Chapter 12.18, Problem 8P

Find $\stackrel{\dot{}}{\theta }=\frac{d\theta }{dt}=\frac{d\theta }{du}\frac{du}{dl}\frac{dl}{dt}$ either from equations (18.10) and (15.2) or from equation (18.7) with $B=0.$ Thus show that $\theta =0$ when $J_1(u)=0$ and $\stackrel{\dot{}}{\theta }=0$ when $J_2(u)=0.$ Show that the successive (variable) quarter periods of the lengthening pendulum are $(v/4g)\left(r_2^2-r_1^2\right)$ or $(v/4g)\left(r_1^2-r_2^2\right),$ where $r_1$ and $r_2$ are successive zeros of $J_1$ and $J_2.$ Use a computer or tables to find the needed zeros and calculate several quarter periods (as multiples of $v/(4g)$ ). Observe that an inward swing takes longer than either the preceding or the following outward swing. [This result is proved by Ll. G. Chambers, Proceedings of the Edinburgh Mathematical Society (2) 12, 17-18 (1960).]