Question

The fundamental frequency in an open organ pipe is equal to the third harmonic of a closed organ pipe. If the length of the closed organ pipe is 20cm, the length of the open organ pipe is
(A) 12.5 cm
(B) 13.2 cm
(C) 16 cm
(D) 8 cm

Answer 1

There are broadly two cases for an organ pipe, one is when the pipe is open at both the ends and the other is when the pipe is closed at one of the ends. If both the ends of the pipe are open then the instrument is said to be an open-end air column. We can find the frequency of the sound that is being produces and the fundamental frequency when the organ pipe is filled with some gas is given by the formula $v=\sqrt{\dfrac{RT}{M}}$ where M is the molar mass of the gas.

Complete step by step answer:
The fundamental frequency is given by the formula $\dfrac{v}{2l}$. This is for the open organ pipe while the third harmonic for closed one is given by $\dfrac{3v}{4l'}$.
According to question,
$\Rightarrow \dfrac{v}{2l}=\dfrac{3v}{4l'}$
$\Rightarrow l=\dfrac{2l'v}{3}$, given the length of the closed organ pipe is 20 cm
$\Rightarrow l=\dfrac{2\times 20}{3}$
$\therefore l=13.2cm$

So, the correct option is B.

Note: Also, the fundamental frequency of an organ pipe depends upon the temperature. the positions of the nodes and antinodes of the standing waves is responsible for different sounds produced from the pipes.The general formula for finding the nth harmonic fundamental frequency is $\dfrac{(2n-1)v}{4l}$, when we are talking about the third harmonic then the value of n is 2, then becomes $\dfrac{(2\times 2-1)v}{4l}=\dfrac{3v}{4l}$