Question 96 of 142

A closed organ pipe of length $L$ and an open organ pipe contain gases of densities ${\rho }_{1}$ and ${\rho }_{2}$ 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 pipe is $\dfrac{x}{3}L\sqrt{\dfrac{{\rho }_{1}}{{\rho }_{2}}}$, where $x$ is _______.
(Round off to the Nearest Integer)

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Answer: 4

${f}_{c}={f}_{0}$
$\dfrac{3{V}_{C}}{4L}=\dfrac{2{V}_{0}}{2{L}^{'}}$
$\dfrac{3{V}_{C}}{4L}=\dfrac{{V}_{0}}{{L}^{'}}$
${L}^{'}=\dfrac{4L}{3}\dfrac{{V}_{0}}{{V}_{C}}=\dfrac{4L}{3}\sqrt{\dfrac{B.{\rho }_{1}}{{\rho }_{2}.B}}$ ($B$ is bulk modulus)
$=\dfrac{4L}{3}\sqrt{\dfrac{{\rho }_{1}}{{\rho }_{2}}}$
$x=4$