Question 8: Waves

Question

The fundamental frequency of an open organ pipe is $$600 \mathrm{~Hz}$$. The first overtone of this pipe has same frequency as first overtone of a closed organ pipe. If speed of sound is $$330 \mathrm{~m} / \mathrm{s}$$, then the length of a closed organ pipe is

Accepted Answer

To solve this problem, we need to understand the relationship between the frequencies of open and closed organ pipes and how their lengths are connected to these frequencies.
Firstly, letÃ¢ÂÂs consider the open organ pipe. The fundamental frequency of an open organ pipe is given by:
$$ f_1 = \frac{v}{2L_1} $$
where $ f_1 = 600 \, 	ext{Hz} $ is the fundamental frequency, $ v = 330 \, 	ext{m/s} $ is the speed of sound, and $ L_1 $ is the length of the open organ pipe.
Hence:
$$ 600 = \frac{330}{2L_1} $$
Solving for $ L_1 $, we get:
$$ L_1 = \frac{330}{2 	imes 600} = \frac{330}{1200} = 0.275 \, 	ext{m} = 27.5 \, 	ext{cm} $$
For the open organ pipe, the first overtone frequency will be:
$$ f_2 = \frac{2v}{2L_1} = \frac{v}{L_1} = \frac{330}{0.275} \, 	ext{m} = 1200 \, 	ext{Hz} $$
Now, considering the closed organ pipe, the first overtone frequency for the closed organ pipe (which corresponds to the third harmonic) is given by:
$$ f'_2 = \frac{3v}{4L_2} $$
We are given that the first overtone of the open organ pipe (which is $ 1200 \, 	ext{Hz} $) is equal to the first overtone of the closed organ pipe. Hence:
$$ 1200 = \frac{3 	imes 330}{4L_2} $$
Solving for $ L_2 $, we get:
$$ 1200 = \frac{990}{4L_2} $$
Rearranging:
$$ 4L_2 = \frac{990}{1200} $$
$$ L_2 = \frac{990}{4800} \, 	ext{m} = 0.20625 \, 	ext{m} = 20.625 \, 	ext{cm} $$
Given the options, there appears to be a minor discrepancy. However, the closest value is:
Option A: 21 cm

Answer

21 cm

Answer

37 cm

Answer

31 cm

Answer

80 cm

Waves

The fundamental frequency of an open organ pipe is $600 \mathrm{~Hz}. The first overtone of this pipe has same frequency as first overtone of a closed organ pipe. If speed of sound is 330 \mathrm{~m} / \mathrm{s}$, then the length of a closed organ pipe is

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