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Question

A closed organ pipe of length 100 cm and an open organ pipe contain gases of densities 9 kg/m3 and 1 kg/m3, respectively. The compressibility of gases is equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open organ pipe is (in m)

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Solution

Let fo be the frequency of first overtone of open organ pipe and fc be the frequency of first overtone for a closed organ pipe.
So fo=2vo2Lo and fc=3vc4Lc
From the data given in the question, fo=fc

so, 2vo2Lo=3vc4Lc
B0ρoLo=34BcρcLc [v=Bρ;B is Bulk modulus of gas]

It is given that, the compressibility of two gases is same B0=Bc=B (say)

1ρoLo=341ρcLc

Substituting the given data, we get
11×Lo=34×19×1

Lo=4 m

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