Question

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

• A. $8/3$
• B. $3/8$
• C. $1/2$
• D. $1/3$

Answer 1

Answer: (B)

For the organ pipe $P_1$ closed at one end the frequency is
$v_1=(2n+1)\frac{v}{4l_1}$
where, variables have their usual meanings.
The first overtone frequency will obtain for $n=1$,
$v_1=(2(1)+1)\frac{v}{4l_1}$
$v_1=\frac{3v}{4l_1}\dots$
For the organ pipe $P_2$ open at both ends, the frequency is $v_2=\frac{nv}{2l_2}$
where, variables have their usual meanings.
The first overtone frequency will obtain for $n=2$,
hence, the third overtone frequency will obtain for $n=4$,
$v_2=\frac{4v}{2l_2}$
These frequencies are in a resonance with a given tuning fork.
Hence, $v_1=v_2$
$\frac{3v}{4l_1}=\frac{4v}{2l_2}$
$\frac{l_1}{l_2}=\frac{6}{16}$
$\frac{l_1}{l_2}=\frac{3}{8}$