(a) Suppose two identical pendulums are coupled by means of a spring with constant k. See Figure 7.R.12. Under the same assumptions made in the discussion preceding Example 3 in Section 7.6, it can be shown that when the displacement angles θ1(t) and θ2(t) are small, the system of linear differential equations describing the motion is
Use the Laplace transform to solve the system when θ1(0) = θ0, θ1‘(0) = 0, θ2(0) = ψ0, θ2‘(0) = 0, where θ0 and ψ0 constants. For convenience let ω2 = g/l, K = k/m.
(b) Use the solution in part (a) to discuss the motion of the coupled pendulums in the special case when the initial conditions are θ1(0) = θ0, θ1‘(0) = 0, θ2(0) = θ0, θ2‘(0) = 0. When the initial conditions are θ1(0) = θ0, θ1‘(0) = 0, θ2(0) = -θ0, θ2‘(0) = 0.
