The fifth harmonic of a closed organ pipe is | vedclass

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(B) The frequency of the $n^{th}$ harmonic of a closed organ pipe is given by $f_{n, closed} = \frac{nv}{4L_{closed}}$, where $n$ is an odd integer $(

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$2$

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(B) The frequency of the $n^{th}$ harmonic of a closed organ pipe is given by $f_{n, closed} = \frac{nv}{4L_{closed}}$, where $n$ is an odd integer $(1, 3, 5, ...)$.For the fifth harmonic, $n = 5$, so $f_{5, closed} = \frac{5v}{4L_{closed}}$.The frequency of the first harmonic (fundamental frequency) of an open organ pipe is given by $f_{1, open} = \frac{v}{2L_{open}}$.Given that the fifth harmonic of the closed pipe is in unison with the first harmonic of the open pipe, we have $f_{5, closed} = f_{1, open}$.Substituting the expressions: $\frac{5v}{4L_{closed}} = \frac{v}{2L_{open}}$.Simplifying the equation: $\frac{5}{4L_{closed}} = \frac{1}{2L_{open}}$.Rearranging for the ratio of lengths: $\frac{L_{closed}}{L_{open}} = \frac{5 	imes 2}{4} = \frac{10}{4} = \frac{5}{2}$.Comparing this with the given ratio $\frac{5}{x}$, we find $x = 2$.

The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is $5 / x$. The value of $x$ is . . . . . . .

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