Chapter 7.1, Problem 20P

The motion of a certain undamped pendulum is described by the equations $\frac{dx}{dt}=y,\frac{dy}{dt}=-4\sin x$.

If the pendulum is set in motion with an angular displacement $A$ and no initial velocity, then the initial conditions are $x\left(0\right)=A,y\left(0\right)=0$.

(a) Let $A=0.25$ and plot $x$ versus $t$. From the graph, estimate the amplitude $R$ and period $T$ 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 pendulum’s motion depend on the initial position $A$? Draw a graph to show each of these relationships. Can you say anything about the limiting value of the period as $A\to 0$?

(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?