Example 9 from Precalculus, 7.6 Modeling with Trigonometric Functions

Modeling Damped Harmonic Motion

Model the equations that fit the two scenarios and use a graphing utility to graph the functions: Two mass-spring systems exhibit damped harmonic motion at a frequency of $0.5$ cycles per second. Both have an initial displacement of 10 cm. The first has a damping factor of $0.5$ and the second has a damping factor of $0.1.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
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 $f = \frac{ω}{2 π}$ of 0.5 cycles per second. Thus,

$\begin{array}{l} \frac{ω}{2 π} = 0.5 \\ ω = (0.5) 2 π \\ = π \end{array}$
The first spring system has a damping factor of $c = 0.5.$ Following the general model for damped harmonic motion, we have

$f (t) = 10 e^{- 0.5 t} \cos (π t)$

Figure 15 models the motion of the first spring system.

Figure 15

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

$f (t) = 10 e^{- 0.1 t} \cos (π t)$

Figure 16 models the motion of the second spring system.

Figure 16

How did it go?

Solution

Solution