When all the finger holes on a clarinet are closed, it can be approximated as a hollow tube with one closed end (the mouthpiece) and one open end. When all the finger holes are closed, the lowest pitch is 186 Hz (the G below middle C). Assuming the speed of sound in air is 340 m/s, how long is the clarinet?

1) 92 cm
2) 46 cm
3) 30 cm
4) 69 cm

Question

10-09-2024
Physics

High School

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When all the finger holes on a clarinet are closed, it can be approximated as a hollow tube with one closed end (the mouthpiece) and one open end. When all the finger holes are closed, the lowest pitch is 186 Hz (the G below middle C). Assuming the speed of sound in air is 340 m/s, how long is the clarinet?

1) 92 cm
2) 46 cm
3) 30 cm
4) 69 cm

Answer :

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ppriyagpt0 ppriyagpt0 · Answer 1 · 2023-10-01T07:54:11-04:00

Using the fundamental frequency formula for a closed-end air column and the given values, the clarinet length is calculated to be approximately 46 cm, which corresponds to option 2 hence option 2) is the correct answer.

When all the finger holes on a clarinet are closed, it produces its lowest pitch, which is known as the fundamental frequency. Given that this lowest pitch is 186 Hz and using the speed of sound in air at 340 m/s, we can calculate the length of the clarinet as a resonator. The clarinet functions as a closed-end air column, which means that the tube has one closed end and one open end.

The fundamental frequency corresponds to a quarter-wavelength fitting in the tube. Using the formula for the fundamental frequency of a closed-end tube, = v / 4Lf, where f is the frequency, vf is the speed of sound, and Lf is the length of the tube, we can rearrange and solve for L as follows: L = v / 4f. By substituting the given values, L = 340 m/s / (4 * 186 Hz)f, which calculates to approximately 0.456 m or 45.6 cm. Therefore, the length of the clarinet is closest to 46 cm which is option 2 hence option 2) is the correct answer.