Parametric Equations for Curtate Cycloid: A Frustrating Journey

In summary, the conversation discusses a problem with finding the right solution for a curtate cycloid curve, despite trying various methods and seeking help. The tutor suggests using a position vector and adding two vectors to find the components involving b and theta.
  • #1
iberhammer
7
0
A wheel radius a rolls along a line without slipping. The curve traced by a point P that is b units from the center (b < a) is called a curtate cycloid (see figure). Use the angle θ to find a set of parametric equations for this curve.




I went through the book, went to the math lab at my university, and still I cannot find the right solution. I had

x=a(θ-sinθ)
y=a(1-cosθ)

but my tutor just said to replace a and b. I tried that on webassign and still, I received a red mark wrong! I could use some help if anyone could spare any? Thank you all :smile:
 

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  • #2
Of course, your answer can't be correct because it doesn't involve b.

Try thinking of the position vector of the moving point of radius b as the sum of two vectors. The first is the vector from the origin to the center of the circle. The x and y components of that vector only involve a. Then think of the vector from the center of the circle of radius b and angle theta. Write its components in terms of b and theta. Then add the two vectors up.
 

1. What are parametric equations for a curtate cycloid?

The parametric equations for a curtate cycloid are x = a(t - sin(t)) and y = a(1 - cos(t)), where a is the radius of the generating circle and t is the parameter that determines the position of the point on the cycloid.

2. How do I graph a curtate cycloid using parametric equations?

To graph a curtate cycloid using parametric equations, first choose a value for the radius a. Then, plug in different values for t into the parametric equations to get corresponding x and y coordinates. Plot these points on a graph and connect them to form a smooth curve.

3. What is the difference between a curtate cycloid and a trochoid?

A curtate cycloid is a type of trochoid, which is a curve traced by a point on a circle as the circle rolls along a straight line. The main difference between a curtate cycloid and a trochoid is that the center of the generating circle for a curtate cycloid is located inside the circle, while the center for a trochoid is located outside the circle.

4. Are there any real-world applications of curtate cycloids?

Yes, there are several real-world applications of curtate cycloids. One example is the shape of the teeth on a gear in a gear train, which allows for smooth rotation and efficient transfer of power. Another example is the shape of the wheels on a roller coaster, which helps to minimize friction and ensure a smooth ride.

5. Why is studying parametric equations for curtate cycloids considered a frustrating journey?

Studying parametric equations for curtate cycloids can be considered frustrating because it involves complex mathematical concepts and calculations. Additionally, it may be difficult to visualize the curve without advanced graphing tools, making it challenging to fully understand its properties and applications.

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