The motion of a certain undamped pendulum is described by

Chapter 9, Problem 23

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The motion of a certain undamped pendulum is described by the equationsdx/dt = y, dy/dt = 4 sin x.If the pendulum is set in motion with an angular displacement A and no initial velocity,then the initial conditions are x(0) = A, y(0) = 0.(a) Let A = 0.25 and plot x versus t. From the graph, estimate the amplitude R andperiod T of the resulting motion of the pendulum.(b) Repeat part (a) for A = 0.5, 1.0, 1.5, and 2.0.(c) How do the amplitude and period of the pendulums motion depend on the initialposition A? Draw a graph to show each of these relationships. Can you say anything aboutthe limiting value of the period as A 0?(d) Let A = 4 and plot x versus t. Explain why this graph differs from those in parts (a)and (b). For what value of A does the transition take place?