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An organ pipe $P_1$, closed at one end and containing a gas of density $\rho_1$ is vibrating in its first harmonic. Another organ pipe $P_2$, open at both ends and containing a gas of density $\rho_2$ is vibrating in its third harmonic. Both the pipes are in resonance with a given tuning fork. If the compressibility of gases is equal in both pipes, the ratio of the lengths of $P_1$ and $P_2$ is (assume the given gases to be monoatomic)
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The correct answer is:
$\frac{1}{6} \sqrt{\frac{\rho_2}{\rho_1}}$
Frequency of closed organ pipe for first harmonic $n_1=\frac{v_1}{4 l_1}$.
Frequency of open organ pipe for third harmonic
$n_3=\frac{3 v_2}{2 l_2}$
At resonance, $\quad n_1=n_3$
or $\quad \frac{v_1}{4 l_1}=\frac{3 v_2}{2 l_2}$
or $\quad \frac{l_1}{l_2}=\frac{1}{6}\left(\frac{v_1}{v_2}\right)$
$\frac{l_1}{l_2}=\frac{1}{6} \sqrt{\frac{B}{\rho_1}} \times \sqrt{\frac{\rho_2}{B}}$
$=\frac{1}{6} \sqrt{\frac{\rho_2}{\rho_1}}$
Frequency of open organ pipe for third harmonic
$n_3=\frac{3 v_2}{2 l_2}$
At resonance, $\quad n_1=n_3$
or $\quad \frac{v_1}{4 l_1}=\frac{3 v_2}{2 l_2}$
or $\quad \frac{l_1}{l_2}=\frac{1}{6}\left(\frac{v_1}{v_2}\right)$
$\frac{l_1}{l_2}=\frac{1}{6} \sqrt{\frac{B}{\rho_1}} \times \sqrt{\frac{\rho_2}{B}}$
$=\frac{1}{6} \sqrt{\frac{\rho_2}{\rho_1}}$
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