(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.

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