Chapter 7.1, Problem 21P

Consider the pendulum equations $\frac{dx}{dt}=y,\frac{dy}{dt}=-6\sin x$.

If the pendulum is set in motion from its downward equilibrium position with angular velocity $\upsilon$, then the initial conditions are $x\left(0\right)=0,y\left(0\right)=\upsilon$.

(a) Plot $x$ versus $t$ for $\upsilon =3$ and also for $\upsilon =6$. Explain the differing motions of the pendulum that these two graphs represent.

(b) There is a critical value of $\upsilon$, which we denote by ${\upsilon }_c$, such that one type of motion occurs for $\upsilon <{\upsilon }_c$ and a qualitatively different type of motion occurs for $\upsilon >{\upsilon }_c$. Determine the value of ${\upsilon }_c$ to two decimal places.