Chapter 5.6, Problem 19E

(a)

To determine

To find: The function of the damped harmonic motion.

Answer to Problem 19E

The function of the damped harmonic motion is $y=2e^{-1.5t}\cos \left(6\pi t\right)$ .

Explanation of Solution

Given Information:

The initial amplitude is $2$ , the damping constant is $1.5$ , and the frequency is $3$ .

Concept used:

The standard form of the function of the damped harmonic motion is given by,

  $y=k{e}^{-ct}\cos \omega t$

The angular velocity is given by,

  $\omega =2\pi f$

The standard form of the function of the damped harmonic motion is given by,

  $y=k{e}^{-ct}\cos \left(2\pi ft\right)$

The function of the damped harmonic motion is obtained as,

  $\begin{array}{c}y=k{e}^{-ct}\cos \left(2\pi ft\right)\\ =2e^{-1.5t}\cos \left(2\pi \times 3t\right)\\ =2e^{-1.5t}\cos \left(6\pi t\right)\end{array}$

Thus, the function of the damped harmonic motion is $y=2e^{-1.5t}\cos \left(6\pi t\right)$ .

(b)

To determine

To find: The graph of the function.

Explanation of Solution

Given Information:

The function model displacement of an object is,

  $y=2e^{-1.5t}\cos \left(6\pi t\right)$

The function model displacement of an object is,

  $y=2e^{-1.5t}\cos \left(6\pi t\right)$

Consider the following graph of the function,