An organ pipe is open at one end and closed at the other. The frequency of the third mode is 300 Hz higher than the frequency of the second mode. If the speed of sound is 345 m/s, then what is the length of the organ pipe?
A) 50.3 cm
B) 57.5 cm
C) 64.5 cm
D) 66.7 cm
E) 70.5 cm

Question

Quizplus · Accepted Answer

For an organ pipe open at one end and closed at the other, the harmonics are odd multiples of the fundamental frequency. Thus, the second mode would be the third harmonic (3f) and the third mode would be the fifth harmonic (5f). Given that the frequency of the third mode is 300 Hz higher than the second, we have 5f - 3f = 300 Hz, leading to 2f = 300 Hz, so f = 150 Hz. The fundamental frequency (f) can be calculated using the formula $f = \frac{v}{4L}$, where $v$ is the speed of sound and $L$ is the length of the pipe. Rearranging for $L$, we get $L = \frac{v}{4f}$. Substituting $v = 345 \, \text{m/s}$ and $f = 150 \, \text{Hz}$, we find $L = \frac{345}{4 \times 150} = \frac{345}{600} = 0.575 \, \text{m}$ or 57.5 cm.

An Organ Pipe Is Open at One End and Closed