Chapter 4.6, Problem 71E

Solution

(a)

To graph : The function and its two damping curves in same window for $0\le t\le 12$ .

Given information :

The oscillations of a spring subject to friction are modeled by the equation $y=0.43e^{-0.55t}\cos 1.8t$ .

Graph :

Consider the given function $y=0.43e^{-0.55t}\cos 1.8t$ .

As $\cos t$ oscillates between $-1\text{ and }1$ , so the given function oscillates between $0.43e^{-0.55t}\left(-1\right)$ and $0.43e^{-0.55t}\left(1\right)$ .

Therefore, the two lines are $y=-0.43e^{-0.55t}$ and $y=0.43e^{-0.55t}$ .

Graph both the lines in the same viewing window $\left[0,12\right]$ by $\left[-0.5,0.5\right]$ .

  

Interpretation :

The given function oscillates between the two lines $y=-0.43e^{-0.55t}$ and $y=0.43e^{-0.55t}$ .

(b)

To determine : Time taken for the spring to be damped so that $-0.2\le y\le 0.2$ .

The time taken for the spring to be damped so that $-0.2\le y\le 0.2$ is where $t>0.5$ .

Given information :

The oscillations of a spring subject to friction are modeled by the equation $y=0.43e^{-0.55t}\cos 1.8t$ .

Explanation :

Consider the graph of given function $y=0.43e^{-0.55t}\cos 1.8t$ .

  

From the graph it is evident that the condition $-0.2\le y\le 0.2$ exists when the value of $t$ is more than $0.5$ .

Therefore, the time taken for the spring to be damped so that $-0.2\le y\le 0.2$ is where $t>0.5$ .