37.An organ pipe P₁ closed at one end vibrating in its first overtone and another pipe P₂open at both ends vibrating in third overtone are in resonance with a given tuning fork. The ratio of the length P₁to that of P₂is --------

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Solution

First overtone of Closed organ pipe P1:

It is equal to 3rd harmonic.

$f_{1} = \frac{3 v}{4 L_{1}} H e r e, v i s s p e e d o f s o u n d a n d L_{1} i s l e n g t h o f p i p e P_{1}$

Third overtone of open organ pipe P2:

It is equal to 4th harmonic.

$f_{2} = \frac{4 v}{2 L_{2}} = \frac{2 v}{L_{2}} H e r e, v i s s p e e d o f s o u n d a n d L_{2} i s l e n g t h o f p i p e P_{2}$

Equate f1 and f2 to find the ratio of lengths: (condition of resonance)

$\frac{3 v}{4 L_{1}} = \frac{2 v}{L_{2}} \frac{3}{4 L_{1}} = \frac{2}{L_{2}} 8 L_{1} = 3 L_{2} \frac{L_{1}}{L_{2}} = \frac{3}{8}$

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37.An organ pipe P1 closed at one end vibrating in its first overtone and another pipe P2open at both ends vibrating in third overtone are in resonance with a given tuning fork. The ratio of the length P1to that of P2is --------

37.An organ pipe P₁ closed at one end vibrating in its first overtone and another pipe P₂open at both ends vibrating in third overtone are in resonance with a given tuning fork. The ratio of the length P₁to that of P₂is --------