A closed organ and an open organ tube are filled by two different… | JEE Main 2025 (22 Jan, Shift 1)

Q46JEE Main 2025 · 22 JAN · SHIFT 1Difficult

Organ Pipes & Resonance Tube

A closed organ and an open organ tube are filled by two different gases having same bulk modulus but different densities $\rho_1$ and $\rho_2$ , respectively. The frequency of $9$ th harmonic of closed tube is identical with $4$ th harmonic of open tube. If the length of the closed tube is 10 cm and the density ratio of the gases is $\rho_1 : \rho_2 = 1 : 16$ , then the length of the open tube is:

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Answer: (b)

Closed tube 9th harmonic: $\frac{9v_1}{4L_c}$ ; open tube 4th harmonic: $\frac{4v_2}{2L_o} = \frac{2v_2}{L_o}$
Equate: $L_o = \frac{8L_c}{9}\frac{v_2}{v_1}$
Same $B$ : $\frac{v_2}{v_1} = \sqrt{\frac{\rho_1}{\rho_2}} = \sqrt{\frac{1}{16}} = \frac{1}{4}$
$L_o = \frac{8 \times 10}{9 \times 4} = \frac{20}{9}\text{ cm}$
So the answer is (b) $\frac{20}{9}\text{ cm}$ .

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