Problem 1. Consider two identical plane pendulums (each of length L and mass m) that are joined by a massless spring (force constant k) as shown. The positions of the pendulums are specified by the angles θ1 and θ2 shown. The natural length of the spring is equal to the distance between the vertical.
Problem 2. Consider the system in the previous problem. Its equation of motion takes the matrix form:
M = -K,
where:
M = [0 w^2 w^3]
[w^2 w^p Vg/L]
[w^s Vk/m]
a) Find the two normal frequencies and normal modes. Hint: remember that a-b = (a+b) - (a-b).
b) (Extra credit) Describe the motion of the two normal modes.