A closed organ pipe of length 10 cm filled with gas of density ρ1 , while an open organ pipe of length ‘L'

Question

A closed organ pipe of length 10 cm filled with gas of density \(\rho_{1}\), while an open organ pipe of length 'L' is filled with gas of density \(\rho_{2}\). Closed organ pipe vibrates in \(7^{\text {th }}\) harmonic while open organ pipe vibrates in \(4^{\text {th }}\) harmonic with same tunning fork. If \(\frac{\rho_{1}}{\rho_{2}}=\frac{1}{3}\), and both gas have same bulk modulus '\(\beta\)'. Find 'L'

(1) \(\frac{80}{7 \sqrt{3}} \mathrm{~cm}\)

(2) \(\frac{20}{7 \sqrt{3}} \mathrm{~cm}\)

(3) \(\frac{40}{7 \sqrt{3}} \mathrm{~cm}\)

(4) \(\frac{10}{7 \sqrt{3}} \mathrm{~cm}\)

Answer 1

Correct option is : (1) \(\frac{80}{7 \sqrt{3}} \mathrm{~cm}\) 

\(7 \frac{\mathrm{~V}_{1}}{4 \mathrm{~L}}=4\left(\frac{\mathrm{~V}_{2}}{2 \mathrm{~L}}\right)\)

\( \frac{7}{4 \mathrm{~L}_{1}} \sqrt{\frac{\beta}{\rho_{1}}}=\frac{4}{2 \mathrm{~L}_{2}} \sqrt{\frac{\beta}{\rho_{2}}}\)

\( \frac{7}{4 \times 10} \sqrt{\frac{\rho_{2}}{\rho_{1}} \frac{1}{2}}=\frac{1}{\mathrm{~L}_{2}}\) 

\( \mathrm{L}_{2}=\frac{2 \times 4 \times 10}{7} \sqrt{\frac{1}{3}}\)

\( \mathrm{L}_{2}=\frac{80}{7 \sqrt{3}} \mathrm{~cm}\)