A closed organ pipe of length L and an open organ pipe contain gases of densities $ρ_{1}$ and $ρ_{2}$ respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open is $\frac{x}{3} L \sqrt{\frac{ρ_{1}}{ρ_{2}}}$ where x is _______. (Round off to the Nearest Integer).

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answer is 4.

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Detailed Solution

Velocity $v = \sqrt{\frac{B}{ρ}} = \sqrt{\frac{1}{K ρ}}$
[ Compressibility, $K = \frac{1}{B (Bulk modulus)}$
And both closed and open organ pipes vibrating in their first overtone with same frequency
$\begin{aligned} f_{closed} = f_{open} \\ \frac{3 v_{1}}{4 L} = \frac{2 v_{2}}{2 L'} \Rightarrow L' = \frac{4}{3} L (\frac{v_{2}}{v_{1}}) \end{aligned}$
$L' = \frac{4}{3} L \sqrt{\frac{ρ_{1}}{ρ_{2}}}$ [ compressibility are equal]
$∴ x = 4$