Two simple pendulum `A` and `B` having lengths `l` and `l//4` respectively are released from the positions as shown in figure. Calculate the time after which the release of the two strings become parallel for the first time. Angle `theta` is very small.

Answer

`(pi)/(3)sqrt((l)/(g))`

Explanation

The angular position of pendulum `1` and `2` are (taking angles to the right of reference line `xx'` to be positive)
`theta_(1) = theta cos((4pi)/(T)t)`, [where `T = 2pi sqrt((l)/(g))`]
`theta_(2) = -theta cos ((2pi)/(T))t = cos ((2pi)/(T) t + pi)`
`F` for the strings to be parallel for the first time
or `cos((4pi)/(T)t) = cos ((2pi)/(T)t + pi)` ltbr. `:. (4pi)/(T)t = 2n pi +- ((2pi)/(T)t + pi)`
for `n = 0, t = (T)/(2)`
for `n = 1, t = (T)/(6), (3T)/(2)`
`:.` Both the pendulum are parallel to each other for the first time after `t = (T)/(6) = sqrt((l)/(g))`