Finding Normal Modes (completely stumped)

[retro10x]

Homework Statement

Two horizontal frictionless rails make an angle θ with each other. Each rail has a bead of mass m on it and the beads are connected by a spring with spring constant k and relaxed length=zero.Assume that one of the rails is positioned a tiny distance above the other so that the beads can pass freely though the crossing. Find the Normal Modes.

Homework Equations

the only force involved is the spring force so:
Fs=-k(Δx)=ma
where Δx=x1sin(θ/2)+x2sin(θ/2)

The Attempt at a Solution

so for each bead (x1,x2):

a1+(k/m)(x1sin(θ/2)+x2sin(θ/2))=0
a2-(k/m)(x1sin(θ/2)+x2sin(θ/2))=0

I guess x1=Ae(iwt) and x2=Be(iwt) and get this:

-Aw^2 +(ksin(θ/2)/m)(A+B)=0
-Bw^2 -(ksin(θ/2)/m)(A+B)=0

Did I set up the equation incorrectly? Finding Normal Modes generally confuses me and this is about as far as I can get, help appreciated!

 

[Spinnor]

Aren't there just two modes. X_1 = X_2 and X_1 = - X_2 ??

They both move together or they both move in oppositely?