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The air column in a closed pipe of length 50 cm is set into vibration to generate a wave at its fundamental frequency. i. Sketch the diagram of the wave generated. ii. Calculate I. its wavelength. II. the frequency of the first overtone. [Speed of sound in air = 340 ms-1]

Physics

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Answer

c(ii)

2.0 m\text{2.0 m}

510 Hz\text{510 Hz}

Step-by-step solution

6 steps

Here's the solution for part (c):

Part c(i)

Sketch the diagram of the wave generated

For a closed pipe at its fundamental frequency, there is a node at the closed end and an antinode at the open end. This corresponds to one-quarter of a wavelength.

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Part c(ii)

2.0 m\text{2.0 m}

Calculate

I. Its wavelength.

  1. State the given length of the pipe and convert to meters.

    The length of the closed pipe, L=50 cm=0.50 mL = 50 \text{ cm} = 0.50 \text{ m}.

  2. Relate the pipe length to the wavelength for the fundamental frequency in a closed pipe.

    For the fundamental frequency in a closed pipe, the length of the pipe is one-quarter of the wavelength (λ\lambda).

    L=λ4L = \frac{\lambda}{4}

  3. Calculate the wavelength.

    λ=4L\lambda = 4L λ=4×0.50 m\lambda = 4 \times 0.50 \text{ m} λ=2.0 m\lambda = 2.0 \text{ m}

    The wavelength is 2.0 m\boxed{\text{2.0 m}}.

    II. The frequency of the first overtone.

  4. State the speed of sound in air.

    Speed of sound in air, v=340 m/sv = 340 \text{ m/s}.

  5. Calculate the fundamental frequency (f1f_1).

    Using the wave speed formula v=fλv = f\lambda, we can find the fundamental frequency:

    f1=vλf_1 = \frac{v}{\lambda} f1=340 m/s2.0 mf_1 = \frac{340 \text{ m/s}}{2.0 \text{ m}} f1=170 Hzf_1 = 170 \text{ Hz}

  6. Determine the frequency of the first overtone.

    For a closed pipe, the overtones are odd multiples of the fundamental frequency. The first overtone is the third harmonic (3f13f_1).

    ffirst overtone=3×f1f_{\text{first overtone}} = 3 \times f_1 ffirst overtone=3×170 Hzf_{\text{first overtone}} = 3 \times 170 \text{ Hz} ffirst overtone=510 Hzf_{\text{first overtone}} = 510 \text{ Hz}

    The frequency of the first overtone is 510 Hz\boxed{\text{510 Hz}}.

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