Solve Cycloid Question to Find Parameter 'a

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In summary, the conversation discusses finding the value of 'a' for a given point (x2, y2) that satisfies the parametric equations for a cycloid. The solution is provided for two specific cases: x2 = (Pi)b, y2 = 2b and x2 = 2(Pi)b, y2 = 0. The question also asks for the corresponding minimum time it takes for the particle to reach the point. The participant has attempted to solve for the first point but has not been successful in finding a solution without the transcendental nature of the equation. They also have a question about how to apply graphical or numerical techniques to solve the problem. However, they later realize that the solution can be easily
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Hi. I was working through a problem and got stuck. The problem asks "To find the value a for a given point (x2, y2) usually requires solution of a transcendental equation. Here are two cases where you can do it more simply: For x2 = (Pi)b, y2 = 2b and again for x2= 2(Pi) b and y2 = 0 find the value of a for which the cycloid goes through the point 2 and find the corresponding minimum times."

To clarify, the parametric equations for a cycloid are

x(t) = a (t - sin[t])
y(t) = a (1- cos[t])

and the question asks to find the constant 'a' for which the cycloid curve contains the point (x2, y2) and then find the time it takes the particle to reach the point.

I haven't been able to figure out how to find the parameter a. I have only attempted the solution for the first point. My thought was to simply eliminate the parameter t and then solve for a in terms of b, but this doesn't eliminate the transcendental nature of the equation. Does anybody have a suggestion?I also have an additional question. I understand that solutions to transcendental equations require graphical or numerical techniques, but I don't understand how that can be applied to this problem (if you were given a certain numeric ordered pair for (x2,y2) for instance). How would this be done?
 
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Wow, nevermind. It is quite easy to see by inspection that a = b ;)
 

1. What is a cycloid and how is it related to the question?

A cycloid is a curve generated by a point on the circumference of a circle rolling along a straight line. In the question, we are trying to find the parameter 'a' that helps define the shape of the cycloid curve.

2. What is the significance of finding the parameter 'a' in a cycloid question?

The parameter 'a' helps determine the size and shape of the cycloid curve. It is an important factor in understanding the properties and behavior of the curve.

3. How do I solve a cycloid question to find the parameter 'a'?

To solve a cycloid question to find the parameter 'a', you can use the equation a = x/y, where x is the distance between the center of the circle and the point on the circumference and y is the distance between the point on the circumference and the line.

4. Are there real-life applications of cycloids and finding the parameter 'a'?

Yes, cycloids and finding the parameter 'a' have practical applications in various fields such as engineering, physics, and mathematics. For example, cycloids can be used to design gear teeth in machinery and to study the trajectory of a projectile.

5. Are there any other methods to find the parameter 'a' in a cycloid question?

Yes, there are other methods to find the parameter 'a' such as using calculus and geometry. These methods may be more complex, but they can provide a more precise and accurate value for 'a' depending on the given information in the question.

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