Question

An open pipe, when closed at one end, can resonate in its third harmonic with a frequency which is $100\,Hz$ more than its fundamental frequency as an open pipe. If the fundamental frequency of the pipe when it is open at both ends is $50\,nHz$ , then the value of $n$ is

Answer 1

Length of the organ pipe is same in both the cases. Fundamental frequency of open pipe is $f_1=\frac{v}{2 l}$ Frequency of third harmonic of closed pipe will be.
$f_2=3 \frac{v}{4 l}$
Given that, $f_2=f_1+100$
$\text { or } f_2-f_1=100$
$\text { or } \frac{3}{4} \frac{v}{l}-\frac{1}{2} \frac{v}{l}=100$
$ \Rightarrow \frac{v}{4 l}=100 $
$\therefore \frac{v}{2 l} \text { or } f_1=200 Hz =50\, nHz$
$ \Rightarrow n=4$
Therefore, fundamental frequency of the open pipe is $200 Hz$. and $n=4$