A 30 cm long organ pipe is filled with air and is open at one end and closed at the other. The speed of sound in air at 0°C is 331 m/s. What is the wavelength of the second mode of vibration?
Question
A 30 cm long organ pipe is filled with air and is open at one end and closed at the other. The speed of sound in air at 0°C is 331 m/s. What is the wavelength of the second mode of vibration?
Solution
The second mode of vibration for a pipe closed at one end is the third harmonic, which is three times the fundamental frequency.
Step 1: Determine the fundamental frequency (first mode of vibration). For a pipe closed at one end, the fundamental frequency corresponds to a wavelength that is four times the length of the pipe.
So, the wavelength (λ1) of the fundamental frequency is λ1 = 4 * length of the pipe = 4 * 0.30 m = 1.2 m.
Step 2: Determine the wavelength of the second mode of vibration. The second mode of vibration for a pipe closed at one end is the third harmonic, which is three times the fundamental frequency. Therefore, the wavelength of the second mode of vibration is one third of the wavelength of the fundamental frequency.
So, the wavelength (λ2) of the second mode of vibration is λ2 = λ1 / 3 = 1.2 m / 3 = 0.4 m.
Therefore, the wavelength of the second mode of vibration is 0.4 m.
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