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

[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. :(

Homework Statement

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 length 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?

Homework Equations

Lagrange's formalism

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 length d has on the springs (it contributes that they are stretched more, right?)

 

[paisiello2]

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

 

[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.

 

[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?

 

[HalfLostAndSearching]

Hi there,

Thank you for reaching out for help with your classical mechanics problem. It sounds like you are working on a problem involving a mass attached to springs on a rotating frame, which is a common problem in introductory mechanics courses. Based on the information provided, it seems like you have a good understanding of the problem and have correctly identified the relevant equations (kinetic and potential energy) to use in finding the Lagrange function.

To incorporate the effect of the rod with length d on the springs, you will need to consider the potential energy contribution from the rod as well. This can be done by breaking up the potential energy into two components - one from the springs and one from the rod. The potential energy from the rod can be calculated using the formula for gravitational potential energy, since the rod is acting as a pivot point for the mass. You can then add this potential energy to the potential energy from the springs to get the total potential energy of the system.

Once you have the Lagrange function, you can use the Euler-Lagrange equations to find the equations of motion for the mass. These equations will tell you how the position of the mass changes with time, taking into account the effects of the rotating frame and the springs. To find the stationary positions of the mass, you will need to set the equations of motion equal to zero and solve for the position of the mass. These positions will correspond to points where the mass is not moving, or is moving with a constant velocity.

To determine the stability of these stationary positions, you can use the second derivative test. This involves taking the second derivative of the potential energy with respect to the position of the mass and evaluating it at the stationary points. If the second derivative is positive, the stationary point is stable, and if it is negative, the stationary point is unstable.

I hope this helps you in solving your problem. Good luck on your exam tomorrow!