Consider 2 pendulums with the same length L, but 2 different masses Ma and Mb. They are coupled by a spring of spring constant k which is attached to the bobs (the masses).
a) find the equations of motion
b) find the frequencies and configurations of the normal modes
2. Relevant equations
θ”(t) = -(g/L)sinθ ≈ -(g/L)θ (using small angle approximation to keep the diffeq linear)
3. The attempt at a solution
a) I came up with:
θa”(t) = -(g/L)θa – (k/Ma)(θa – θb)
θb”(t) = -(g/L)θb – (k/Mb)(θb – θa)
b) I tried using θa(t) = θAcos(ωt – ø) and θb(t) = θBcos(ωt – ø) to solve the differential equation but I could not obtain a symmetrical matrix since the masses of the bobs are different.