(a) Use the Laplace transform and the information given in Example 3 to obtain the solution (8) of the system given in (7).

(b) Use a graphing utility to graph θ₁(t) and θ₂(t) in the tθ-plane. Which mass has extreme displacements of greater magnitude? Use the graphs to estimate the first time that each mass passes through its equilibrium position. Discuss whether the motion of the pendulums is periodic.

(c) Graph θ₁(t) and θ₂(t) in the θ₁θ₂-plane as parametric equations. The curve defined by these parametric equations is called a Lissajous curve.

(d) The positions of the masses at t = 0 are given in Figure 7.6.5(a). Note that we have used 1 radian ≈ 57.3°. Use a calculator or a table application in a CAS to construct a table of values of the angles θ₁ and θ₂ for t = 1, 2, . . . , 10 s. Then plot the positions of the two masses at these times.

(e) Use a CAS to find the first time that θ₁(t) = θ₂(t)vand compute the corresponding angular value. Plot thevpositions of the two masses at these times.

(f) Utilize the CAS to draw appropriate lines to simulate the pendulum rods, as in Figure 7.6.5. Use the animation capability of your CAS to make a “movie” of the motion of the double pendulum from t = 0 to t = 10 using a time increment of 0.1. Express the coordinates (x₁(t), y₁(t)) and (x₂(t), y₂(t)) of the masses m₁ and m₂, respectively, in terms of θ₁(t) and θ₂(t).

Figure 7.6.5

Transcribed Image Text:

(a) t= 0 (b) t = 1.4 (c) t = 2.5 (d) t = 8.5 %3D