Foucault Pendulum: Deriving Equations of Motion

[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:

$$\ddot{x}=-\frac{g}{l}x+2(\omega)cos\theta\dot{y}$$
$$\ddot{y}=-\frac{g}{l}y-2(\omega)cos\theta\dot{x}$$

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

 

[tiny-tim]

Welcome to PF!

Hi ian2012! Welcome to PF!

(have a theta: θ and an omega: ω )

They come from http://books.google.com/books?id=0a...nepage&q=pendulum equations of motion&f=true" …

r'' = -(g/l) r - 2ωcosθk Λ r'​

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 http://www.cleonis.nl/physics/phys256/foucault_pendulum.php" 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.