Parametric equation of a Curtate cycloid

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In summary, we discussed finding a set of parametric equations for a curtate cycloid, which is the resulting curve traced out by a reflector attached to a spoke of a bicycle wheel as it rolls without slipping on a flat surface. Using vectors, we found the equations x=at-|a-b|sin(t) and y=a-|a-b|cos(t). The correctness of this answer cannot be verified due to it being an even numbered problem, but it appears to be correct.
  • #1
Townsend
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The problem says;

Suppose that a bicycle wheel of radius a rolls along a flat surface without slipping. If a reflector is attached to a spoke of the wheel at a distance b from the center of the resulting curve traced out by the reflector is called a curtate cycloid.

I need to find a set of parametric equations for the curtate cycloid.

Using vectors I found the following set of parametric equations

[tex]
x=at-|a-b|sin(t)[/tex]

[tex]
y=a-|a-b|cos(t)
[/tex]

I would appreciate it if someone could tell me if my answer is correct since this is an even numbered problem I cannot check my answer. And don't worry...I don't have to turn this in or anything so it's not cheating... :smile:
 
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  • #2
looks good to me
 
  • #3
mathmike said:
looks good to me

Hey thanks Mike.. :smile:
 

1. What is a curtate cycloid?

A curtate cycloid is a type of parametric equation that describes the path of a point on the circumference of a circle as it rolls along a straight line. It is also known as a trochoid.

2. How is a curtate cycloid different from a regular cycloid?

A curtate cycloid is a shortened version of a cycloid, meaning the circle used to create the curve is smaller than the one used for a regular cycloid. This results in a curve that is less steep and has a smaller amplitude.

3. What is the parametric equation for a curtate cycloid?

The parametric equation for a curtate cycloid is x = a(t - sin t) and y = a(1 - cos t), where a is the radius of the circle and t is the parameter representing the angle of rotation.

4. What are some real-world applications of curtate cycloids?

Curtate cycloids can be seen in the motion of a rolling wheel or the shape of a bead on a rotating wire. They are also used in the design of gears and cams in machinery.

5. Are there any interesting properties of curtate cycloids?

Yes, curtate cycloids have several interesting properties. They are self-intersecting, meaning the curve crosses itself at certain points. They also have a constant normal force, meaning the force acting on an object moving along the curve is always perpendicular to the curve at that point.

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