Question

The first overtone frequency of a closed organ pipe is equal to the second overtone frequency of an open organ pipe. If the length of the open pipe is $80\text{cm}$, what is the length of the closed organ pipe?

• A. $20\text{cm}$
• B. $40\text{cm}$
• C. $60\text{cm}$
• D. $80\text{cm}$

Answer 1

The correct option is B $40\text{cm}$

Modes of vibration of air column in a closed organ pipe is given by,
$f_n=\frac{(2n+1)v}{4L_C}$

So for first overtone or second harmonic
$f_1=\frac{3v}{4L_c}$

Modes of vibration of air column in an open organ pipe is given by,
$f_n=\frac{nv}{2L_o}$

For second overtone or third harmonic
$f_3=\frac{3v}{2L_o}$

From the data given in the question,

$\frac{3v}{4L_c}=\frac{3v}{2L_o}$

$\Rightarrow 2L_c=L_o$

Given that,
Length of the open organ pipe, $L_o=80\text{cm}$
$\Rightarrow L_c=\frac{80}{2}=40\text{cm}$.

Thus, option (b) is the correct answer.