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10 October, 06:24

The end of a stopped pipe is to be cut off so that the pipe will be open. If the stopped pipe has a total length L, what fraction of L should be cut off so that the fundamental mode of the resulting open pipe has the same frequency as the fifth harmonic (and=5) of the original stopped pipe?

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  1. 10 October, 06:31
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    4/10 of L.

    Explanation:

    A stopped pipe is a pipe with one closed end and one opened end. it is also called a closed pipe.

    The fundamental mode of a stopped pipe is also called its fundamental frequency, and is f₁=v/4L.

    Where f₁=fundamental frequency, v = velocity of sound, L = Length of pipe.

    The fifth harmonic of the stopped pipe f₅ = 5v/4L ... (1)

    For an open pipe,

    Fundamental mode is also called fundamental frequency f₁₀=v/2l₀ ... (2)

    Where f₁₀ = fundamental frequency of a closed pipe, v = velocity of sound and l₀=length of the resulting open pipe.

    from the question, the fundamental mode of the resulting open pipe = The fifth harmonic of the original stopped pipe.

    ∴ f₅=f₁₀

    ⇒5v/4L = v/2l₀

    Equating v from both side of the equation,

    ⇒ 5/4L = 1/2l₀

    Cross multiplying the equation,

    5*2l₀ = 4L * 1

    10l₀ = 4L

    Dividing both side of the equation by the coefficient of l₀ i. e 10

    10l₀/10 = 4L/10

    ∴ l₀ = 4/10 (L)

    ∴ 4/10 of L must be cut off
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