An organ pipe $\mathrm{P}_1$ closed at one end has vibrating air column in its first overtone and another…
An organ pipe $\mathrm{P}_1$ closed at one end has vibrating air column in its first overtone and another pipe $\mathrm{P}_2$ open at both ends has vibrating air column in the third overtone, which are in resonance with a given tuning fork. The ratio of the length of pipe $P_1$ to that of $P_2$ is
3:8
1:2
1:8
5:8
Solution
Since, both pipes are in resonance with a given tuning fork, therefore, they will have the same frequency of vibration.
For the organ pipe closed at one end.
First overtone $(n=3)$, .
For organ pipe open at both ends,
Third overtone $(n=4)$,
From (1) and (2)
$\begin{aligned}
& \frac{3 v}{4 l_1}=\frac{4 v}{2 l_2} \\
& \Rightarrow \frac{l_1}{l_2}=\frac{3}{8}
\end{aligned}$
For organ pipe open at both ends,
Third overtone $(n=4)$,
From (1) and (2)
$\begin{aligned}
& \frac{3 v}{4 l_1}=\frac{4 v}{2 l_2} \\
& \Rightarrow \frac{l_1}{l_2}=\frac{3}{8}
\end{aligned}$