Visualizing Parametric Equations for the Cycloid
Play with the applet to get a feel for it.
- It is color coordinated
so that things with the same color are related.
- Try to figure out the relationships.
Set the applet as follows
- Set a = 1 and set b = 1.
(When a and b are equal we get the cycloid.)
- Check the 'trace' box.
- Check the 'velocity vector' box.
The following questions will guide you through a visual exploration of the cycloid and related curves.
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Describe a real world system that this applet models.
- What is the yellow dot in your model?
- Notice that the radius of the wheel and the length of the blue strut
can be adjusted.
- Are there values of these parameters for which the
yellow dot sometimes moves to the left? What are they?
-
Are there values for which the yellow dot is sometimes stationary?
What are they?
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Select 'trace' and animate the system.
- For what values of the parameters does the yellow curve cross the x-axis?
- When it does cross the axis,
what is the direction of the tangent vector? Please explain.
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When 'trace' is selected, a purple line is drawn. What does it
represent, in the real world system you described above?
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Carry out the following vector manipulations:
- Express the directed segment joining the origin to the left end of this
purple line as a vector.
- Express the directed segment along the purple line from the left end to
the right end as a vector. (It will depend upon theta).
- Express the blue segment directed from the center of the circle to the
end as a vector. (This will also depend upon theta).
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Now express the position of the yellow dot as a vector-valued function
of theta.
Exploring the cusp of the cycloid
Set the parameters a and b back to 1 and turn trace on.
- Grab the circle with your mouse and drag it to the right.
- The 'cusps' are the sharp points on the cycloid. What happens
to the velocity vector at these points?
- Now set b to 3.5 and drag the circle to the right.
What happened to the cusp? What happens to the velocity vector at
the bottom of the trajectory?
- Repeat this with b = .5.