A closed organ and an open organ tube filled by two different gases having same bulk modulus but different densities and respectively. The frequency of 9th harmonic of closed tube is identical with 4th harmonic of open tube. If the length of the closed

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Solution

Q.
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 [2025]

1 $\frac{20}{7} cm$

2 $\frac{15}{7} cm$

3 $\frac{20}{9} cm$

4 $\frac{15}{9} cm$

Ans.
(3)

9th harmonic of closed pipe = $\frac{9 v_{1}}{4 l_{1}}$

4th harmonic of open pipe = $\frac{2 v_{2}}{l_{2}}$

$∴ \frac{9 v_{1}}{4 l_{1}} = \frac{2 v_{2}}{l_{2}}$

$∴ \frac{9}{4 l_{1}} \sqrt{\frac{B}{ρ_{1}}} = \frac{2}{l_{2}} \sqrt{\frac{B}{ρ_{2}}}$

$\Rightarrow \frac{l_{2}}{l_{1}} = \frac{8}{9} \sqrt{\frac{ρ_{1}}{ρ_{2}}}$

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

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