Algebra

A hanging spring is compressed 3 inches from its rest position and released at t = 0 seconds. It returns to the same position after 0.8 seconds.

I need help Finding:

a) the amplitude of the motion

b) the period of the motion

c) the frequency of the motion

d) a function that models the displacement, y, of the end of the spring from the rest position at time, t.

e) the displacement from the rest position at t= 3 min rounded to the tenths place

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  1. (a)
    compressed 3 inches from its rest position
    the amplitude is 3

    (b)
    returns to the same position after 0.8 seconds
    the period is 0.8 seconds
    If it had not been at its maximum compression when released, then things would have been a bit more complicated. But it only returns that far after a full period.

    (c) frequency = 1/period

    (d) cos(t) has its maximum at t=0. So, if y is the height of the end of the spring, which is hanging down, its maximum is when t=0. So,
    y = cos(kt)
    cos(kt) has period 2π/k. Since our period is 0.8 (or 4/5), we need 2π/k = 4/5. k=5π/2

    y = cos(5π/2 t)

    (e) now just plug in t=180 (3 minutes is 180 seconds)

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