An organ pipe P 1 closed at one end and vibrating in its first overtone and other pipe P 2 open at both ends vibrating in its third overtone are in resonance with a given tuning fork. The ratio of

Question

An organ pipe P 1 ​ closed at one end and vibrating in its first overtone and other pipe P 2 ​ open at both ends vibrating in its third overtone are in resonance with a given tuning fork. The ratio of length of P 1 ​ and P 2 ​ will be

TestHub · Accepted Answer

For an organ pipe closed at one end, the frequency of the first overtone (which corresponds to the third harmonic) is given by $f_{1} = \frac{3 v}{4 L _{1}}$ .

For an organ pipe open at both ends, the frequency of the third overtone (which corresponds to the fourth harmonic) is given by $f_{2} = \frac{4 v}{2 L _{2}}$ .

Since both pipes are in resonance with the same tuning fork, their frequencies must be equal:

$f_{1} = f_{2}$

$\frac{3 v}{4 L _{1}} = \frac{4 v}{2 L _{2}}$

$\frac{3}{4 L _{1}} = \frac{2}{L _{2}}$

$\frac{L _{1}}{L _{2}} = \frac{3}{8}$

The ratio of the lengths of $P_{1}$ and $P_{2}$ is $3 : 8$ .

Answer: $3 : 8$

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