Chapter 8.5, Problem 22AYU

(a)

To determine

To develop:A model that relates the distance d of the object from its rest position after t seconds.

Answer to Problem 22AYU

The equation of the model is $d\left(t\right)=15e^{-\frac{0.75t}{40}}\cos \left(t\sqrt{{\left(\frac{\pi }{3}\right)}^2-\frac{0.5625}{1600}}\right)$ .

Explanation of Solution

Given:

Mass $m=20\text{ grams}$ , distance $a=15\text{ centimeters}$ , damping factor $m=0.75\text{grams}/\text{second}$ and time period $T=6\text{ seconds}$ .

The motion of the object is simple harmonic.

Calculation:

The general equation of a simple harmonic motion is given as follows where the variables have their usual meaning as given in the data:

  $d\left(t\right)=a{e}^{-\frac{bt}{2m}}\cos \left(t\sqrt{{\omega }^2-\frac{b^2}{4m^2}}\right)$

To determine the value of the angular frequency, use the following formula:

  $\begin{array}{c}T=\frac{2\pi }{\omega }\\ \omega =\frac{2\pi }{T}\\ =\frac{2\pi }{6}\end{array}$

Substitute the value of the angular frequency and of the variables m , a , b and T in the general equation of a simple harmonic motion and simplify it to a reduced form as shown below:

  $\begin{array}{c}d\left(t\right)=a{e}^{-\frac{bt}{2m}}\cos \left(t\sqrt{{\omega }^2-\frac{b^2}{4m^2}}\right)\\ =15e^{-\frac{0.75t}{2\left(20\right)}}\cos \left(t\sqrt{{\left(\frac{2\pi }{6}\right)}^2-\frac{{\left(0.75\right)}^2}{4{\left(20\right)}^2}}\right)\\ =15e^{-\frac{0.75t}{40}}\cos \left(t\sqrt{{\left(\frac{\pi }{3}\right)}^2-\frac{0.5625}{1600}}\right)\end{array}$

(b)

To determine

To plot: The graph of the equation $d\left(t\right)=15e^{-\frac{0.75t}{40}}\cos \left(t\sqrt{{\left(\frac{\pi }{3}\right)}^2-\frac{0.5625}{1600}}\right)$ for 5 oscillations, using a graphing utility.

Answer to Problem 22AYU

The graph is shown in Figure 1.

Explanation of Solution

Plot the curve of the equation $d\left(t\right)=15e^{-\frac{0.75t}{40}}\cos \left(t\sqrt{{\left(\frac{\pi }{3}\right)}^2-\frac{0.5625}{1600}}\right)$ in a graphing utility for 5 oscillations, as shown in the following figure:

  

Figure 1