Mathematical Investigations 3 Writing Activity
Name
Swingin' into Spring
In this activity, you will use a camera (perhaps the one in your laptop) to film a swinging
object, use Logger Pro to gather data from your movie clip, and analyze the data using techniques
and functions we have studied in MI. This project provides a connection between trigonometric
functions, function transformations, and other functions studied previously in MI. This activity
will provide you the opportunity to work with a partner to explore a simple yet important
physical application of sinusoidal behavior. The work will also serve to strengthen and broaden
your skills in Excel, and refresh your knowledge of data analysis, curve fitting, and functions.
(5 pts) Part I – Data Collection
Equipment:
• camera
• Vernier software (Logger Pro)
• swing set and human riding the swing OR ring stand, string, tennis ball
• measuring stick or measured distance for reference
Process:
Record the swinging object that is swinging to an approximately constant height for at least 3
full swings. You will be using your video to gather data of time, horizontal distance and
vertical distance of the person/weight.
Logger Pro Instructions:
You may obtain your video movie by using your laptop computer if it has a camera or you may
use a separate video camera. Remember to position your camera so that it is viewing the
swinging from the side (that is a line from your camera to the swing is perpendicular to the plane
of the swinging motion. To get the best data keep your camera as steady as possible. Do NOT
rotate the camera to track the swinging object.
If you are using your pc camera use the following three steps:
________________________________________________
Click through the following: LoggerPro->insert->videocapture->startcapture.
End video capture by clicking on “stop”.
Close the video capture window.
If you are using a separate camera then click through the following: LoggerPro->insert>video and select the video you previously copied to you pc.
The remaining steps are the same for either camera option.
Play video by clicking on play in video display.
Get video analysis options by clicking on “enable/disable video analysis”.
Advance to desired portion of video using play and stop.
“Toggle trails” off to avoid the clutter of seeing too many points on the image. This will
make it easier to see the object you are trying to digitize.
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Click on “add point” and then click on desired point for each frame (frames advance
automatically with each click (digitize at least two swings).
Choose an appropriate origin for your coordinate axes by clicking on “set origin” and
clicking on the image at the desired location. Choosing an appropriate origin is useful for all
investigations but only necessary if you decide to analyze the angular motion. You may at any
time select a new origin and your x and y values will be recalculated. Therefore make sure you
save this Logger Pro file.
Set the scale by clicking on “set scale” and drawing a line on the reference object in the
image (such as a meter stick). In the pop-up window type in the length and units.
Open a new Excel file.
Expand the data window so that the time, x, and y columns are visible. Then select all
the time, x, and y data using the mouse. Copy the selected data to the Excel sheet using
copy/paste.
(25 pts) Part II – Sinudoidal Analysis (use Excel spreadsheet):
Horizontal Position vs. time:
Create a chart (Scatter Plot) of horizontal distance versus time You should see a sinusoidal
pattern to the data. It is now your task to determine a sinusoidal function of the form
x(t )  A sin  B  t  C    D or x(t )  A cos  B  t  C    D
that models your data. Make your initial choices for A, B, C, and D carefully, providing
mathematical justification for each. You should also include the interpretation of each
parameter in the context of this physical situation.
Using your values for A, B, C, and D, plot the curve you calculated on the same axes as the
data points  t , x  . Add a column to your spreadsheet containing values of your function
x t  .
Next, you want to compare the x(t)-values with your position data. Begin by computing and graphing
the residuals, or differences between distance values from your model, x(t), and your data points,
as well as the sum of the squares of the residuals (as you did in the light intensity project).
Organize your findings in a clear, concise manner, and discuss. How well does your initial
model fit the data?
If your curve does not appear to be a good fit, adjust your values of A, B, C, and D until you
get an accurate match. Make changes in your model (try various values for A, B, C, and D),
until you are satisfied with your model. You can make these adjustments using the sliders
in the Excel file provided. Note, you want to make the sum of the squares of the residuals as
small as possible (you should discuss why this is desirable in your write-up). Keep track of the
various models you try, as you need to discuss them and the reasoning behind each in your writeup.
(25 pts) Part III – Sinudoidal Analysis (use Excel spreadsheet):
Vertical Position vs time:
Create a chart of vertical distance versus time. You should see a sinusoidal pattern to the
data. It is now your task to determine a sinusoidal function of the form
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y (t )  A sin  B  t  C    D or y (t )  A cos  B  t  C    D
that models your data. Make your initial choices for A, B, C, and D carefully, providing
mathematical justification for each. Note that these values of the parameters need not
be the same as for x  t  . You should also include the interpretation of each parameter in
the context of this physical situation.
Using your values for A, B, C, and D, plot the curve you calculated on the same axes as the
data points  t , y  . Add a column to your spreadsheet containing values of your function
y t  .
Next, you want to compare the y(t)-values with your position data. Begin by computing and graphing
the residuals, or differences between distance values from your model, y(t), and your data points,
as well as the sum of the squares of the residuals (as you did in the light intensity project).
Organize your findings in a clear, concise manner, and discuss. How well does your initial
model fit the data?
If your curve does not appear to be a good fit, adjust your values of A, B, C, and D until you
get an accurate match. Make changes in your model (try various values for A, B, C, and D),
until you are satisfied with your model. You can make these adjustments using the sliders in
the Excel file provided. Note, you want to make the sum of the squares of the residuals as
small as possible (you should discuss why this is desirable in your write-up). Keep track of the
various models you try, as you need to discuss them and the reasoning behind each in your writeup.
(25 pts) Part IV – The Extensions
You are next invited to explore your own curiosity about the motion described above.
Identify another relationship to model. Identify your variables, generate appropriate data, model
the relationship with a mathematical equation, and graph.
(25 pts) Part V – The complete model and write-up
Now that you have determined sinusoidal models for both horizontal and vertical position,
you will need to make one more graph that shows the original data (markers only) of both
horizontal position vs. time and vertical position vs. time, the best fit curve (using "curves
only") for horizontal position and the best fit curve (again using "curves only") for
vertical position. Be sure that your graph is adequately and appropriately labeled.
Finally, write up a report about your work. Each team should submit a typed report which
must include the following, in order:
a. A cover sheet containing the name of the problem, the names of the group’s members,
your teacher’s name, course, and date.
b. Include a strong introduction. This should grip the reader and make them want to
read on! It should include a restatement of the explorations, an overview of your
approach to the investigation and conclusions that you reach.
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c. The main body of the paper containing a detailed analysis of the problem, along with
statements of any assumptions you made in deriving your models and a definition of
variables, including units and reasonable domains. Also include proposed solutions,
including any you tried that you didn’t like and how you fixed them (e.g., your choice
of D was too small when you compared the graphs, so you updated your model with a
better choice of D) Here is where you should include your techniques and
justifications for choices of A, B, C, and D, all your relevant graphs (inline is
preferred), and the results of any computations that justify that you have found the
best fitting models for your swing data. (This includes horizontal and vertical
position, and your chosen extension.)
d. You will be submitting a word document to turnitin.com and an Excel file via
Moodle.
e. A Bibliography should be included if you use any additional references.
In writing up your report, pay attention to clarity and analysis. Include a strong introduction
and conclusion. You should probably think of it as a narrative leading the reader through
your solution to the problem from data collection to final model. Don’t just simply list
the parts of this assignment and your responses to them. Explain how the parts fit
together to make a whole.
How you choose to organize your work will affect your score. Plan ahead and give this some
thought! There are 100 points available for accurately reporting the mathematics you
have done in this project and meeting interim deadlines. An additional 25 points are
available for good presentation!
You may use computers, software packages, libraries, or any other inanimate sources in
creating your report. Note, texting and email are not considered inanimate sources.
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DEADLINES (5 points each):
1. Thursday, April 26, 2012: Data collected and shared with your teacher by
class
2. Monday, April 30, 2012: Rough draft of Parts II-III and written proposal
for extension to be explored
3. Monday, May 7. In class check of progress
4. Wednesday, May 9. Word document due to turnitin.com and Excel file due
to Moodle by 11:00 pm.
Note: Save all files as yournames.docx or yournames.xlsx
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For more information on residual analysis, you should visit
http://www.mathworks.com/access/helpdesk/help/toolbox/curvefit/index.html?/access/helpdesk/
help/toolbox/curvefit/bq_5ka61_1.html&http://www.google.com/search?q=residual+analysis&hl=en&client=firefoxa&rls=org.mozilla:en-US:official&hs=uO7&start=40&sa=N
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