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On closing one end of an open pipe, the ...

On closing one end of an open pipe, the frequency of fifth harmonic of closed pipe becomes larger by 200 Hz than the fundamental frequency of open pipe. Calcuate the fundamental frequency of open pipe.

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To solve the problem, we need to analyze the frequencies of the open pipe and the closed pipe. ### Step 1: Define the fundamental frequency of the open pipe The fundamental frequency (ν₀) of an open pipe is given by the formula: \[ ν₀ = \frac{V}{2L} \] where \( V \) is the speed of sound in air and \( L \) is the length of the pipe. ### Step 2: Define the frequency of the closed pipe When one end of the open pipe is closed, it becomes a closed pipe. The fundamental frequency (ν₁) of a closed pipe is given by: \[ ν₁ = \frac{V}{4L} \] The frequency of the 5th harmonic (ν₅) of the closed pipe is: \[ ν₅ = 5 \cdot ν₁ = 5 \cdot \frac{V}{4L} = \frac{5V}{4L} \] ### Step 3: Establish the relationship between the frequencies According to the problem, the frequency of the 5th harmonic of the closed pipe is larger by 200 Hz than the fundamental frequency of the open pipe. Therefore, we can write: \[ ν₅ = ν₀ + 200 \text{ Hz} \] Substituting the expressions for \( ν₅ \) and \( ν₀ \): \[ \frac{5V}{4L} = \frac{V}{2L} + 200 \] ### Step 4: Solve for the speed of sound To solve for \( V \), we first find a common denominator and rearrange the equation: \[ \frac{5V}{4L} - \frac{2V}{4L} = 200 \] This simplifies to: \[ \frac{3V}{4L} = 200 \] Now, multiply both sides by \( 4L \): \[ 3V = 800L \] Thus, we have: \[ V = \frac{800L}{3} \] ### Step 5: Substitute back to find the fundamental frequency of the open pipe Now, we substitute \( V \) back into the formula for the fundamental frequency of the open pipe: \[ ν₀ = \frac{V}{2L} = \frac{800L/3}{2L} = \frac{800}{6} = \frac{400}{3} \text{ Hz} \] Calculating this gives: \[ ν₀ \approx 133.33 \text{ Hz} \] ### Final Answer The fundamental frequency of the open pipe is approximately **133.33 Hz**. ---
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Knowledge Check

  • An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is termed to be higher by 100 Hz, than the fundamental frequency of the open pipe. The fundamental frequency of the open pipe is :

    A
    200 Hz
    B
    150 Hz
    C
    100 Hz
    D
    250 Hz
  • An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is termed to be higher by 100 Hz, than the fundamental frequency of the open pipe. The fundamental frequency of the open pipe is :

    A
    200 Hz
    B
    150 Hz
    C
    100 Hz
    D
    250 Hz
  • An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by 100 Hz then the fundamental frequency of the open pipe. The fundamental frequency of the open pipe is

    A
    `200 Hz`
    B
    `300 Hz`
    C
    `240 Hz`
    D
    `480 Hz`
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