This problem extends to a damped pendulum. The equations of motion are dx dt = y, dy dt

Chapter 9, Problem 22

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This problem extends to a damped pendulum. The equations of motion are dx dt = y, dy dt = 4 sin x y, where is the damping coefficient, with the initial conditions x(0) = 0, y(0) = v. G a. For = 0.25, plot x versus t for v = 2 and for v = 5. Explain these plots in terms of the motions of the pendulum that they represent. Also explain how they are related to the corresponding graphs in 21a. b. Estimate the critical value vc of the initial velocity where the transition from one type of motion to the other occurs. c. Repeat part b for other values of and determine how vc depends on . 2