Two simple pendulums A and B having length $l$ and  $\frac{l}{4}$respectively are released at time  from the position as shown in the figure. The time after which the two strings become parallel to each other for the first time is $t=\frac{\pi }{b}\sqrt{\frac{l}{g}}$where b is positive integer. Find the value of b (Angle  $\theta$ is very small and assume that the pendulums do not disturb each other’s motion).

The angular position of pendulum 1 and 2 are (taking angles to the right of reference line  to be positive)

${\theta }_1=0\cos \left(\frac{4\pi }{T}t\right)[\text{where}T=2\pi \sqrt{\frac{l}{g}}]$

${\theta }_2=-0\cos \left(\frac{2\pi }{T}t\right)=\cos (\frac{2\pi }{T}t+\pi )$

For the strings to be parallel for the first time

or $\cos \left(\frac{4\pi }{T}t\right)=\cos (\frac{2\pi }{T}t+\pi )$

$\therefore \frac{4\pi }{T}t=2n\pi \pm (\frac{2\pi }{T}t+\pi )$

for  $\mathrm{n}=0,t=\frac{T}{2}$

for  $\mathrm{n}=1,$ $t=\frac{T}{6},\frac{3T}{2}$

$\therefore$ Both the pendulum are parallel to each other for the first time after

$t=\frac{T}{6}=\frac{\pi }{3}\sqrt{\frac{l}{g}}$