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A closed organ tube and an open organ tube are filled with two different gases of the same bulk modulus but densities ρ₁ and ρ₂. The ninth harmonic of the closed tube is identical with the fourth harmonic of the open tube. If the closed tube is 10 cm long and ρ₁:ρ₂=1:16, the length of the open tube is

Asked in JEE Main 22nd Jan 1st Shift 2025 · Closed and open pipes

Answer: (2) (20)/9 cm

Step-by-step solution

v=√B/ρ with the same B, so (v₁)/(v₂)=√(ρ₂)/(ρ₁)=4.

Closed tube ninth harmonic: (9v₁)/(4L_c). Open tube fourth harmonic: (4v₂)/(2Lₒ)=(2v₂)/(Lₒ).

(9×4v₂)/(4×10)=(2v₂)/(Lₒ) gives (9v₂)/(10)=(2v₂)/(Lₒ), so Lₒ=(20)/9 cm.

Why the other options are wrong

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