Chapter 4.4, Problem 77E

Solution

a.

To find: The scatter plot $d$ and to compute the amplitude of the block’s motion.

The amplitude of the block’s motion is $7.1$ .

Given information:

The minimum value is $7.2$ .

Calculation:

The scatter plot is,

  

From the above graph,

The maximum value of $d$ is $M=21.4$

The minimum value of $d$ is $m=7.2$

The amplitude is given by,

  $\begin{array}{c}A=\frac{M-m}{2}\\ =\frac{21.4-7.2}{2}\\ =\frac{14.2}{2}\\ A=7.1\end{array}$

Therefore, amplitude of the block’s motion is $7.1$ .

b.

To find: The period of the block’s motion from the scatter plot.

The period of the block’s motion from the scatter plot is $8.3$ .

Given information:

The minimum value is $7.2$ .

Calculation:

The scatter plot is,

  

Let the period is the measure of time, which takes one cycle to complete.

So, from the graph it is known that,

The cycle repeats from $0$ to $8.3$ .

Therefore, the period of the block’s motion from the scatter plot is $8.3$ .

c.

To find: The model for the motion of the block as a sinusoidal function.

The sinusoidal function for the motion of block is $d\left(t\right)=-7.1\sin \left(\frac{2\pi t}{8.3}\right)+14.3$ .

Given information:

The minimum value is $7.2$ .

Calculation:

The block attains a sinusoid with,

Amplitude $A=7.1$

  $b=\frac{2\pi }{8.3}$

To find the vertical shift $C$:

  $\begin{array}{c}C=\frac{M+m}{2}\\ =\frac{21.4+7.2}{2}\\ C=14.3\end{array}$

The general sinusoidal function is given by,

  $d\left(t\right)=-A\sin \left(bt\right)+c$

Now substitute the values of $a,b,c$ in above equation to get,

  $d\left(t\right)=-7.1\sin \left(\frac{2\pi t}{8.3}\right)+14.3$

Therefore, the sinusoidal function for the motion of block is $d\left(t\right)=-7.1\sin \left(\frac{2\pi t}{8.3}\right)+14.3$ .

d.

To graph: The function with the scatter plot given.

Given information:

The minimum value is $7.2$ .

Graph:

  

Interpretation:

The graph for the given scatter plot is plotted.