A closed organ pipe of length $L$ and an open organ pipe contain gases of densities $\rho_1$ and $\rho_2$,…
A closed organ pipe of length $L$ and an open organ pipe contain gases of densities $\rho_1$ and $\rho_2$, respectively. The compressibility $\left( = \frac{1}{\text{bulk modulus}} \right)$ of gases are equal in both the pipes. Both the pipes are vibrating in their first overtones with same frequency. The length of the open organ pipe is
$\frac{L}{3}$
$\frac{4L}{3}$
$\frac{4L}{3} \sqrt{\frac{\rho_1}{\rho_2}}$
$\frac{4L}{3} \sqrt{\frac{\rho_2}{\rho_1}}$
Solution
Frequency of first overtone of closed pipe = Frequency of first overtone of open pipe
$\Rightarrow \frac{3v_1}{4L_1} = \frac{v_2}{L_2}$
$\Rightarrow \frac{3}{4L_1}\sqrt{\frac{B}{\rho_1}} = \frac{1}{L_2}\sqrt{\frac{B}{\rho_2}}$
$\Rightarrow \text{Length of the open pipe, } L_2 = \frac{4L_1}{3}\sqrt{\frac{\rho_1}{\rho_2}} = \frac{4L}{3}\sqrt{\frac{\rho_1}{\rho_2}}$