Movement of a Ring on a Parabolic Track in Lagrangian Mechanics

[mooeypoo]

I'm trying to solve a rather complicated question I was faced with using Lagrangians (this isn't homework, it's a challenge I want to solve, I just figured this is the best forum for that type of question).

 

In order to do that, I've simplified the physical situation to show a flat ring being pulled (from the center, for simplicity) on a track shaped as a negative parabola. The ring itself represents a part of a spring, so the pulling force is relative to the spring constant.

 

I've simplified it a bit for starters, see the following schematics:

 

 

m would be the mass of the ring. The ball at the far end represents whatever mass is pulling on the ring and creating the elastic energy of the spring that pulls the ring through the parabola.

(The ring represents a single rotation of a spring.)

 

So I started by stating the location x and y of the center of the ring:

 

[math]y = h-x^2[/math]

[math]x = x[/math]

 

and so my x is a function of time, while y depends on x, So:

 

[math]\dot{x} = 1[/math]

[math]\dot{y} = -2x\dot{x}[/math]

 

And my energies:

 

[math]T = \frac{1}{2}m ( \dot{x}^2 +\dot{y}^2 ) = \frac{1}{2}m ( 1 + 4x^2\dot{x}^2 )[/math]

 

[math]U = mg(h-x^2) + \frac{k}{2} ( x^2 + (h-x^2)^2 ) = mgh - mgx^2 + \frac{k}{2} ( x^2 + h^2 -2hx^2 + x^4 )[/math]

 

And so, the Lagrangian:

[math]L = T-U = \frac{1}{2}m ( 1 + 4x^2\dot{x}^2 ) - mgh - mgx^2 - \frac{k}{2} ( x^2 + h^2 -2hx^2 + x^4 )[/math]

 

First off, seeing as I learned Lagrangians 3 semesters ago, I'm not too sure I'm doing this right, so assistance would be much appreciated.

 

Second, I now have a problem with representing the moment of inertia for the disk. I know how to do that with a turning wheel (or ball) but I am not sure how to do that for the ring that is moving in this direction.

 

 

Is my initial representation correct? and.. well.. how do I represent the moment of inertia if I want to treat the ring as a rigid body moving at that "track"?

 

 

Thanks!

 

~moo

[Xittenn]

Composite bodies

 

If a body can be decomposed (either physically or conceptually) into several constituent parts, then the moment of inertia of the body about a given axis is obtained by summing the moments of inertia of each constituent part around the same given axis.

 

-Wikipedia