Chapter 4.5, Problem 86E

a.

To determine

To find is y a function of t , explain.

Answer to Problem 86E

y is a function of t because t will produce a single y value.

Explanation of Solution

Given information: The motion of an oscillating weight suspended from a spring was measured by a motion detector. The data were collected, and the approximate maximum displacement from equilibrium ( y =2) are labeled as (0.125,2.35) and minimum labeled as (0.375,1,65).The distance y from the motor detector is measured in centimeter and the time t is measured in seconds.

Calculation:

y is a function of t because t will produce a single y value.

b.

To determine

To approximate the amplitude and period.

Answer to Problem 86E

So, amplitude is 0.35 cm and period is 0.5 sec.

Explanation of Solution

Given information: The motion of an oscillating weight suspended from a spring was measured by a motion detector. The data were collected, and the approximate maximum displacement from equilibrium ( y =2) are labeled as (0.125, 2.35) and minimum labeled as (0.375,1.65). The distance y from the motor detector is measured in centimeter and the time t is measured in seconds.

Concept Used:

The amplitude is half the vertical length from the low to the high point.

Calculation:

The amplitude is half the vertical length from the low to the high point.

  $a=\frac{2.35-1.65}{2}=0.35cm$

.

The period is the length from one peak to the next peak. They give the point of the peak and one valley , so can double the horizontal distance of these.

  $2(0.375-0.125)=0.5\sec$ .

So, amplitude is 0.35 cm and period is 0.5 sec.

c.

To determine

To find a model for the data.

Answer to Problem 86E

  $y=0.35\sin (4\pi t)+2.$

Explanation of Solution

Given information: The motion of an oscillating weight suspended from a spring was

measured by a motion detector. The data were collected, and the approximate maximum

displacement from equilibrium ( y =2) are labeled as (0.125,2.35) and minimum labeled as (0.375,1.65). The distance y from the motor detector is measured in centimeter and the time t is measured in seconds

Calculation:

  $y=a\sin (bx-c)+d.$

Equilibrium is y =2, which is the middle line that the sine curve goes back and forth from . For the parent $y=\sin x,$ it is y =0 so we will have to shift up by d =2.

Find b from the period equation and the period.

  $\begin{array}{l}\frac{2\pi }{b}=0.5\\ b=\frac{2\pi }{0.5}=4\pi .\end{array}$

Assuming the weight started at equilibrium ( c =0), the model is ,

  $y=0.35\sin (4\pi t)+2.$

d.

To determine

To graph the model by using graphing utility.

Answer to Problem 86E

The graph of the model looks like the given data.

Explanation of Solution

Given information: The motion of an oscillating weight suspended from a spring was

measured by a motion detector. The data were collected, and the approximate maximum displacement from equilibrium ( y =2) are labeled as (0.125, 2.35) and minimum labeled as (0.375,1,65). The distance y from the motor detector is measured in centimeter and the time t is measured in seconds

Calculation:

The graph of the model $y=0.35\sin (4\pi t)+2.$ is given below.

  

Where x -axis shows time t and y -axis shows displacement y .

It looks like the given data.