What Are the Normal Modes of a Mass-Spring System in 2-D?

[macker1]

Homework Statement

A mass M is confined to move on a smooth, flat, two-dimensional surface. Label the locations on this surface using the Cartesian coordinates (x,y). The mass is attached to two identical springs, each of length l and spring constant k. One spring has one end fixed to the point (-L, 0) and the other spring has one end fixed to the point (L, 0)

1) Find the normal modes of this system when l<L.
2) At t=0, the mass is released at rest from the point (0.1L, -0.2L). What is the position of the mass for all subsequent times?

Homework Equations

The Attempt at a Solution

I used the Lagrangian method to find the equations of motion (see attachment). I'm sure there is something I could do to simplify these two equations by making some sort of approximation. But I'm not sure what to do. Any help would be appreciated. Thanks!

 

[BigJDubs]

[1]: https://i.stack.imgur.com/q6U9y.jpgFor 1), you can solve the equation of motion for the normal modes by solving two uncoupled equations of motion. The equations would take the form:m*x_ddot + k*(x-L) = 0m*y_ddot + k*(y+L) = 0where x and y are the coordinates of the mass, m is the mass of the object and k is the spring constant.By solving these equations, you will get the normal modes as a function of time:x(t) = A*cos(wt) + Ly(t) = B*sin(wt) - Lwhere A and B are constants determined from the initial conditions and w is the angular frequency of the system.For 2), you can use these normal modes to find the position of the mass at any subsequent time. Using the initial conditions given, you can determine the values of A and B. Then, you can plug in any time t to find the position of the mass. The position would be given by:x(t) = A*cos(wt) + Ly(t) = B*sin(wt) - L