Question

An organ pipe $P_1$ closed at one end is vibrating in its first overtone. Another pipe $P_2$ open at both ends is vibrating in its third overtone. They 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)

3/8

For the organ pipe $P_1$ closed at one end the frequency is

${\nu }_1=(2n+1)\frac{v}{4l_1}$

where, variables have their usual meanings.

The first overtone frequency will obtain for n=1,

${\nu }_1=\left(2\right(1)+1)\frac{v}{4l_1}$

${\nu }_1=\frac{3v}{4l_1}$............................(1)

For the organ pipe $P_2$ open at both ends, the frequency is

${\nu }_2=\frac{nv}{2l_2}$

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,

${\nu }_2=\frac{4v}{2l_2}$ ..........................(2)

These frequencies are in a resonance with a given tuning fork. Hence,

${\nu }_1={\nu }_2$

$\frac{3v}{4l_1}=\frac{4v}{2l_2}$

$\frac{l_1}{l_2}=\frac{6}{16}$

$\frac{l_1}{l_2}=\frac{3}{8}$