Pendulum 2

Problem 2.

Approximate the motion of simple pendulum in case of $\theta(0)=0$, $\theta'(0)=v$.

Show Solution

We have seen in Problem 1 the differential equation of the motion of the simple pendulum is $$\theta\,''(t)+\frac{g}{\ell}\sin(\theta(t))=0.$$ Substituting $t:=0$ we obtain $$\theta\,''(0)+\frac{g}{\ell}\sin(\theta(0))=0.$$ It follows $$\theta\,''(0)=0.$$ Taking the derivative of the equation we have $$\theta\,'''(t)+\frac{g}{\ell}\cos(\theta(t))\theta\,'(t)=0.$$ Substituting $t:=0$ we obtain $$\theta\,'''(0)+\frac{g}{\ell}\theta\,'(0)=0.$$ It follows $$\theta\,'''(0)=-\frac{g}{\ell}v.$$ Then we get $$\begin{align*} \theta(t) &=\theta\,(0)+\theta\,'(0)t+\frac{1}{2!}\theta\,''(0)t^2+\frac{1}{3!}\theta\,'''(0)t^3+\ldots\\ &=vt-\frac{g}{6\ell}vt^3+\ldots\qquad\blacksquare \end{align*}$$