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A Magic Parabola

Haiduke Sarafian
Organization: The Pennsylvania State University
Department: Department of Physics
Nenette Sarafian
Organization: University of Virginia

2005 Wolfram Technology Conference
Conference location

Champaign IL

We consider a soft metal strip of mass a and length a and bend it into a parabola. Applying Mathematica, we determine the coordinate of its center of mass. By bending the strip continually, mathematically, we reshape the strip in a family of parabolas with a common vertex and axis. For each parabola, we evaluate the coordinates of the corresponding center of mass. We animate the process depicting each parabola along with its corresponding center of mass and focus. Mathematica proves its ingenious usefulness providing a road map to unveiling the hidden features of the problem at hand; animation of movement of the center of mass and the corresponding focus along the common axis reveals a peculiar situation. Among all possibilities, we identify a unique case, the one with the overlapping center of mass and focus. We write a Mathematica program, enabling automation and animation of the entire process conducive to evaluation of the needed parameter. Utilizing this parameter, we display the unique, magic parabola along with its coincidental center of mass and the focus. The authors built a one-to-one scaled replica of the magic parabola and utilized two pieces of string as well as a laser beam to verify the correctness of the prediction. The article includes two digital photos of one such replica: one with the strings and the other with the laser beam.

*Education > College
*Education > Precollege
*Mathematics > Algebra > Polynomials

TechConf2005_MagicParabola_Sarafian.zip (5.1 MB) - ZIP archive