Associated Topics || Dr. Math Home || Search Dr. Math

### Cycloid

```
Date: 01/30/98 at 18:04:34
From: Chris
Subject: What is a cycloid and what does it do?

Hi,

My school is having an Ed-Fair (education fair) in a few weeks. I
decided to do my project on math. My math teacher suggested I might
want to do a project on cycloids. I took her recommendation and now I
am stuck.

This is what I know so far: A cycloid is the curve traced by a fixed
point on the radius of a circle (or it's extension) while rolling
across the x axis (given that it doesn't slip). Also, two equations
can be made from cycloids. To understand these equations you should
know x = the distance of the curve along the x axis, r = radius of
circle, y = the distance of the curve along the y axis, theta = the
angle between the line of the radius and the vertical line through
the center of the circle. The equations are x = r(theta - sin * theta)
and y = r(1 - cos * theta).

Could you please tell me the purpose of cycloids and more about them?
```

```
Date: 01/31/98 at 08:03:38
From: Doctor Pete
Subject: Re: What is a cycloid and what does it do?

Hi,

So far, you're doing fine.  Some ideas that would help when presenting

* draw a picture of a cycloid.  Include the coordinate axes and the
circle.  How would you visually interpret the values x, y, r, and
theta?

* generalize cycloids.  That is, what would happen if you took a
point along the radius of the circle, but not "on" the circle's
circumference?  What is the curve traced by this point?  What if
this point is inside the circle?  Outside?

* What kinds of curves would you get if you rolled circles around
other circles, as opposed to a straight line?  How do these curves
change as the relative sizes of these circles change?  In
particular, what curve do you get when you roll two circles of
equal size around each other?

If you want more specific information about cycloids, here is some
stuff I know.

Take a marble and a curved ramp. If you roll the marble down the ramp,
it takes a certain amount of time for it to reach the bottom. Now, how
should the ramp be curved so that the marble will roll to the bottom
in the shortest amount of time? The answer is that the ramp should be
half a cycloid (turned upside-down). It makes sense, because initally
you want the marble to go fast, so the ramp should be steep, but once
it's going fast, you want it to move forward, so you need to curve it
out since the steepness doesn't take it very far.

This problem was considered by Bernoulli and was referred to as the
brachistochrome problem.  Isaac Newton was the first to solve it
correctly; it is a classic use of the calculus of variations.

-Doctor Pete,  The Math Forum
Check out our web site!  http://mathforum.org
```
Associated Topics:
High School Calculus
High School Conic Sections/Circles
High School Geometry
High School Projects

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search