Roulettes
Part 1: Warm-up Exercise
In this project we investigate the path traced out by a point on one wheel rolling
on another wheel, a situation encountered in the design of gears. We start with
a simple example that will later help us determine correct formulas.
- Find two coins of the same size
-- pennies are fine. Start by placing your two coins heads up and side-by-side
with Lincoln (or other important historical figure) facing in the same direction
on both coins, as in the following picture.
- Hold the left coin fixed, and carefully roll the other coin around it counterclockwise
without slipping. When you have rotated through an angle of pi/2
on the fixed coin, which way is Abe facing? In particular, which of the following
pictures is correct?
- When the rolling coin has traveled to the left of the fixed coin (an angle
of pi on the fixed coin), which way is Abe facing? How
many complete turns has the rolling coin made?
- How many complete turns does
the rolling coin make when it goes all the way around the fixed coin? Record
your result. In Part 3, when you write formulas for one circle rolling on
another, this test will be your reality check. When the two circles have the
same radius, you should get the same answer as you got here for the number
of turns of the rolling circle.
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