A small wheel rolls around a large fixed wheel without slipping. What path does a point on the edge of the small wheel trace out as it goes? You can answer this quaestion with vectors and an understanding of radian measure. The Spirograph toy makes such plots, called "cycloids." Figure 43.1 is a sample path.

Figure 43.1: An epicycloid

This is not trivial. Vectors help, and here is a special case to show you why. Suppose the large wheel has radius 3 and the small wheel has radius 1. How many turns must the small wheel make while traveling once around the large wheel? This is a famous question because it appeared on a national college entrance test as a multiple-choice question. The reason it is famous is that the correct answer was not among the choices.

Here is the obvious solution: The big wheel has three times the circumference of the small wheel (you can use ), so the small one must mark off three revolutions to cover the same distance and travel all the way around the large one. Obvious, but wrong. The small wheel turns four times. Run the computer animation in EpiCyAnimate to see for yourself (Figure 43.2).

43.1 Epicycloids

We will use vectors to analyze the motion of a point on the little wheel. This will give us parametric equations for the whole path but also show us why the small wheel turns four times in the 3-to-1 ratio case. You will need to know that the length of a circular segment of angle and radius r is

For example, the length all the way around a circle of radius r is what we get when we go an angle , so .

We begin with one vector pointing to the center of the small wheel. We know from the sine-cosine discussion that

if measures the angle from the x-axis and R=r_b+r_s, the sum of the radii of the large and small wheels.

We also have a second vector, , pointing from the center of the small wheel to the tracing point on its rim. Again using sine-cosine, we have

where is the angle measured around the small wheel starting from the horizontal and r_s is the radius of the small wheel.

The vector is a displacement; it does not start at the origin. The position vector interpretation is simply that

is the arrow from the origin to the reference point on the small wheel. This may be written in components as

where the components of C and W are given by the formulas above. The two vectors start in the position shown in figure 43.3 with both pointing to the right of their centers.
Figure 43.2: Starting Position

After a small increase in , the vectors look as in Figure 43.4.

Figure 43.3: Position after Rotation

The length of portion along the big wheel where contact has been made is . The portion contacted along the small wheel is because the contact point lags behind the original horizontal reference line. "No slipping" means that equal lengths are marked off along both wheels, so

Consider the case where r_b=3 and r_s=1. When so that the small wheel has gone all the way around the large wheel, we have . The small wheel turns four times a full turn. The computer program Cycloid on our website contains the formulas for the epicycloid we have just derived and uses them to plot the curve. We want you to extend the idea of using parametric circles and vector addition together in the next exercise.

43.2 Cycloids

The Cycloid
A wheel of radius r_s=1 rolls to the right along a straight line at speed v without slipping. The vector pointing to the center of the wheel has the form

where t is the time.

Figure 43.4: Start of a Cycloid

The vector pointing from the center of the wheel to a reference point on the wheel has the form

In what position does the wheel begin?

Figure 43.5: After a Small Change

The distance moved along the line at time t is . How much is the distance marked along the wheel in terms of ? Equate these, assuming no slipping, and find in terms of t. Modify the computer program Cycloid to plot this curve, known as a cycloid.