Calculus Volume 3
Calculus Volume 3
16th Edition
ISBN: 9781938168079
Author: Gilbert Strang, Edwin Jed Herman
Publisher: OpenStax
Textbook Question
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Chapter 1.1, Problem 1.5SP

Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid. called the curtate and prolate cycloids.

First. let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 1.13).

As we have discussed, we have a lot of flexibility when parameterizing a curve. In this case we let our parameter c represent the angle the tire has rotated through. Looking at Figure 1.13, we see that after the tire has rotated through an angle of t, the Position of the center of the wheel, C = ( x c , y c ) , is given by

Chapter 1.1, Problem 1.5SP, Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a , example  1

Furthermore, letting A = ( x A , y A ) denote the position of the ant, we note that

x C x A = a sin t and y C y A = a cos t .

Then

   x A = x c a sin t = a t a sin t = a ( t sin t ) y A = y C a cos t = a a cos t = a ( 1 cos t ) .

Chapter 1.1, Problem 1.5SP, Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a , example  2

Figure 1.13 (a) The ant clings to die edge of the bicycle tile as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t.

Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t.

After a while the am is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel. the ant has changed his path of motion. The new path has less up-and-down motion and is called a mutate cycloid (Figure 1.14). As shown in the figure. we let b denote the distance along the spoke flow the center of the wheel to the ant. As before, we let t represent the angle the fire has rotated through. Additionally, we let C = ( x C , y C ) represent the position of the center of the wheel and A = ( x A , y A ) represent the position of the ant.

Chapter 1.1, Problem 1.5SP, Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a , example  3

Figure 1.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant's path of motion after he climbs closer to the center of the wheel. This is called a mutate cycloid. (c) The new Setup, now that the ant has moved closer to the center of the wheel.

5. What do you notice about your answer to pan 3 and your answer to part 4?

Notice that the ant is actually traveling backward at times (the “loops” in the graph), even though the train continues to move forward. He is probably going to be really dizzy by the time he gets home!

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Chapter 1 Solutions

Calculus Volume 3

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