Coupled Oscillators: Masses m and 2m in 3l_0 String

[trelek2]

The problem is:
A mass m and a mass 2m are attached to a light string of unstretched length 3l _{0}, so as to divide it into 3 equal segments. The string is streched between rigid supports a distance 3l \textgreater 3l _{0} apart and the masses are free to oscillate longitudinally. The oscillations are of small enough amplitude that the string is never slack. The tension in each segment is k times the extension. The masses are initially displaced slightly in the same direction so that mass m is held at a distance \sqrt{3} -1 further from its equilibrium position than the mass 2m. They are released simultaneously from rest.

The task is to show that they oscillate in phase and explain why.

I have found the general solution and applied the initial conditions and found the solution for this particular case to be:
{x _{1} \choose x _{2} } = {-1 - \sqrt{3} \choose 1} ( \frac{- \sqrt{3} }{2}(a+1)+ \frac{1}{2})cos \omega _{1} t+ {1- \sqrt{3} \choose 1} (a+ \frac{ \sqrt{3} }{2} (a+1)- \frac{1}{2} )cos \omega _{2}t
where I set a to be the initial displacement of mass 2m.
Is my answer correct and how do I show that the oscillate in phase?

 

[Bob S]

What are the length(s) of the two vertical string(s) between the masses and the horizintal string?

A friend of mine spent many many hours building a very tall grandfather clock, It had a long pendulum and a weight hanging on a chain to supply mechanical power. He put in in a prominent place in his living room, on a high-pile wall-to-wall carpet. He told me that he was having a problem with the clock. Whenever the weight on the chain dropped to a point where it was level with the pendulum, it slowly began swinging back and forth, in sync with the pendulum. He asked me what was wrong with the clock and how to fix it. He was an engineer. Should I show him the above equation?

 

[trelek2]

There are no vertical strings in this case. This just one horizontal string with two masses attached to it.