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

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 $T=2\pi \sqrt{\frac{l}{g}}]$
${\theta }_2=-\theta \cos \left(\frac{2\pi }{T}t\right)=\cos \left(\frac{2\pi }{T}t+\pi \right)$

For the strings to be parallel for the first time, ${\theta }_1={\theta }_2$
$\cos \left(\frac{4\pi }{T}t\right)=\cos \left(\frac{2\pi }{T}t+\pi \right)$
$\therefore \frac{4\pi }{T}t=2n\pi \pm \left(\frac{2\pi }{T}t+\pi \right)$
for $n=0,t=\frac{T}{2};$
for $n=1,t=\frac{T}{6},\frac{3T}{2}$
Both the pendulum are parallel to each other for the first time after
$t=\frac{T}{6}=\frac{\pi }{3}\sqrt{\frac{l}{g}}=1s$