- Be familiar with the graphs of sine and cosine functions.
- Know how the amplitude, period, phase shift, and vertical shift are related to the equation and graph of a sine function.
Ohio Standards Alignment
- Each student will need a copy of the activity pages and a sheet of graph paper.
Sine curves, periodic motion, period, amplitude, phase shift, vertical shift, connections, representation
This problem relates the circular motion of a point on a Ferris wheel to its up-and-down motion relative to the ground. Students can measure the height of a particular point as the Ferris wheel rotates, plot the height versus time, and observe that the graph resembles that of a sine function. Students can determine the period, amplitude, phase shift, and vertical shift for their graph and construct a mathematical function y = a sin (bx + c) + d to fit the data.
- Collect periodic data and model these data by a sine or cosine function.
- Apply concepts of period, amplitude, phase shift, and vertical shift.
- Explore simple harmonic motion.
When you ride on a Ferris wheel, you move in a circle relative to the hub of the wheel.
How would you describe your motion relative to the ground?
What kind of a function could you use to represent this motion?
If the Ferris wheel rotates at a constant speed, at what positions are you rising or falling the fastest?
Students will investigate the motion of a point on a Ferris wheel that is 80 feet in diameter, with its hub 45 feet above the ground. It takes 36 seconds for the wheel to make one complete revolution. The starting position (t = 0) is one-fourth the way around the wheel from its lowest point.
Using Figure A on the activity pages, students can measure the height of the designated seat at three-second intervals, plot the height versus time, and find a sine curve that will fit the data. They will need to determine the period, amplitude, phase shift, and vertical shift of the function. If graphing calculators are available, they can use trigonometric regression to check their work.
- Students can get an intuitive notion of vertical velocities by analyzing the height-versus-time data. They can check their answers with the answer key.
- The problem is posed so that the phase shift is zero. If we change the problem so that t = 0 when the seat is at its lowest point on the Ferris wheel, then the phase shift leads to a more complicated function. This might be a good place to discuss advantages and disadvantages of choosing the initial point in an experiment.
- Many natural phenomena exhibit periodic behavior and can be approximated by sinusoidal functions. Listed below are a few examples. The last two can be investigated with graphing calculators and CBL technology.
Back-and-forth motion of a pendulum or swing
Up-and-down motion of a weight bouncing on a spring
Hours of daylight at a particular location over the course of a year
Depth of the tide at a particular location over the course of a day
Vibration of a guitar string
Flickering of a fluorescent light
The web page Sinusoids: Applications and Modeling, by David Roth, includes a Ferris wheel, animation, some history of Ferris wheels, and suggestions for other sources of periodic data that can be modeled using a sine or cosine curve.
Paul Foerster has a section on "Sinusoidal Functions as Mathematical Models" in his textbook, Precalculus with Trigonometry Concepts and Applications, published by Key Curriculum Press (2003).
Developed by David Kullman as an adaptation of Sinusoids: Applications and Modeling from the Demos with Positive Impact site.