Foucault's Pendulum

ian2012

Consider a pendulum which is free to move in any direction and is sufficiently long and heavy that it will swing freely for several hours. Ignoring the vertical component both of the pendulum's motion and of the Coriolis force, the equations of motion for the bob are:

[tex]\ddot{x}=-\frac{g}{l}x+2(\omega)cos\theta\dot{y}[/tex]
[tex]\ddot{y}=-\frac{g}{l}y-2(\omega)cos\theta\dot{x}[/tex]

I've found these equations from 'Classical Mechanics - Kibble & Berkshire, 5th Edition'. I don't understand how they are derived?

tiny-tim

Hi ian2012! Welcome to PF!

(have a theta: θ and an omega: ω )

They come from equation (5.25) …


which in turn comes from the definition of the Coriolis force (ignoring the vertical component).

What don't you understand about that?

Cleonis

For me the thing that gives me difficulty in following Foucault pendulum derivations is that the author usually jumps from notation to notation. I see authors switching between index notation, vector notation and parametric notation.

I suggest you comb the internet and textbooks that you can get hold of for derivations, and piece together a picture that you comprehend.

There is a Foucault pendulum equation-of-motion derivation on my own website. I don't claim my derivation is better than the others, but I think it will contribute, as I discuss aspects that other authors tend to gloss over. Also there is a Foucault pendulum Java applet simulation on my website, and in all there are three Foucault related simulations.

The applets feature true simulations, not animations.
- An animation depicts the mathematics of the analytic solution to the equation of motion.
- A simulation takes as input the raw differential equation that relates acceleration to the force(s) that act(s), and then performs numerical analysis to obtain a trajectory.

Cleonis
http://www.cleonis.nl

ian2012

Thanks Cleonis.