Discussion
A closed organ pipe and an open organ pipe have the same length. The ratio of the frequencies in their third mode of vibrations is
Answer: C
The ratio is 5 : 6.
A closed pipe supports only odd harmonics, \(\frac{v}{4L}, \frac{3v}{4L}, \frac{5v}{4L}, \ldots\), so its third mode is \(\frac{5v}{4L}\). An open pipe supports all harmonics, \(\frac{v}{2L}, \frac{2v}{2L}, \frac{3v}{2L}, \ldots\), so its third mode is \(\frac{3v}{2L}\).
\[ \frac{5v/4L}{3v/2L} = \frac{5}{6} \]
The other ratios do not match these two frequencies. Taking the closed pipe's third harmonic \(3v/4L\) instead of its third mode would give 1 : 2, and 3 : 5 compares the third and fifth harmonics of one pipe.
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