Modeling the Cycloid

Cycloid Movie

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Activity: Modeling the Cycloid

Objective: To give students a visualization of the Cycloid Curve.

For this model, we took a cylindrical can, marked a point on the circumference of its bottom, rolled it along a flat surface, and made a video of the rolling can. Theory tells us that the point on the can's rim traces a path called a cycloid. The parametric equations that model the cycloid are

x(t) = r(t - sin(t)), y(t) = r(1 - cos(t))

where r is the radius of the circular bottom of the can and t is the angle of rotation of the rolling can. The software package "GraphicConverter" was used to convert the video into a sequence of individual frames and the frames were imported into TEMATH. The first image was positioned over a pair of coordinate axes. Using TEMATH's Line tool, students can measure the radius r of the can from one of the frames of the video. The standard parametric equations for the cycloid assume that x(0) = y(0) = 0, that is, the cycloid "begins" at the origin. However, the first frame of the video doesn't necessarily show the marked point at the origin. You will need to find the first picture in the sequence where the red dot on the bottom of the can touches the x-axis. Then, this frame must be moved so that the point is aligned with the origin, or, as we prefer, TEMATH's Line tool can be used to measure the horizontal shift between the origin and the point on the can. If the horizontal shift is measured, then the parametric equation for x becomes

x(t) = r(t - sin(t)) + h,

where h is the horizontal shift. Once the radius r and the horizontal shift h are measured, the parametric equations are entered into TEMATH and plotted on top of the image of the rolling can. To check how well the theory models the experiment, students display the images in a rapid sequence simulating a video. If the measured parameters are accurate, students will see the point on the rolling can tracing the plotted parametric cycloid.

Click the buttons to see the cycloid

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