Finding a Sine Function that Models Damped Harmonic Motion

Calculate the value of $\omega$ and substitute the known values into the model.

1. ⓐ As period is $\frac{2\pi }{\omega },$ we have
   $$\begin{array}{l}\frac{\pi }{6}=\frac{2\pi }{\omega }\hfill \\ \omega \pi =6(2\pi )\hfill \\ \omega =12\hfill \end{array}$$

   The damping factor is given as 10 and the amplitude is 7. Thus, the model is $y=7e^{-10t}\sin \left(12t\right).$ See Figure 19.

   

   Figure 19
2. ⓑ As frequency is $\frac{\omega }{2\pi },$ we have
   $$\begin{array}{l}20=\frac{\omega }{2\pi }\hfill \\ 40\pi =\omega \hfill \end{array}$$

   The damping factor is given as $0.2$ and the amplitude is $\mathrm{0.3.}$ The model is $y=0.3e^{-0.2t}\sin \left(40\pi t\right).$ See Figure 20.

   

   Figure 20