​A closed organ and an open organ tube filled by two different gases having same bulk modulus but different densities ρ1 and ρ2 respectively. ​

Question

A closed organ and an open organ tube filled by two different gases having same bulk modulus but different densities \(\rho_{1}\) and \(\rho_{2}\) respectively. The frequency of \(9^{th}\) harmonic of closed tube is identical with \(4^{th}\) harmonic of open tube. If the length of the closed tube is 10 cm and the density ratio of the gases is \(\rho_1 : \rho_2 = 1:16\)  then the length of the open tube is :

(1) \(\frac{20}{7}cm\)

(2) \(\frac{15}{7}cm\)

(3) \(\frac{20}{9}cm\)

(4) \(\frac{15}{9}cm\)

Answer 1

Correct option is:  (3) \(\frac{20}{9}cm\)

\(9^{\text {th }} \text{harmonic of closed pipe} =\frac{9 \mathrm{~V}_{1}}{4 \ell_{1}}\)

\( 4^{\text {th }} \text{ harmonic of open pipe} =\frac{2 \mathrm{~V}_{2}}{\ell_{2}}\)

\( \therefore \frac{9 \mathrm{~V}_{1}}{4 \ell_{1}}=\frac{2 \mathrm{~V}_{2}}{\ell_{2}}\)

\( \therefore \frac{9}{4 \ell_{1}} \sqrt{\frac{\mathrm{~B}}{\rho_{1}}}=\frac{2}{\ell_{2}} \sqrt{\frac{\mathrm{~B}}{\rho_{2}}} \Rightarrow \frac{\ell_{2}}{\ell_{1}}=\frac{8}{9} \sqrt{\frac{\rho_{1}}{\rho_{2}}}\)

\( \ell_{2}=\ell_{1} \times \frac{8}{9} \times \frac{1}{4}=\frac{20}{9} \mathrm{~cm}\)