Can the Foucault Experiment's Principles Be Derived Through Calculus?

[JeffOCA]

Hello

The Foucault experiment is based on the non rotation of the oscillation plane of a classical pendulum. Maybe a stupid question but I'd like to know if there is a simple manner to derive it. By the conservation of angular momentum ?

Thanks for your help...

 

[Born2bwire]

I have never gone to such lengths. It is a simple demonstration of geometric phase and can be derived by geometry.

The simplest example is say you are at the north pole and you have your pendulum swinging in a protective box. This box is always held in the same orientation in regards to the north-south direction. Initially you face towards Chicago, and feeling like you need some real food, you walk from the north pole to Chicago to get some Hot Doug's sausages (MMMMmmmm). When you get to Hot Doug's, your pendulum is swinging along the line connecting Chicago to the north pole. Now keep going until you hit the equator and walk east until you get to the west coast of Africa. When you turn, you keep the pendulum swinging along the north-south line. Equitorial Guinea isn't so hot so you go back to the north pole. When you reach the north pole, you have now completed a full closed path; going from the north pole to the equator via Chicago, traveling along the equator to Africa, and then going back to the north pole. Now since you kept the pendulum swinging from north to south, the path that it swings when you return from Africa is different from the path that it was swinging when you left toward Chicago.

Despite having traveled a closed path, your pendulum's axis has shifted from along the line from the north pole to Chicago to the north pole to some point in Africa (ok, I don't know Africa too well). This is called a geometric phase and is the principle of the Foucault's pendulum. In our example, you physically walked around the Earth. With a Foucault's pendulum, you do not move the pendulum, but the pendulum is moved about the sphere in space by the actual rotation of the Earth. The solid angle that subtends the closed path is related to the actual shift in the pendulum over the closed path.

There is an explanation following these lines in Griffith's Quantum Mechanics textbook I think because geometric phase comes up in Quantum Mechanics too (as Berry's phase for example) but it also plays a role in many other physical problems. I think however, you can derive these results using Coriolis forces in a rotating reference frame.

 

[Cleonis]

Hello
The Foucault experiment is based on the non rotation of the oscillation plane of a classical pendulum. Maybe a stupid question but I'd like to know if there is a simple manner to derive it. By the conservation of angular momentum ?
Thanks for your help...

I assume that you are referring to the general case, a Foucault pendulum that can be located at any latitude. (The special case of a Foucault pendulum located at one of the poles is trivial.)

The standard way of deriving the equation of motion is to find the equation for the motion with respect to the Earth. Then some terms drop away against each other, and the result is pretty clean.
(The derivation does not refer to conservation of angular momentum.)

On my website there is a http://www.cleonis.nl/physics/phys256/foucault_pendulum.php" about the Foucault pendulum, including discussion of the derivation.
Most discussions available on the internet are very abstract, just mathematical operations. I tried to make each step tangible, relating it to the physics taking place.

Also three of the Java applets that are available on my website involve the Foucault pendulum. Those applets have two side-by-side display panels, the left panel shows the motion with respect to the inertial coordinate system, the right panel shows the motion with respect to the co-rotating coordinate system.

- "[URL pendulum simulation
[/URL]. In this 3D simulation the calculation is simplified to the case of motion confined to the surface of a sphere.

- http://www.cleonis.nl/physics/ejs/foucault_rod_simulation.php" that is exhibited by the University at Buffalo, State University of New York.

- http://www.cleonis.nl/physics/ejs/circumnavigating_pendulum_simulation.php" . This 2D simulation focusus on the case of how a pendulum is affected when the suspension point is circumnavigating a central axis.

Cleonis
http://www.cleonis.nl

 

[Cleonis]

It is a simple demonstration of geometric phase and can be derived by geometry.

With a Foucault's pendulum, you do not move the pendulum, but the pendulum is moved about the sphere in space by the actual rotation of the Earth. The solid angle that subtends the closed path is related to the actual shift in the pendulum over the closed path.

There is an explanation following these lines in Griffith's Quantum Mechanics textbook I think because geometric phase comes up in Quantum Mechanics too (as Berry's phase for example) but it also plays a role in many other physical problems. I think however, you can derive these results using Coriolis forces in a rotating reference frame.

Using the concept of geometric phase to account for the Foucault pendulum precession will give a good approximation, but ultimately it breaks down.

As the http://www.physics.buffalo.edu/ubexpo/Flash/PendulumModelRotation.html" at the Buffalo exhibition illustrates, the motion of the pendulum bob is not confined to the surface of a sphere, and the geometric phase approximation hinges on confinement to the surface of a sphere.

The geometric phase approximation works well when the length of the pendulum is negligable relative to the radius of circumnavigating motion.
The more general explanation of the Foucault effect is not limited to that, it applies both for the terrestrial case and the tabletop device.

Cleonis

 

[JeffOCA]

I assume that you are referring to the general case, a Foucault pendulum that can be located at any latitude. (The special case of a Foucault pendulum located at one of the poles is trivial.)

Yes

The standard way of deriving the equation of motion is to find the equation for the motion with respect to the Earth. Then some terms drop away against each other, and the result is pretty clean.

I read your article but I don't find where is the derivation of the non rotation of the oscillation plane...

Thanks,
JF

 

[Cleonis]

Yes
I read your article but I don't find where is the derivation of the non rotation of the oscillation plane...

I'm not sure whether I even understand your question.

For comparison, there is no such thing as a derivation of Newton's first law. One can view the first law as an emperical finding, or as an assumption, but being a first principle there's no deriving it.

Likewise, there is no such thing as a derivation of the fact that a polar pendulum will keep swinging in the same direction. it just will; first principles can't be derived.

Cleonis

 

[JeffOCA]

I'm not sure whether I even understand your question.

For comparison, there is no such thing as a derivation of Newton's first law. One can view the first law as an emperical finding, or as an assumption, but being a first principle there's no deriving it.

Likewise, there is no such thing as a derivation of the fact that a polar pendulum will keep swinging in the same direction. it just will; first principles can't be derived.

Cleonis

Sorry if I was not clear enough (I'm not a native english ). I reformulate my question : a pendulum swings always in the same direction (non rotation of the oscillation plane) even if we rotate the pendulum itself.
Is this a principle of mechanics (which cannot be proved) or a result which can be derived by some calculus ?

I hope it's more clear now.

Thanks,
Jeff

 

[Cleonis]

Sorry if I was not clear enough (I'm not a native english ). I reformulate my question : a pendulum swings always in the same direction (non rotation of the oscillation plane) even if we rotate the pendulum itself.
Is this a principle of mechanics (which cannot be proved) or a result which can be derived by some calculus ?

It's the first: it's a principle (as far as we know).

How it was discovered is a nice story.
Foucault was working on something else, and he had clamped a rod of 5mm thick or so and a foot long or so in the chuck of a lathe. He twanged the rod, so it vibrated. Foucault happened to turn the lathe, by hand, and he noticed that the plane of swing of the rod kept the same direction. (See also the following page with pictures made by William Tobin of http://www.cleonis.nl/physics/phys256/foucault_pendulum_intro.php" .)

Reporting about this observation Foucault noted the implication: the plane of swing is in a sense "immaterial". When the rod rotates along its long axis the immaterial plane keeps it direction.

Foucault's observation is not really surprising. For example, take the trajectory of a cannonball moving at great velocity, while spinning along some axis. The cannonball will still move in a straight line, just as a non-spinning ball would. (That is, in vacuum, a spinning ball in the atmosphere has different aerodynamics than a non-spinning ball.) One can say that the direction of linear momentum of the ball as a whole is immaterial.

Still, in the case of a vibrating rod it's somewhat counterintuitive. One could suspect that the bending of the rod will come in somewhere, but it seems not.

Cleonis