## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## April 24, 2007

### Tilt-A-Whirl Chaos

Much of the fun of an amusement park ride is its stomach-churning, mind-jangling unpredictability. The Tilt-A-Whirl, for example, spins its passengers in one direction, then another. The ride's cars sometimes hesitate between moves and at other times swing suddenly from one motion to another. You never know what to expect next.

These surprising movements arise from a simple geometry. A rider sits in one of seven cars, each mounted on a circular platform and free to pivot about the center of the platform. The platforms, in turn, move at a constant speed along a track with three identical hills, which tilt the platforms. So the platform movements are perfectly regular, but the cars whirl around on their own quite unpredictably.

The only thing that a Tilt-A-Whirl operator can adjust is the speed at which the platforms travel around the track. When the platforms move at very low speeds, each car completes one backward turn as its platform goes over a hill. At high speeds, a car gets slammed to its platform's outer edge and stays in that position. In either case, the motion is predictable. It's only at speeds somewhere in between these extremes that a car's motion becomes complicated and unpredictable. So it's important for a Tilt-A-Whirl operator to make sure the ride runs at the proper speed, about 6.5 revolutions per minute.

At just the right speed, it becomes nearly impossible to predict exactly what will happen from one moment to the next during a ride, or from one ride to the next. Because a car's motion depends on the weight of its passengers and where they are sitting, the thrills and chills are different each trip.

Do you like the Tilt-A-Whirl? You can find similar thrills at the amusement park by looking for rides with cars that are free to rotate or shift back and forth as they follow a fixed track.

Muse, May/June 2001, p. 34.