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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=34√BcρcLc [∵v=√Bρ;B is Bulk modulus of gas]

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

⇒1√ρoLo=341√ρcLc

Substituting the given data, we get

1√1×Lo=34×1√9×1

⇒Lo=4 m

So fo=2vo2Lo and fc=3vc4Lc

From the data given in the question, fo=fc

so, 2vo2Lo=3vc4Lc

⇒√B0ρoLo=34√BcρcLc [∵v=√Bρ;B is Bulk modulus of gas]

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

⇒1√ρoLo=341√ρcLc

Substituting the given data, we get

1√1×Lo=34×1√9×1

⇒Lo=4 m

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