JEE Main 2025 Jan 22 Shift 1, Physics Q11: Stationary Waves in Air Column

JEE Main2025Jan 22, Shift 1Physics

A closed organ and an open organ tube filled by two different gases having same bulk modulus but different densities $ρ_{1}$ and $ρ_{2}$ respectively. The frequency of 9th harmonic of closed tube is identical with 4th harmonic of open tube. If the length of the closed tube is 10 cm and the density ratio of the gases is $ρ_{1} : ρ_{2} = 1 : 16$ , then the length of the open tube is :

A
$\frac{20}{7}$ cm
B
$\frac{15}{7}$ cm
C
$\frac{20}{9}$ cm
D
$\frac{15}{9}$ cm

Solution

Closed tube admits only odd harmonics; its 9th harmonic frequency is $f_{C} = \frac{9 v _{1}}{4 ℓ _{1}}$ .

Open tube 4th harmonic: $f_{O} = \frac{4 v _{2}}{2 ℓ _{2}} = \frac{2 v _{2}}{ℓ _{2}}$ .

Equating: $\frac{9 v _{1}}{4 ℓ _{1}} = \frac{2 v _{2}}{ℓ _{2}}$ , so $ℓ_{2} = \frac{8 ℓ _{1}}{9} \cdot \frac{v _{2}}{v _{1}}$ .

With identical bulk modulus, $v = B / ρ$ , giving $\frac{v _{2}}{v _{1}} = \frac{ρ _{1}}{ρ _{2}} = \frac{1}{16} = \frac{1}{4}$ .

$ℓ_{2} = 10 \times \frac{8}{9} \times \frac{1}{4} = \frac{20}{9}$ cm.

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JEE Main 2025 Jan 22 Shift 1, Physics Q11: Stationary Waves in Air Column

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