Question

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 organ pipe is

• A. $\frac {L}{3} $
• B. $\frac {4L}{3} $
• C. $\frac {4L}{3} \sqrt {\frac {\rho_1}{\rho_2}} $
• D. $\frac {4L}{3} \sqrt {\frac {\rho_2}{\rho_1}} $

Answer 1

Answer: (C)

Frequency of first over tone of closed pipe $=$ frequency of first overtone of open pipe
$ \Rightarrow \frac{3 v}{4 L_{1}}=\frac{v}{L_{2}} \Rightarrow \frac{3}{4 L_{1}} \sqrt{\frac{\gamma P}{\rho_{1}}}=\frac{1}{L_{2}} \sqrt{\frac{\gamma P}{\rho_{2}}} $
$\therefore L_{2}=\frac{4}{3} L \sqrt{\frac{\rho_{1}}{\rho_{2}}}:\left[\therefore V=\sqrt{\frac{\gamma P}{p}}\right] $