Question

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

Answer 1

(a)
<u>compressed 3 inches from its rest position</u>
the amplitude is 3

(b)
<u>returns to the same position after 0.8 seconds</u>
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)