The following standing wave below is a 5 m string that vibrates up and down as the four harmonic (4 bumps). The string vibrates 48 cycles in

Question

The following standing wave below is a 5 m string that vibrates up and down as the four harmonic (4 bumps). The string vibrates 48 cycles in 16 seconds. Determine the string’s speed

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Khải Quang 3 years 2021-08-12T21:21:53+00:00 1 Answers 7 views 0

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  1. haidang
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    2021-08-12T21:23:21+00:00
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    Answer:

    7.5 m/s

    Explanation:

    Standing waves are waves that do not propagate, so they are just oscillations of the medium over fixed positions.

    Standing waves are produced for example in a string, which is tied at its ends.

    The wavelength of the fundamental mode of vibration of a string is equal to twice the length of the string:

    \lambda=2L

    where L is the length of the string.

    Here, the string vibrates in its fourth harmonic – this means that the wavelength is actually 1/4 of the wavelength of the fundamental mode:

    \lambda_4=\frac{\lambda}{4}

    Here, the length of the string is

    L = 5 m

    So the wavelength of the 4th harmonic is:

    \lambda_4=\frac{\lambda}{4}=\frac{2L}{4}=\frac{2(5)}{4}=2.5 m

    The frequency of the wave is equal instead to the ratio between the number of cycles and the time taken:

    f=\frac{N}{t}

    where here

    N = 48

    t = 16 s

    Substituting,

    f=\frac{48}{16}=3 Hz

    Now we can find the string’s speed by using the wave equation; we find:

    v=f\lambda=(3 Hz)(2.5 m)=7.5 m/s

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