Question

Two simple pendulums $A$ and $B$ having lengths $l$ and $l / 4$ respectively are released from the position as shown in figure. The time after which the release of the two strings become parallel for the first time is ___

Answer 1

The angular position of pendulum $1$ and $2$ are
(taking angles to the right of reference line $x x^{\prime}$ to be positive)
$\theta_{1}=\theta \cos \left(\frac{4 \pi}{T} t\right) $ [where $\left.T=2 \pi \sqrt{\frac{l}{g}}\right]$
$\theta_{2}=-\theta \cos \left(\frac{2 \pi}{T} t\right)=\cos \left(\frac{2 \pi}{T} t+\pi\right)$

For the strings to be parallel for the first time, $\theta_{1}=\theta_{2}$
$\cos \left(\frac{4 \pi}{T} t\right)=\cos \left(\frac{2 \pi}{T} t+\pi\right)$
$\therefore \frac{4 \pi}{T} t=2 n \pi \pm\left(\frac{2 \pi}{T} t+\pi\right)$
for $n=0, t=\frac{T}{2} ;$
for $n=1, t=\frac{T}{6}, \frac{3 T}{2}$
Both the pendulum are parallel to each other for the first time after
$t=\frac{T}{6}=\frac{\pi}{3} \sqrt{\frac{l}{g}}=1 \,s$