(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm trying to derive an expression for the angular frequency of a set of springs hanging vertically down with masses in between them. 3 cases:

2 equal masses and springs.

2 different masses and equal springs

3 equal masses and springs

2. Relevant equations

ω0=√(k/m)

F= -kx

3. The attempt at a solution

i know its a combination of 2 fundamental modes

and that the answer I'm aiming for for the first case is (ω1^2)=((3+√5)/2)(ω0)^2 and (ω2^2)=((3-√5)/2)(ω0)^2

But i can't get there.

I did out a force diagram for the general case of 2 uneven masses and 2 springs which i hoped would do for case 1 and 2 (case1 just being a special case of case2) and came up with 2 equations of motion

ma1=-2kx1 +kx2 +(m1+m2)g

ma2=-kx2 +kx1 + m2g

I really don't know where to go from here, can someone please point me in the right direction. Im looking for someone to give me the derivation but you know that feeling when you hit a brick wall??? ya :P

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# Derivation of angular frequency of coupled Oscillator-vertical spring

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