Online Museum Collections in the Mathematics Classroom – Kinetic Models and Calculus

Classroom Application #8: Kinetic Models and Calculus

The cycloid is a favorite topic in many calculus classes. The classic cycloid is the path traced by a point on the circumference of a circle as the circle rolls along a straight path. However, the cycloid is one member of a whole class of curves called trochoids: the paths generated by a point anywhere on the radius or extension of the radius of a circle as the circle rolls along another curve, usually a line or another circle.

The National Museum of American History (NMAH) houses several kinematic models published by Martin Schilling in the late 19th century that depict various kinds of trochoids. The images of these items, such as this one showing three epitrochoids, can be used in a calculus classroom. They can be shown simply as a visual aid during the discussion of trochoids and the mathematics behind them, or the images can be given to students as part of a project to explore and discuss the mathematics and uses of trochoids.

Kinematic Model of Hypotrochoids by Martin Schilling, series 24, model 3, number 331, ca 1900, Smithsonian Institution negative number DOR2013-50214.

To further illustrate the development of the various trochoids, a Spirograph® toy provides the perfect in-class demonstration. The Spirograph comes with circles, bars, discs and other fun shapes that can be rolled around or inside each other. One can produce the same curves shown in the three NMAH Schilling models that produce trochoids. The collection also includes a Schilling model that produces the involute of a circle, another popular calculus topic. The involute is easily modeled in class with a disc and piece of string.