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11 April, 08:08

At a certain harbor, the tides cause the ocean surface to rise and fall a distance d (from highest level to lowest level) in simple harmonic motion, with a period of 12.8 h. How long does it take for the water to fall a distance 0.250d from its highest level

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  1. 11 April, 10:52
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    2.13 h

    Explanation:

    Simple harmonic motion is:

    x = A sin (2π/T t + φ) + B

    where A is the amplitude, T is the period, φ is the phase shift, and B is the midline.

    This can also be written in terms of cosine:

    x = A cos (2π/T t + φ) + B

    Here, A = d/2, T = 12.8, φ = 0, and B = d/2. I'll use cosine so that the highest level is at t=0.

    x = d/2 cos (2π/12.8 t) + d/2

    When x = d - 0.250 d = 0.750 d:

    0.750 d = d/2 cos (2π/12.8 t) + d/2

    0.250 d = d/2 cos (2π/12.8 t)

    0.500 = cos (2π/12.8 t)

    π/3 = 2π/12.8 t

    12.8/6 = t

    t = 2.13

    It takes 2.13 hours to fall 0.250 d from the highest level.
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