A closed organ pipe of length L and an open organ pipe contain gases of densities \rho _1\text{and}\rho _2 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 \frac{x}{3}L \sqrt{\frac{\rho _1}{\rho _2}} where x is ________ (Rounded off to the nearest integer)
Option: 1 4
Option: 2 5
Option: 3 6
Option: 4 7

Answers (1)

\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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