7.6 Modeling with trigonometric equations  (Page 4/14)

1. $y=5\sin \left(3t\right)$
   1. The maximum displacement is equal to the amplitude, $\left|a\right|,$ which is 5.
   2. The period is $\frac{2\pi }{\omega }=\frac{2\pi }{3}.$
   3. The frequency is given as $\frac{\omega }{2\pi }=\frac{3}{2\pi }.$
   4. See [link] . The graph indicates the five key points.

      

2. $y=6\cos \left(\pi t\right)$
   1. The maximum displacement is $6.$
   2. The period is $\frac{2\pi }{\omega }=\frac{2\pi }{\pi }=2.$
   3. The frequency is $\frac{\omega }{2\pi }=\frac{\pi }{2\pi }=\frac{1}{2}.$
   4. See [link] .

      

3. $y=5\cos \left(\frac{\pi }{2}\right)t$
   1. The maximum displacement is $5.$
   2. The period is $\frac{2\pi }{\omega }=\frac{2\pi }{\frac{\pi }{2}}=4.$
   3. The frequency is $\frac{1}{4}.$
   4. See [link] .

      

At time $t=0,$ the displacement is the maximum of 10 cm, which calls for the cosine function. The cosine function will apply to both models.

We are given the frequency $f=\frac{\omega }{2\pi }$ of 0.5 cycles per second. Thus,

The first spring system has a damping factor of $c=\mathrm{0.5.}$ Following the general model for damped harmonic motion, we have

[link] models the motion of the first spring system.

The second spring system has a damping factor of $c=0.1$ and can be modeled as

[link] models the motion of the second spring system.