# Problem - Lagrange function for mass on springs attached on a frame

1. Apr 24, 2014

### asynja

Hello! I have a problem in classical mechanics that I'm unable to solve. Any help would be much appreciated since we have a partial exam tomorrow. :(

1. The problem statement, all variables and given/known data

There's a picture of the problem in the attachment

A mass m, which is on a light rod (lenght d), is attached to a square frame with sides 2d and to two springs with coefficient k, as shown on the picture. The lenght of unstreched springs is d. We put the frame on the surface and spin it around the point s with angular speed ω. We assume that there is no friction between mass and the surface and that springs can slide up and down the frame (they are not fixated).
Find the Lagrange function and the equation of motion for the mass.
Where are the stationary positions? For which coefficients k is the stationary position stable when the springs are the least streched?

2. Relevant equations

Lagrange's formalism

3. The attempt at a solution

T=1/2mω2 , V=kx2*2 , but then I don't know how to take into the account the effect that rod with lenght d has on the springs (it contributes that they are stretched more, right?)

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2. Apr 24, 2014

### paisiello2

I think it only acts as a constraint to keep the mass on a semi-circle.

3. Apr 25, 2014

### BiGyElLoWhAt

The d's make me think you need to take into account GPE. The way I was taught to setup a lagrangian is to NOT start in polar, but to start in cartesian and work you're way over to polar (for the position of the mass and thus velocities). There are "mysterious" cross-terms that pop up when you do it that way. Maybe there won't be in this system, but I don't know without actually setting it up.
Also, $V_{spring}=\frac{1}{2}k\Delta x^2$ and 2 springs gives you $k\Delta x^2$, not $2k\Delta x^2$

paisiello is correct, since the rod wasn't given a mass, you can simply act as though it's an imaginary rod, but I'm not even sure that you need to account for it in your constraints. You just need d for you moment of inertia.

4. Apr 25, 2014

### voko

How many degrees of freedom does the system have? What sort of coordinates can describe those degrees of freedom conveniently? Can you express the kinetic and the potential energies of the system in those coordinates?