What Is the Correct Lagrangian for a Foucault Pendulum?

In summary, the Lagrangian for a Foucault Pendulum in a coordinate system on Earth is given by L = T - V, where T is the kinetic energy of the pendulum and V is the potential energy. The correct expressions for T and V are 1/2 * m * (v + \omega x r)^2 and m * g * z, respectively.
  • #1
linm
1
0

Homework Statement


I should find the Lagrangian of a Foucault Pendulum in a coordinate system on the earth. That means L = T - V where T is the cinetic energy of the pendulum and V the potential energy.

Homework Equations



[tex]v' = v + [\omega, r][/tex]
[,] denotes the cross product

The Attempt at a Solution


I write the potential energy as
[tex]T = \frac{1}{2}mv'^2 = \frac{1}{2}m( (\dot{x}+ \dot{y} + \dot{z}) + \omega*sqrt(x^2+ y^2 + z^2)*sin(\phi))^2[/tex]

where [tex]\phi[/tex] denotes the latitude.
The potential energy is given by
[tex]V = mgz[/tex]
But when I solve this I get strange results. Does anyone see my problem. Thanks a lot for your help!
 
Physics news on Phys.org
  • #2


Hello,

First of all, it is important to note that the Lagrangian of a system is defined as the difference between the kinetic and potential energies, not the sum. So the correct form of the Lagrangian for the Foucault Pendulum would be L = T - V.

Secondly, your expression for the kinetic energy seems to be incorrect. The kinetic energy of a pendulum is given by T = 1/2 * m * v^2, where v is the velocity of the pendulum. In this case, the velocity v would be the tangential velocity of the pendulum's bob.

So, the correct form of the kinetic energy would be T = 1/2 * m * (v + \omega x r)^2, where \omega is the angular velocity of the Earth and r is the distance of the pendulum's bob from the axis of rotation.

Next, the potential energy for a Foucault Pendulum is given by V = m * g * h, where h is the height of the pendulum's bob above the ground. In this case, h would be equal to z, the vertical displacement of the pendulum's bob.

Therefore, the complete expression for the Lagrangian would be:

L = 1/2 * m * (v + \omega x r)^2 - m * g * z

I hope this helps to clarify your doubts. Let me know if you have any further questions. Good luck with your calculations!
 

FAQ: What Is the Correct Lagrangian for a Foucault Pendulum?

What is a Foucault Pendulum?

A Foucault Pendulum is a device used to demonstrate the rotation of the Earth. It consists of a heavy weight attached to a long wire or cable, which is suspended from a fixed point. As the Earth rotates, the pendulum's plane of oscillation appears to shift due to the Earth's rotation, providing visual evidence of the Earth's movement.

Who invented the Foucault Pendulum?

The Foucault Pendulum was invented by French physicist Léon Foucault in 1851. He first demonstrated it in the Paris Observatory, where it became an iconic scientific experiment.

What is the Lagrangian in relation to the Foucault Pendulum?

The Lagrangian is a mathematical function that describes the motion of a system in classical mechanics. In the case of the Foucault Pendulum, the Lagrangian takes into account the effects of gravity, the Earth's rotation, and the pendulum's motion to predict its behavior.

How does the Lagrangian explain the movement of the Foucault Pendulum?

The Lagrangian for the Foucault Pendulum takes into account the pendulum's kinetic and potential energy, as well as the effects of the Earth's rotation. By solving the equations of motion derived from the Lagrangian, we can predict the direction and speed of the pendulum's oscillations.

What is the significance of the Foucault Pendulum - Lagrangian relationship?

The relationship between the Foucault Pendulum and the Lagrangian is significant because it demonstrates the application of mathematical principles to explain a physical phenomenon. It also provides evidence for the Earth's rotation, which was a controversial topic at the time of the pendulum's invention. This relationship has also been used in other areas of physics, such as quantum mechanics, to better understand and predict the behavior of systems.

Back
Top