Normal modes of Mass and two Spings

[xago]

Homework Statement

Consider a mass M whose motion is confined to a flat, smooth two-dimensional surface. Label the locations in 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 of its ends fixed to the point (-L, 0) and the other spring has one of its ends fixed to the point (L, 0). Find the normal modes of this system when ℓ < L.

Homework Equations

F=ma , T =-kx

The Attempt at a Solution

Well first I've my goal is to find the EOM of the system then plug in the solution to the EOM to find the normal modes. Since ℓ < L , then the springs are stretched in equilibrium and provide constant tension of T = -2k(L- ℓ). Taking into account when the mass is displaced a distance x from equilibrium also provides additional tension T = -2kx. Therefor my EOM is

ma = -2k(L- ℓ) + (-2kx)

However when I plug in a solution of the form x(t) = Acos(wt) it doesn't seem easy to solve.
Can someone please tell me if I am on the right track and if my EOM is correct?

 

[vela]

You also need to consider motion in the vertical direction.