A Project on Cycloids
Date: 01/16/99 at 19:36:59 From: Jenny Subject: Cycloids and their formulas Hi, My math class was assigned to research and write a math fair project. I chose to do cycloids, something I knew nothing about and thought I would try. However, they are confusing me. I know that the parametric formulas are x = a(theta - sin*theta) and y = a(1 - cos*theta), where (a) is the radius of the circle that is rolling on the straight line. I read that theta is the variable angle through which the circle rolls, but how can I find theta? Can I substitute all the values from 0 to 2pi as degree measures in? Now, how is this formula used and what does it figure out? Would it find all of the points on a cycloid? If I used any value for (a), would the values of x and y give me the coordinated to graph the cycloid? Also, does this equation have anything to do with time? Also, for one cusp of the cycloid to be drawn, theta would take the angle measures of 0-360 degrees. Thanks.
Date: 01/17/99 at 06:25:48 From: Doctor Jaime Subject: Re: Cycloids and their formulas Hello Jenny, For each value of theta that you choose, you get a point (x,y) of the cycloid. But you must be aware that if you are taking degrees, you get a complete cycloid when x goes from 0 to 360*a units, and if you choose radians you get a complete cycloid when x goes from 0 to 2pi*a units. Yes, if you take for theta the values from 0 to 2pi when computing cos and sin in radians, you get a complete cycloid, and if you give values for theta higher than 2pi you get more cycloids (and the same for negative values of theta). You can also use degrees, but then the cycloid will occupy the space when x goes from 0 to 360*a and other intervals with equal length. Note that for each value of a, you get a cycloid with different size. The cycloid is related to time in interesting ways. First, it is a Brachistochrone, that is a curve of descent in minimum time. It also has the tautochrone property; that is, the property that a particle P sliding on a cycloid will exhibit simple harmonic motion and the period will be independent of the starting point. Note that to draw the cusp of a cycloid, you must label the graph in this case with 0-360 intervals. At these sites you can find information on the cycloid, including its history: the MacTutor History of Mathematics Archives: http://www-history.mcs.st-and.ac.uk/history/Curves/Cycloid.html from Xah Lee: http://www.best.com/~xah/SpecialPlaneCurves_dir/Cycloid_dir/cycloid.html from the University of Melbourne's School of Physics http://www.ph.unimelb.edu.au/lecdem/mg3.htm from Alexander Bogomolny: http://www.cut-the-knot.org/pythagoras/cycloids.shtml Hope this helps. - Doctor Jaime, The Math Forum http://mathforum.org/dr.math/
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