Use your calculator or computer to find a sinusoidal

Chapter 4, Problem 390

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Modeling the Motion of a Pendulum

As a simple pendulum swings back and forth, its displacement can be modeled using a standard sinusoidal equation of the form

              y = a cos (b(x - h)) + k

where y represents the pendulum’s distance from a fixed point and x represents total elapsed time. In this project, you will use a motion detection device to collect distance and time data for a swinging pendulum, then find a mathematical model that describes the pendulum’s motion.

Collecting the Data

To start, construct a simple pendulum by fastening about 1 meter of string to the end of a ball. Set up the Calculator Based Laboratory (CBL) system with a motion detector or a Calculator Based Ranger (CBR) system to collect time and distance readings for between 2 and 4 seconds (enough time to capture at least one complete swing of the pendulum). See the CBL/CBR guidebook for specific setup instructions. Start the pendulum swinging in front of the detector, then activate the system. The data table below shows a sample set of data collected as a pendulum swung back and forth in front of a CBR.

Use your calculator or computer to find a sinusoidal regression equation to model this data set (see your grapher’s guidebook for instructions on how to do this). If your calculator/computer uses a different sinusoidal form, compare it to the modeling equation you found earlier, y = a cos (b(x - h)) + k.