Question

An organ pipe $P_1$ closed at one end is vibrating in its first overtone and another organ pipe $P_2$ opened at both ends is vibrating in its third overtone are in resonance with a given tuning fork. The ratio of length of organ pipe $P_1$ and $P_2$ is

• A. $\frac{1}{3}$
• B. $\frac{3}{8}$
• C. $\frac{2}{3}$
• D. $\frac{8}{3}$

Answer 1

The correct option is B $\frac{3}{8}$

Both pipes are in resonance with a given tuning fork, therefore, both pipes will have the same frequency of vibrations.
For organ pipe closed at one end,
First overtone $(n=3)$,
$f=\frac{3v}{4l_1}$ ..........(1)
For organ pipe open at both ends,
Third overtone $(n=4)$,
$f=\frac{4v}{2l_2}$ ..........(2)
From (1) and (2),
$\frac{3v}{4l_1}=\frac{4v}{2l_2}$
$\Rightarrow \frac{l_1}{l_2}=\frac{3}{8}$