harty 05:50, 10 January 2006 (UTC)

[edit] The foucault pendulum and the coriolis effect

After some thought, I added a brief paragraph about the Coriolis effect and referred to the article in Wikipedia. I hope that this is helpful to clarifying the information as this is the effect that is at the heart of the behavior of the Foucault pendulum. I think the Foucault pendulum is elucidating to the Coriolis effect. [...] .David Harty 15:21, 22 January 2006 (UTC)

Indeed the Coriolis effect is involved in the Foucault pendulum. Unfortunately, the current wikipedia Coriolis effect article presents a very confusing picture, and is unhelpful in helping people to understand what is taking place. --Cleonis | Talk 19:35, 22 January 2006 (UTC)

The external links are helpful. David Harty 22:31, 22 January 2006 (UTC)

Several of those external links were added by me. The article by the meteorologist Anders Persson is was a big eye-opener for me. the coriolis effect (PDF-file 870 KB).
I wrote an article about the Eötvös effect and I have written an article about Rotational-vibrational coupling. Both articles describe things that are related to the coriolis effect. --Cleonis | Talk 23:07, 22 January 2006 (UTC)

[edit] Relative Motion of the Plane of the Pendulum Swing to the Surface of the Earth

In order to observe the rotation of the Earth in relation to the plane of the pendulum swing there must be a basic difference in the two types of motion that are being compared. This basic difference is then manifested as (1) being able to 'observe' the change in position of the Earth in relation to the pendulum swing and (2) the time to observe a complete 'relative rotation' decreases with the sine of the latitude (decreases with an increase in the angle of alignment with the Earth's axis of rotation).

At the North Pole:

The central axis of the pendulum aligns with the axis of rotation of the Earth. The central axis of the pendulum is always determined by the force of gravity directed towards the center of the Earth.

A pendulum bob at rest at the North Pole still has spin on the bob.

If a pendulum bob is hanging vertically at the North Pole and held in place, the bob is stationary but is rotating (spinning) with the Earth. Once the bob is released (but not swinging) it will continue to rotate (spin) unless one stops the rotation (spin) by forcing a spot on the side of the bob to always line up or point to one star. If the rotation of the bob is stopped then the connection wire will twist one turn every day unless there is a connection that is free to rotate (spin) (at either end of the wire or the support structure). If the bob at the North Pole is allowed to continue to rotate (spin) then it will and the wire won't twist one turn in one day.

The Foucault pendulum connection is constructed such that the pendulum is free to swing in any direction.

This is not the same thing as the support connection being free to rotate (spin). The swing of the pendulum is different then the rotation (spin) of the bob.
If a pendulum is hanging at the North Pole, before the bob is released, the bob is stationary but is rotating (spinning) with the Earth. Once the bob is released it will continue to rotate (spin) unless one stops the rotation (spinning) by forcing a spot on the side of the bob to always line up or point to one star.

If the bob is displaced from the central axis of the pendulum in preparation for swinging and held in place, then the bob will revolve about the central axis of the pendulum along with the rotation of the Earth and has an angular velocity equal to that of the Earth's angular velocity.

Before the bob is released there is a force that is exerted through the holding point of the bob that causes the bob to revolve about the pendulum axis and rotate (turn) with the Earth. This is because the holding point is attached to the surface of the Earth just like the structure of the pendulum is attached to the Earth.

Once the bob is displaced from the central axis of the pendulum and then released there no longer is a force acting on the bob that causes it to revolve about the central axis of the pendulum and rotate (turn) with the Earth.

As observed from an end-view of the swinging bob, the swing of the bob will always line up or swing towards one star (just like the axis of the Earth points at one star for the time periods considered) as the bob swings through the central axis of the pendulum. There can be a slight ellipsoid swing if the initial conditions of angular motion are not cancelled but there is no longer a force acting on the bob causing it to have an angular velocity after the bob is released. The plane of the swing of the pendulum bob is now independent of the surface of the earth which was imparting a force to the bob before it was released (through the holding point). As noted previously, the bob is still spinning with the Earth (a spot of the bob will spin with the Earth), even though the bob is no longer turning with the Earth. Thus the Earth continues to turn underneath the swing of the pendulum while the swing of the pendulum remains in a fixed plane that doesn't rotate (turn).

The point of significance is that the force imparting an angular velocity to the pre-released bob is no longer acting on the swinging bob. At the North Pole, this force takes one day for the direction of the force to complete a full circle since it takes one day for the Earth to rotate.


At the equator:

The central axis of the pendulum is perpendicular with the axis of rotation of the Earth. The central axis of the pendulum is always determined by the force of gravity directed towards the center of the Earth.

A pendulum bob at rest at the Equator is still rotating with the Earth and there is no spin on the bob.

The pendulum is moving with the rotation of the Earth when located at the equator, as is the support structure, so one can't see the rotation of the Earth in relation to the pendulum. The observation of the relative motion of the Earth in relation to the pendulum depends on the location of the surface of the Earth where the initial conditions are established.
If a pendulum bob is hanging vertically at the Equator and held in place, the bob is stationary relative to the Earth and is rotating (turning) with the Earth. Once the bob is released (but not swinging) it continues to rotate (turn) with the Earth.

If the bob is displaced from the central axis of the pendulum in preparation for swinging and held in place, then the bob is still rotating (turning) with the Earth with the same angular velocity equal to that of the Earth's angular velocity. This is the same angular velocity when at rest. Since the central axis of the pendulum is perpendicular with the axis of rotation of the Earth this is not the same as the North Pole where the central axis is aligned with the axis of the Earth. The bob is not revolving about the axis of the pendulum when held in place.

Before the bob is released there is a force that is exerted through the holding point of the bob that causes the bob to rotate (turn) with the Earth. This is because the holding point is attached to the surface of the Earth just like the structure of the pendulum is attached to the Earth.

Once the bob is displaced from the central axis of the pendulum and then released there is still the same force acting on the bob that causes it to rotate (turn) with the Earth.

As observed from an end-view of the swinging bob, the swing of the bob will not line up or swing towards one star as the bob swings through the central axis of the pendulum. There will not be a slight ellipsoid swing in relation to the Earth since the initial conditions of angular motion are not changed and there is still a force acting on the bob (transmitted through the support structure, pendulum wire, and gravity) causing it to have an angular velocity after the bob is released. The plane of the swing of the pendulum bob is independent of the surface of the earth but is not independent of the pendulum system which is still imparting the same force to the bob as before it was released through the single support point of the pendulum. As noted previously, there is no spin on the bob (a spot of the bob does not change with respect to the Earth) and the bob is not revolving about the axis of the pendulum. Thus the Earth continues to turn underneath the swing of the pendulum and the swing of the pendulum continues to turn with the Earth since there is still a force acting on the bob of the pendulum swing.

The point of significance is that the same forces imparting an angular velocity to the pre-released bob are still acting on the swinging bob. At the Equator, the relative motion of the Earth is not observable because there is no change in the force imparting an angular velocity to the bob. This is because the central axis of the pendulum is perpendicular with the axis of rotation of the Earth.

For a separate, imaginative arrangement, if one could imagine a large pendulum structure that is mounted at the North Pole and free to not rotate with Earth (e.g., mounted on a platform that is free of the rotation (spin) of the Earth) but has long arms that allows the pendulum to swing at the Equator then the Earth's surface would move underneath the pendulum. The Earth doesn't rotate (turn) under the pendulum swing like at the North Pole but the equatorial plane rotates perpendicular to the pendulum swing. This is a very large pendulum and an idealized situation.

At intermediate latitudes:

The rotation of the Earth is observable in relation to the plane of the pendulum swing but the time to observe a full rotation depends on the latitude of the location. The time to observe a full rotation is equal to one day at the North Pole with the time increasing with decreasing latitude and not observable at the Equator (infinite length of time).

The time increases because the central axis of the pendulum is aligned with the axis of rotation of the Earth at the North Pole and then the angle of misalignment increases as the latitude decreases to the point of perpendicularity at the Equator.
The angular velocity that is imparted to the pendulum bob about the axis of the pendulum prior to release decreases with the cosine of the degree of misalignment of the central axis of the pendulum in comparison to the axis of rotation of the Earth (zero degrees of misalignment at the North Pole, cosine of zero degrees equals 1; 90 degrees of misalignment at the Equator, cosine of 90 degrees equals 0).
This is equivalent to stating that the angular velocity that is imparted to the pendulum bob prior to release decreases with the sine of the latitude of the location (the sine of 90 degrees latitude equals 1; the sine of zero degrees latitude equals 0).
When the bob is released there is no longer a force acting on the bob causing it to revolve about the central axis of the pendulum. That force that is no longer applied is less than that applied at the North Pole where axis are fully aligned.
The time to observe a complete rotation of the Earth is inversely proportional to the angular velocity that is not imparted to the pendulum bob.

The statements above are thus equivalent to the inverse sine law for the observed time for a full rotation of the pendulum in relation to the rotation of the Earth.

Final Note: There is only one point of connection to the Earth for the swinging pendulum and that point of connection doesn't move in relation to the Earth.

[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.

[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.

70.58.144.193 08:32, 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.

David Harty 11:39, 17 March 2007 (UTC)