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 pipe is $\frac{ x }{3} L \sqrt{\frac{\rho_{1}}{\rho_{2}}}$ where $x$ is _________. (Round off to the Nearest Integer)

Answer 1

$f _{ c }= f _{0} $
$\frac{3 V _{ C }}{4 L }=\frac{2 V _{0}}{2 L ^{\prime}} $
$\frac{3 V _{ C }}{4 L }=\frac{ V _{0}}{ L ^{\prime}}$
$L ^{\prime}=\frac{4 L }{3} \frac{ V _{0}}{ V _{ C }}=\frac{4 L }{3} \sqrt{\frac{ B \cdot \rho_{1}}{\rho_{2} \cdot B }}\,\,\,$( B is bulk modulus)
$=\frac{4 L }{3} \sqrt{\frac{\rho_{1}}{\rho_{2}}} $
$x =4$