Chapter 6.4, Problem 252E

The following exercises deal with Fresnel integrals.

252. [T] An equivalent formula for the period of a pendulum with amplitude ${\theta }_{\max }$ is T( ${\theta }_{\max }$ ) = $2\sqrt{2}\sqrt{\frac{L}{g}}{\displaystyle {\int }_0^{{\theta }_{\max }}\frac{d\theta }{\sqrt{\cos \theta }-\cos ({\theta }_{\max })}}$where L is the pendulum length and g is the gravitational acceleration constant. When $\begin{array}{l}\\ {\theta }_{\max }=\frac{\pi }{3}\end{array}$ we get $\frac{1}{\sqrt{\cos t-1/2}}\approx \sqrt{2}\left(1+\frac{t^2}{2}+\frac{t^4}{3}+\frac{181t^6}{720}\right)$ . Integrate this approximation to estimate in terms of L and g. Assuming g = 9.806 meters per second squared, find an approximate length L such that

   $T\left(\frac{\pi }{3}\right)$ = 2 seconds.