3. \"The general motion of a pendulum in a coupled-pendulum problem a is the sup

Question

3. "The general motion of a pendulum in a coupled-pendulum problem a is the superposition of two modes. Assume modes are present with equal amplitudes and
assume that there is zero. Examine oscillation amplitudes of each pendulum as functions
of time."

I've derived the equations of motion for a pair of pendulums coupled by a spring with equal string length and a mass of m1 and m2 and spring constant k as

m1x1'' = -m1(g/l)x1 - k(x1 - x2)
m2x2'' = -m2(g/l)x2 - k(x2 - x1)

They can be decoupled when added or subtracted from each other, so after making the substitutions (w1)^2 = g/l, (w2)^2 = (g/l - 2k/M), q1 = x + y and q2 = x - y, the k(x - xo) values cancell out and I get:

q^2 = -w^2phi

After using euler's identity, I get this:

q1 = A1cos(w1t + phi1)
q2 = A2cos(w2t + phi2)

I'm not too sure if I did this one properly either, but if my ODE solution is correct, how do I use this to find the actual eigenmodes and amplitude without an initial value problem?

Explanation / Answer

your equations seem you to be correct.to find eigenmodes and amplitude you definitely need an initial value because depending upon the intial force the amplitude will change.

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