Foucault Pendulum Question

In summary, the conversation is discussing the tangential velocity of a pendulum released at the North Pole and its potential effects on the experiment. The conversation also touches on the idea of analyzing the situation from an inertial frame and the use of Lagrangean in finding a solution.
  • #1
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If the pendulum on the North pole is released to start it via burning the separation string, (to avoid adding an additional tangental force), wouldn't the pendulum at the time of release, already have a tangential velocity? does that effect the outcome of the experiment ? since the plane of movement is supposed to be fixed in space (an inertial frame of refence), if it has a some velocity at release as it would, should be considered , correct?
 
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  • #2
Of course, seen from an observer in the rest frame of the center of the Earth (which can be taken as an inertial frame for this purpose), the pendulum has a tangential velocity since it has none in the rest frame of the observer on Earth. This you have to take into account whenever you want to solve the Foucault-pendulum equation of motion in the center-of-Earth inertial frame. Interestingly enough, I've never seen a solution of the problem in this frame. Maybe it's worth while studying it.

Note that in general due to the centrifugal force in the surface-of-the-Earth frame the stable equilibrium position of the pendulum is not along the direction of the Earth's radius through the point where the pendulum is fixed either, though in your special case where the fixed point is above the North Pole that's the case!
 
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  • #3
The point here is, how to we give credibility to this experiment when those that might not understand it, will object to the error possibly being in one of the control variables. now, my knee jerk answer might be that the tangential velocity is so small and would only add a path shape variant and would not effect the fixed plane motion by adding precession to it. With both being very slight, it is expected to object to this as being a possible reason for the apparent precession around the 360 degree of position points.
 
  • #4
Hi.
On North or South pole, set the ceiling rotating at rate of 1 cycle/24H with reverse direction against the Earth rotation around the pole axis ,i,e, still in a inertial reference frame. Set the end of separation string on the ceiling and burn it. I hope it would help you to solve your case.
 
  • #5
that's what I was thinking, but what if you don't... can it add the amount of precession to match the Earth rotation? (or some factor.) doesn't seem like other tests completed have seen any issues with extreme errors.
 
  • #6
Hi.
Say a pendulum in a reference frame of reference. It moves to and fro in a plane. Let it have initial velocity tangent to the plane when it is at the extreme point. How this initial tangential velocity changes the motion? This motion in the frame of reference easily would be interpreted in a rotating frame of reference. I set the question to investigate how tangential speed effects but have not investigated it by myself.
 
  • #7
mitochan said:
Say a pendulum in a reference frame of reference. It moves to and fro in a plane. Let it have initial velocity tangent to the plane when it is at the extreme point. How this initial tangential velocity changes the motion? This motion in the frame of reference easily would be interpreted in a rotating frame of reference. I set the question to investigate how tangential speed effects but have not investigated it by myself.
The situation is most easily analyzed from the inertial frame. We have an object moving under the influence of a field that goes as ##F=-k\vec{r}##. This force law can equally well be broken down into components: ##F_x=-kx## and ##F_y=-ky##. The forces in the two component directions are independent and give rise to a simple differential equation. The solution to the equation is simple harmonic motion.

Since the solution for x and the solution for y are independent, periodic and share the same period. It follows that the orbit is a closed path. It does not precess -- assuming zero drag and the ideal ##F=-k\vec{r}## force law.
 
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  • #8
Thanks for advice on the case of my previous post.

I think Lagrangean of the case is
[tex]L=\frac{ml^2}{2}(\dot{\theta}^2+\dot{\phi}^2sin^2\theta)+mgl \ cos \theta[/tex]
where parameters are those of polar coordinates and gravity accerelation works +z direction. It seems tiresome to get solution.
 
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  • #9
Correction: spherical coordinates 
 

1. What is a Foucault Pendulum?

A Foucault Pendulum is a device used to demonstrate the Earth's rotation. It consists of a long pendulum with a heavy weight at the end that swings back and forth in a fixed plane. As the Earth rotates underneath it, the pendulum appears to change direction, demonstrating the Earth's rotation.

2. How does a Foucault Pendulum work?

A Foucault Pendulum works due to the Coriolis effect, which is the apparent deflection of moving objects when viewed from a rotating frame of reference. As the Earth rotates, the pendulum's plane of swing remains fixed while the Earth moves underneath it, causing the pendulum to appear to change direction.

3. Who invented the Foucault Pendulum?

The Foucault Pendulum was invented by French physicist Léon Foucault in 1851. He first demonstrated it at the Paris Observatory, where it remains on display today.

4. What is the significance of the Foucault Pendulum?

The Foucault Pendulum is significant because it provides visual proof of the Earth's rotation, which was a controversial concept at the time of its invention. It also helped to develop the understanding of the Coriolis effect and its impact on moving objects.

5. Can the Foucault Pendulum be used to determine one's location on Earth?

No, the Foucault Pendulum is not accurate enough to determine one's location on Earth. It only demonstrates the Earth's rotation and does not provide any information about specific locations on the planet.

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