# Coupled oscillators

by trelek2
Tags: coupled, oscillators
 P: 88 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?