The first overtone of the vibrating air column of a closed pipe has the same frequency as the second…
The first overtone of the vibrating air column of a closed pipe has the same frequency as the second overtone of an open pipe $1.5 \mathrm{~m}$ Jung. The length of the closed pipe is
1.0 m
0.75 m
0.5 m
1.25 m
Solution
Frequency is given by $\frac{v}{\lambda}$, where $v$ is the speed of the sound and $\lambda$ the length of the pipe.
For the second overtone of an open pipe:
$\begin{aligned}
& \frac{3}{2} \lambda=L_O \\
& \therefore f_O=\frac{3 v}{2 L_O}
\end{aligned}$
For the first overtone of the closed pipe:
$\begin{aligned}
& \frac{3}{4} \lambda=L_C \\
& \therefore f_C=\frac{3 v}{4 L_C}
\end{aligned}$
If they have the same frequency:
$\frac{3 v}{2 L_O}=\frac{3 v}{4 L_C}$
$\Rightarrow L_C=\frac{L_O}{2}=\frac{1.5}{2} \mathrm{~m}$