Answer: D(t) = ![8.e^{-0.4t}.cos(\frac{\pi }{6}.t )](https://tex.z-dn.net/?f=8.e%5E%7B-0.4t%7D.cos%28%5Cfrac%7B%5Cpi%20%7D%7B6%7D.t%20%29)
Explanation: A harmonic motion of a spring can be modeled by a sinusoidal function, which, in general, is of the form:
y =
or y = ![a.cos(\omega.t)](https://tex.z-dn.net/?f=a.cos%28%5Comega.t%29)
where:
|a| is initil displacement
is period
For a Damped Harmonic Motion, i.e., when the spring doesn't bounce up and down forever, equations for displacement is:
or ![y=a.e^{-ct}.sin(\omega.t)](https://tex.z-dn.net/?f=y%3Da.e%5E%7B-ct%7D.sin%28%5Comega.t%29)
For this question in particular, initial displacement is maximum at 8cm, so it is used the cosine function:
period =
12 =
ω = ![\frac{\pi}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpi%7D%7B6%7D)
Replacing values:
![D(t)=8.e^{-0.4t}.cos(\frac{\pi}{6} .t)](https://tex.z-dn.net/?f=D%28t%29%3D8.e%5E%7B-0.4t%7D.cos%28%5Cfrac%7B%5Cpi%7D%7B6%7D%20.t%29)
The equation of displacement, D(t), of a spring with damping factor is
.