Modeling Gravity and Friction: A STEM Activity Using Trigonometry

By: John Lamb, Ph.D.The University of Texas at Tylerhttp://math.uttyler.edu/ut3mc/stem

How this startedDiscovery 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 researchGalileo TautochronePendulum

Galileo

Tautochrone (curve of equal decent)

Cycloid

Pendulum

Guiding QuestionWhat function would model the

effect gravity and friction has on a ball as it travels on a curved ramp?

Field Trip To Discovery Science PlaceI 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

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35

Series1

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25 30 35Series1

Raw Data

Cleaned Data

Modeling Function

Middle and High School Math and Science TeachersAt 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.

MaterialsDecaying Ball Materials2 CBRs2 TI-84sBall8’ corner molding (cove) piece2 Ring StandsVelcro StrapsDuct TapeMeasuring TapeComputerTI-Connect SoftwareExcel

Some ExamplesShow Excel documents

Then I wrote to present hereI was accepted…YayBut 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

Show iPad Video

Now, Let’s ExperimentPhase 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 ExploreMake 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 Transfer your data from your calculator to your

computer.◦ 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 MathematicsUsing the calculators and L3 and L4Plot 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 emailIf you have questions or want the

powerpoint and any handouts we have used, please visit our STEM website

http://math.uttyler.edu/ut3mc/stem

My email is jlamb@uttyler.edu