Chapter 4.4, Problem 78E

Solution

a.

To find: The amplitude of the arrow’s motion.

The amplitude of the arrow’s motion is $12.7$ .

Given information:

The motion detector is situated at $60\text{ cm}$ away.

The turntable is $25.4\text{cm}$ in diameter.

Calculation:

The maximum value of amplitude is $M=84.7$

The minimum value of amplitude is $m=60$

The amplitude is given by,

  $\begin{array}{c}A=\frac{M-m}{2}\\ =\frac{25.4}{2}\\ A=12.7\end{array}$

Therefore, amplitude of the arrow’s motion is $12.7$ .

b.

To find: The period of the arrow’s motion by analyzing the given data.

The period of the arrow’s motion by analyzing the given data is $1.8$ .

Given information:

The motion detector is situated at $60\text{ cm}$ away.

The turntable is $25.4\text{ cm}$ in diameter.

Calculation:

From the given data, let’s take two data points where $d$ is minimum.

The distance $d=60$ is minimum when $t=1.8$ and $t=3.6$ .

Let’s consider a sinusoid function, where the period goes from minimum to maximum.

So, the period is given by

  $\begin{array}{c}p=3.6-1.8\\ =1.8\end{array}$

Therefore, the period of the arrow’s motion is $1.8$ seconds.

c.

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

The sinusoidal function for the motion of arrow is $d\left(t\right)=-12.7\cos \left(\frac{10\pi t}{9}\right)+72.7$ .

Given information:

The motion detector is situated at $60\text{ cm}$ away.

The turntable is $25.4\text{ cm}$ in diameter.

Calculation:

The sinusoidal function for the given graph is of the form,

  $y=a\sin \left(b\left(x-h\right)\right)+k$

Or

  $y=a\cos \left(b\left(x-h\right)\right)+k$

The block attains a sinusoid with,

Amplitude $A=12.7$ , taken from part (a)

Period,

  $\begin{array}{c}p=\frac{2\pi }{\left|b\right|}\\ 1.8=\frac{2\pi }{\left|b\right|}\\ \left|b\right|=\frac{2\pi }{1.8}\\ b=\frac{10\pi }{9}\end{array}$

To find the vertical shift $k$:

  $\begin{array}{c}k=\frac{M+m}{2}\\ =\frac{85.4+60.0}{2}\\ k=72.7\end{array}$

The general sinusoidal function is given by,

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

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

  $d\left(t\right)=-12.7\cos \left(\frac{10\pi t}{9}\right)+72.7$

Therefore, the sinusoidal function for the motion of arrow is $d\left(t\right)=-12.7\cos \left(\frac{10\pi t}{9}\right)+72.7$ .

d.

To graph: The function with the scatter plot given.

Given information:

The motion detector is situated at $60\text{ cm}$ away.

The turntable is $25.4\text{ cm}$ in diameter.

Graph:

  

Interpretation:

From part (c), it is found that the sinusoid function $d\left(t\right)$ models the motion of the arrow is given by,

  $d\left(t\right)=-12.7\cos \left(\frac{10\pi }{9}t\right)+72.7$

Therefore, the graph for the given scatter plot is plotted.