The Surprise Attack in Mathematical Problems by Louis A. Graham

Author:Louis A. Graham [Graham, Louis A.]
Language: eng
Format: azw3
Publisher: Dover Publications
Published: 2017-03-17T04:00:00+00:00

THE HULA HOOP

We are reminded of hula days by Marlow Sholander, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, who invites us to consider a vertical girl whose waist is circular, not smooth, and temporarily at rest. Around the waist rotates a hula hoop of twice its diameter. Show that after one revolution of the hoop, the point originally in contact with the girl has traveled a distance equal to the perimeter of a square circumscribing the girl’s waist.

Solution. Writes Craig Redmond, associate research engineer, American Machine and Foundry Company, Niles, Illinois: “Let D = hoop diameter, d = girl waist diameter (see Fig. 47). Hula hoop rotating about vertical, fixed girl is identical to girl rolling inside fixed hoop. A moving point on the girl will describe a degenerate cycloidal path, namely, a straight line across the hoop diameter. The reasoning is as follows: Since the ratio of diameters is 2:1, the ratio of circumferences is the same; hence when the girl has rolled one-quarter of the way around the hoop, the original contact point on her coincides with the center of curvature of the hula hoop; at half way around, the original point has traversed the hoop diameter. Continued rolling of girl reverses direction of travel of point along hoop diameter to initial position. The perimeter of circumscribing square equals four girl diameters or two hula hoop diameters which is the total displacement of initial point of contact between hula hoop and the aforementioned vertical girl.”

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