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 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

The correct option is C $\frac{4L}{3}\sqrt{\frac{{\rho }_1}{{\rho }_2}}$

Both piper vibrate in first overtone.

$f_c=f_o$

$\frac{3V_c}{4L}=\frac{2V_o}{2l_o}$

$\frac{3V_c}{4L}=\frac{V_o}{l_o}$

$\therefore \frac{V_o}{V_c}=\frac{3l_o}{4L}$

Also,

$\frac{V_o}{V_c}=\sqrt{\frac{{\rho }_1}{{\rho }_2}}$

Then,

$\sqrt{\frac{{\rho }_1}{{\rho }_2}}=\frac{3l_o}{4L}$

$l_o=\frac{4L}{3}\sqrt{\frac{{\rho }_1}{{\rho }_2}}$