Question

A closed organ pipe of length ${}^{\prime }{L}^{\prime }$ 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. If the frequencies of their first overtones are same, then the length of the open organ pipe is

• A. $\frac{4L}{3}\sqrt{\frac{{\rho }_2}{{\rho }_1}}$
• B. $\frac{4L}{3}\sqrt{\frac{{\rho }_1}{{\rho }_2}}$
• C. $\frac{4L}{3}$
• D. $\frac{L}{3}$

Answer 1

Answer: (B)

For closed organ pipe,
$f=f_2=3f_1$
$=\frac{3\times V}{4L}=\frac{3}{4L}\times \sqrt{\frac{\gamma p}{{\rho }_1}}$
For open organ pipe,
$f=f_2=2f_1=\frac{2v}{2L^{\prime }}=\frac{V}{L^{\prime }}$
$=\frac{1}{L^{\prime }}\times \sqrt{\frac{\gamma p}{{\rho }_2}}$
For same sound frequency,
$\frac{3}{4L}\times \sqrt{\frac{\gamma P}{{\rho }_1}}=\frac{1}{L^{\prime }}\times \sqrt{\frac{\gamma p}{{\rho }_2}}$
$L^{\prime }=\frac{4L}{3}\times \sqrt{\frac{{\rho }_1}{{\rho }_2}}$