Chapter 5.6, Problem 23E

(a)

To determine

To find: The function of the damped harmonic motion.

Answer to Problem 23E

The function of the damped harmonic motion is $y=7e^{-10t}\sin \left(12t\right)$ .

Explanation of Solution

Given Information:

The initial amplitude is $7$ , the damping constant is $10$ , and the time period is $\frac{\pi }{6}$ .

Concept used:

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

  $y=k{e}^{-ct}\sin \omega t$

The angular velocity is given by,

  $\omega =2\pi f$

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

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

The function of the damped harmonic motion is obtained as,

  $\begin{array}{c}y=k{e}^{-ct}\sin \left(\frac{2\pi }{p}t\right)\\ =7e^{-10t}\sin \left(\frac{2\pi }{\pi /6}t\right)\\ =7e^{-10t}\sin \left(12t\right)\end{array}$

Thus, the function of the damped harmonic motion is $y=7e^{-10t}\sin \left(12t\right)$ .

(b)

To determine

To find: The graph of the function.

Explanation of Solution

Given Information:

The function model displacement of an object is,

  $y=7e^{-10t}\sin \left(12t\right)$

The function model displacement of an object is,

  $y=7e^{-10t}\sin \left(12t\right)$

Consider the following graph of the function,