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
$\Rightarrow \frac{3v}{4L_1}=\frac{v}{L_2}\Rightarrow \frac{3}{4L_1}\sqrt{\frac{\gamma P}{{\rho }_1}}=\frac{1}{L_2}\sqrt{\frac{\gamma P}{{\rho }_2}}$
$\therefore L_2=\frac{4}{3}L\sqrt{\frac{{\rho }_1}{{\rho }_2}}:\left[\therefore V=\sqrt{\frac{\gamma P}{p}}\right]$