![]() The tight petals of this curve correspond to the surprising acceleration riders feel (when whipping into retrograde) on such a benevolent-seeming ride!Īnother familiar family of curves found in the Ferrisferris wheel can be seen by tracing the height of a rider from the ground over time. Second, passengers on many of these rides trace paths through space that describe classical geometric curves, just as the Ferrisferris wheel provides a physical interpretation of an epicycloid. (As far as I know, they've yet to invent an attraction that actually dilates its rider, although with many rides, one frequently feels somewhat dilated on disembarking.) Building geometric models of such mechanisms is an excellent introduction to the basic transformations. First, many of these machines-merry-go-rounds, Ferris wheels, Mad Hatter's teacups, and so forth-have motions easily described by a short series of rotations, reflections, and translations. Modeling amusement park rides with simple geometric graphics can serve a double purpose in the classroom. Conversely, if the outer wheel rotated more slowly about its center than around the inner wheel, we'd say it skidded as it rolled. In terms of a physical interpretation of the epicycloid, multiple petals per revolution implies that the rolling circle is "spinning out" as it drives around the fixed circle. That the Ferrisferris wheel requires several revolutions to "close" the curve, as is about to happen in Figure 12, expresses a fundamental relationship between the diameters of the two wheels (one of which is only implicit in the construction), and the number of petals in the rose characterizes the speed of the rolling circle. This complex, many-petaled rose is a form of epicycloid, the curve traced by a point on a circle that rolls on the outside of a fixed circle. To plot the locus of the moon, simply trace a point on one of the passenger cars while the Ferrisferris wheel spins, as shown in Figure 12. The endpoints of this axis are two planets orbiting the sun, and the passenger cars are moons of their respective planets, rotating about them as the wheels spin on their hubs. In terms of the previous astronomical analogy, the midpoint of the axis represents the sun (a fixed "center of the universe"). Kristin provides an animation button (not shown) that sets the entire contraption in motion. Passenger cars hang from brackets mounted at regular intervals along the rotating wheels. Two independent wheels rotate on the ends of a structural axis, which in turn spins slowly around its midpoint. A novel interpretation of this last question-finding the locus of one body (the moon) as it orbits another orbiting body (the earth as it rotates about the sun)-crossed my desk recently in the following sketch, submitted by Kristin Wallace (a high school classmate of Nathan Reed's).įigure 11 illustrates a double-Ferris wheel amusement park ride. ![]()
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