# A closed organ pipe of length L and an open organ pipe contain gases of densities  respectively. The compressebility 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  where x is ________ (Rounded off to the nearest integer) Option: 1 4 Option: 2 5 Option: 3 6 Option: 4 7

$\begin{array}{l} \mathrm{f}_{\mathrm{c}}=\mathrm{f}_{0} \\ \\ \frac{3 \mathrm{~V}_{\mathrm{C}}}{4 \mathrm{~L}}=\frac{2 \mathrm{~V}_{0}}{2 \mathrm{~L}^{\prime}} \\ \\ \frac{3 \mathrm{~V}_{\mathrm{C}}}{4 \mathrm{~L}}=\frac{\mathrm{V}_{0}}{\mathrm{~L}^{\prime}} \end{array}$

$\mathrm{L}^{\prime}=\frac{4 \mathrm{~L}}{3} \frac{\mathrm{V}_{0}}{\mathrm{~V}_{\mathrm{C}}}=\frac{4 \mathrm{~L}}{3} \sqrt{\frac{\mathrm{B} \cdot \rho_{1}}{\rho_{2} \cdot \mathrm{B}}} =\frac{4 \mathrm{~L}}{3} \sqrt{\frac{\rho_{1}}{\rho_{2}}}$

where  B is the bulk modulus

So x=4

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