### PDF of PowerPoint - The University of Texas at Tyler

```Modeling Gravity and Friction:
A STEM Activity Using Trigonometry
By: John Lamb, Ph.D.
The University of Texas at Tyler
http://math.uttyler.edu/ut3mc/stem
How this started
Discovery Science Place had a “Golf Ball
Room” with spiraling ramps, rollercoaster
simulations, and a station that illustrated
gravity and friction for young children.
 As a secondary mathematics enthusiast, I
wanted to know if the behavior of the ball
on the gravity ramp could be modeled?

This led to some research
Galileo
 Tautochrone
 Pendulum

Galileo
Tautochrone
(curve of equal
decent)
Cycloid
Pendulum
Guiding Question

What function would model the effect
gravity and friction has on a ball as it
travels on a curved ramp?
Field Trip To Discovery Science
Place

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I had three independent study pre-service
mathematics education students and we
went to the discovery Science Place with
calculators, CBRs, and laptops.
I had an conceptual idea of how to collect
the data and answer the guiding question,
but I tried to guide the students toward
discovering what they needed to do to
collect accurate data that would model the
behavior of the golf ball on the ramp.
We were mildly successful, only because of
limited time.
Here is what we came up with
Cleaned Data
1.4
1.5
1.2
1
1
0.8
0.5
0.6
0
Series1
0
0.4
0.2
5
10
15
20
-0.5
0
0
5
10
15
20
25
30
35
-1
-1.5
Raw Data
Modeling Function
1.4
1.2
1
0.8
Series1
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
25
30
35
Middle and High School Math and
Science Teachers
At the end of that semester, I presented the idea
to some middle and high school math and science
teachers from the East Texas area.
 I couldn’t take them all to the Discovery Science
Place, so I had to bring the ramp to them.
 I wanted to do something that I believe a math
and science teacher operating on a small (literally
microscopic) budget could do.
 I therefore spent about \$5 and build a ramp using
mostly supplies a school would already have.

Materials

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

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Decaying Ball Materials
2 CBRs
2 TI-84s
Ball
8’ corner molding (cove) piece
2 Ring Stands
Velcro Straps
Duct Tape
Measuring Tape
Computer
TI-Connect Software
Excel
Some Examples

Show Excel documents
Then I wrote to present here
I was accepted…Yay
 But I now had the problem of getting my
ramp not just across town, but across the
country.
 So I turned to the internet and found
these stringless pendulums you have at
the table

Now, Let’s Experiment
Phase I:
 Make sure your CBRs on the left and
right are the same distance away from the
center of the ramp.

◦ What is the distance from your CBR to the
ball resting at the center of the ramp?
Engage and Explore
Make Sure you get smooth graphs!

Using correct timing, collect distance over
time data using both CBRs for 15 seconds for
the ball bearing and 10 seconds for the golf
ball.You want to collect 300 data points.
◦ List the minimum points (ordered pairs) found
from each side of the ramp from either the golf
ball or ball bearing.
Example Ordered Pairs
More Exploration
• What observations can you make about the
period and amplitude of the graphs you found?
(hint: you may need to use the distance and
points found earlier)
Let me Wave My Hands at Phases II

◦ Using the data from the CBR where the ball started on the
ramp, you need to transform the data so it is reflected and
translated to the first quadrant representing the upper half of
the sinusoidal function. This is done by subtracting the initial
CBR distance then multiplying by negative one. Then remove all
the negative distance points.
◦ Using the data from the other CBR, you need to translate the
points down by the initial CBR distance to the fourth quadrant
representing the lower half of the sinusoidal function. Then
remove all the positive distance points.
◦ Now combine both data sets, order them by time, and remove
any points that share the same time value. Then graph the
cleaned data to see the damping trig function.

Transfer the data from the computer to the calculator.
The real Mathematics
Using the calculators and L3 and L4
 Plot the points on the calculator and then
try and determine the damping trig function
that best fits your data. The damping function
will be in the form f(x)=a*cos(b*x+c)+d
 (hint: f(x)=a*e^(b*x)*cos(2*pi/period*x+c)+d where a, b, c,
and d are unknowns)
 What is your damping function that models
the movement of the ball?

Visit Website or email

If you have questions or want the
powerpoint and any handouts we have