Question

Difference in frequencies between $3^{\text{rd }}$ overtone of closed pipe and $5^{\text{th }}$ harmonic of the same pipe is $400Hz$. Further, $3^{\text{rd }}$ harmonic of this closed pipe is equal to $6^{\text{th }}$ harmonic of another organ pipe.
If speed of sound is $330m/s.$ The lengths of closed pipe and open pipe are :

• A. 0.4125 m, 1.65 m
• B. 3.3 m, 1.65 m
• C. 0.825 m, 0.825 m
• D. 1.65 m, 0.825 m

Answer 1

Answer: (A)

For closed pipe,
$3^{rd}$ overtone $=7^{\text{th }}$ harmonic $=n_7=\frac{7v}{4l_c}$
$5^{\text{th }}$ harmonic $n_5=\frac{5v}{4l_c}$
$n_7-n_5=\frac{2v}{4l_c}=400Hz$
$\Rightarrow n_0=\frac{v}{4l_c}=200Hz$
(fundamental frequency of closed pipe)
Now $3^{\text{rd }}$ harmonic of closed pipe is equal to $6^{\text{th }}$ harmonic of open pipe
$\Rightarrow \frac{v}{4l_c}=\frac{6v}{2l_0}$
$\Rightarrow \frac{v}{l_0}=\frac{v}{4l_c}=200Hz$
Fundamental frequency of open pipe $=\frac{v}{2l_0}=100Hz$
Further, $l_c=\frac{330}{4\times 200}=0.4125m$
$l_0=\frac{330}{2\times 100}=1.65m$