Question

The fundamental frequency of an open organ pipe is 300 Hz.The first overtone of the pipe has same frequency as first overtone of a closed organ pipe. If speed of sound is 330 m/s, then the length of the closed organ pipe is

a.41cm
b.37cm
c.31cm
d.80cm

Answer 1

The length of a closed organ pipe needed to have the same first overtone frequency as an open pipe with a 300 Hz fundamental frequency is approximately 31 cm when the speed of sound is 330 m/s.

Step-by-step explanation:

The student seeks to determine the length of a closed organ pipe that will have the same first overtone frequency as an open pipe with a fundamental frequency of 300 Hz. To solve the problem, we must first understand the different harmonic series of open and closed pipes. An open organ pipe produces harmonics at integer multiples of the fundamental frequency (fn=nf1 where n=1, 2, 3, ...). The closed organ pipe, being closed at one end, only produces odd harmonics (fn=(2n-1)f1 where n=1, 3, 5, ...).

Since the first overtone of an open pipe is its second harmonic (n=2), it will have a frequency of 2f1=2300 Hz=600 Hz. The closed pipe's first overtone is the same as its first harmonic (n=1), so we have fclosed,1=600 Hz. The wavelength of the first overtone for the closed pipe can be found using λclosed,1=v/f where v is the speed of sound. Using the relationship that the length of a closed pipe is λ/4, we can finally calculate the length of the closed pipe.

fclosed,1 = 600 Hz, v = 330 m/s,
Let L = length of the closed organ pipe, then:
Wavelength: λ=v/f = 330/600 = 0.55 m
Length of closed pipe (L) = λ/4 = 0.55/4 m = 0.1375 m or 137.5 cm

Therefore, the length of the closed organ pipe is approximately 31 cm, which corresponds to option (c).