Example 12 from Precalculus, 7.6 Modeling with Trigonometric Functions

Modeling the Oscillation of a Spring

A spring measuring 10 inches in natural length is compressed by 5 inches and released. It oscillates once every 3 seconds, and its amplitude decreases by 30% every second. Find an equation that models the position of the spring $t$ seconds after being released.

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
The amplitude begins at 5 in. and deceases 30% each second. Because the spring is initially compressed, we will write A as a negative value. We can write the amplitude portion of the function as

$A (t) = 5 {(1 - 0.30)}^{t}$

We put ${(1 - 0.30)}^{t}$ in the form $e^{c t}$ as follows:

$\begin{array}{l} 0.7 = e^{c} \\ c = \ln .7 \\ c = - 0.357 \end{array}$

Now let’s address the period. The spring cycles through its positions every 3 seconds, this is the period, and we can use the formula to find omega.

$\begin{array}{l} 3 = \frac{2 π}{ω} \\ ω = \frac{2 π}{3} \end{array}$
The natural length of 10 inches is the midline. We will use the cosine function, since the spring starts out at its maximum displacement. This portion of the equation is represented as

$y = \cos (\frac{2 π}{3} t) + 10$

Finally, we put both functions together. Our the model for the position of the spring at $t$ seconds is given as

$y = - 5 e^{- 0.357 t} \cos (\frac{2 π}{3} t) + 10$

See the graph in Figure 21.

Figure 21

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

A mass suspended from a spring is raised a distance of 5 cm above its resting position. The mass is released at time $t = 0$ and allowed to oscillate. After $\frac{1}{3}$ second, it is observed that the mass returns to its highest position. Find a function to model this motion relative to its initial resting position.

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solution

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Solution

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