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Q.

An organ pipe open at one end is vibrating in first overtone and is in resonance with another pipe open at both ends and vibrating in third harmonic. The ratio of length of two pipes is $\frac{x}{y}$ , then $x + y =$

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answer is 3.

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Detailed Solution

For an organ pipe open at one end,
frequency of 1st overtone, $n_{1} = \frac{3 v}{4 l_{1}}$
For the organ pipe open at both ends,
frequency of 3rd harmonic, $n_{2} = \frac{3 v}{2 l_{2}}$
$\begin{array}{l} as n_{1} = n_{2} \\ \frac{3 v}{4 l_{1}} = \frac{3 v}{2 l_{2}} \\ \frac{l_{1}}{l_{2}} = \frac{2}{4} = \frac{1}{2} = \frac{x}{y} (given) \\ x + y = 3 \end{array}$

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