A closed pipe and an open pipe of same length produce 2 beats, when they are set into vibration…
A closed pipe and an open pipe of same length produce 2 beats, when they are set into vibration simultaneously in their fundamental mode. If the length of the open pipe is halved and that of closed pipe is doubled, and if they are vibrating in the fundamental mode, then the number of beats produced is
4
7
2
8
Solution
For closed organ pipe, fundamental mode, $n_1 = \frac{v}{4l}$ ...(i)
and for open organ pipe, $n_2 = \frac{v}{2l}$ ...(ii)
According to the question, $n_2 - n_1 = 2\text{ (beats)}$
$\Rightarrow \frac{v}{2l} - \frac{v}{4l} = 2 \Rightarrow \frac{2v - v}{4l} = 2 \Rightarrow v = 8l$ ...(iii)
When open organ pipe is halved and closed organ pipe is doubled, let the two produce $n$ beats.
$\frac{v}{l} - \frac{v}{8l} = n \Rightarrow \frac{8v - v}{8l} = n$
$\Rightarrow \frac{7v}{8l} = n \Rightarrow n = \frac{7(8l)}{8l} = 7$