How to get an A+ on the 1st Semester Project:
For the full 40 points, you may do the following. Up to 10 more points of extra credit may be earned if
you build something cool or go beyond in some way (especially mathematically).
Gather Ideas and Build or Take a Picture / Video (10 points)
1. Take a picture or video of something that is sinusoidal. Be sure to include a ruler or
something that is a length reference. Make sure the ruler is positioned in the same plane as
the sinusoid.
2. Import the picture (of if it’s a video, take a frame) into Geometer’s sketchpad.
Analyze (10 points)
3. Rotate the picture and scale it until the sinusoid fills most of the screen.
4. Show Grid (click on “Graph”)
5. Adjust the grid so that the ruler measurement matches the grid.
6. Move the grid (or the picture) so that you have an untranslated cosine function.
7. Figure out the equation for the graph.
o Remember: y = Acos(Bx) where A is the Amplitude and B is the number of cycles
in 2π .
o To find A, just look at your graph and estimate the amplitude.
o To find B it’s easy to use a proportion. Figure out how long one cycle is by looking
€ graph and you can fill out the following proportion:
at your
€
1
B
=
length of 1 cycle 2π
8. Graph your equation. Click on “Graph”, “Plot New Function”. After you enter your
equation,, hit enter, then click to be in Radian mode. If your graph doesn’t match very
closely, you can readjust the A and B values until you are satisfied. Also, you will want to
€ to not be in terms of pi. You can click on “Graph”, “Grid Form” and deselect
get the x-axis
“Trigonometric Axis”.
9. Print Preview – Be sure to fit it to the page, then print!
Compute (10 points)
Part I
10. For the function you found above, show algebraically how to evaluate it for a given x. In
GS you can show this point. Also describe in words the meaning of the answer.
11. For the function you found above, show algebraically how to solve it for a given y. Also
describe in words the meaning of the answers. (you will have an infinite number of answers
that you will need to describe) In GS, graph a horizontal line for your y value and verify
that your algebra is correct.
Part II
12. Take a second function that is different than the one above. In a new window in Geometer’s
Sketchpad graph both functions individually. Color code them so it is easy to tell which
graph goes with which equation.
13. On the same axes, graph a third function which is the two functions added together. Color
code the third function also. Your third equation will look like:
o y = A1 cos B1 x + A2 cos B2 x
14. The equation from #13 above is a sum of two sinusoids. Turn it into a product of two
sinusoids using a technique from chapter 5.
15. Finally, plot the equation you found in #14 on the same axes as the other three. Show that it
€ is equivalent to the sum of sinusoids equation.
Semester 1 Harmonic Motion Project
Gather Ideas & Build (10 Points)
Find something interesting to build or to observe that exhibits harmonic motion.
Analyze (10 Points)
Describe and analyze 2 different sinusoids (they can both be from the same observed or built thing).
One must have all four Transformations y = C + Acos B(θ − D) . Some form of technology must be
used to analyze. For example: Geometer’s Sketchpad, Excel, etc.
€
Compute (10 Points)
What would it look like if you could add your two sinusoids? Show the math, show the resulting
graphs. Turn the addition into multiplication. Verify the multiplication graph vs. the addition graph –
are they really the same?
Evaluate – Given an “x” find a “y”
Solve – Given a “y” find an “x”. (find multiple x’s – it’s periodic!)
Show your mathematical prowess. The above is just one way to show your understanding of
trigonometry.
Present (10 points)
4-8 minute Presentation on your findings. Either bring in your built/observed item, or show a video/
picture of it for the group.
Bonus for clever presentations
1 or 2 people per “Group”
Some Ideas:
pendulum wave effect
Damped pendulum
Simple harmonic motion
MIT Physics Demo -- Spray Paint Oscillator
Sound analysis
Vibrating string analysis
Standing Waves Part I: Demonstration
Twirling chain analysis
The simplest motor in the world
Harmonic series (music) – build tubes
Tides at Cherry point, etc (NOAA.com)
Something harmonic as videoed on your iPhone, etc.
Standing waves in a tube