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Comparison of the Runtime and the Kinematics of a Bead Sliding along an Equal Arc Length Cycloid and Catenary with and without Kinetic Friction under Gravity's Pull

Haiduke Sarafian
Organization: The Pennsylvania State University
Department: Department of Physics

2006 Wolfram Technology Conference
Conference location

Champaign IL

It is well established that the runtime of a bead sliding along a concave frictionless cycloid under gravity’s pull is the shortest [1]. These studies focus on the runtime and hence ignore the kinematics of the bead such as time-dependent coordinates, speed, acceleration, surface reaction, and the attack angle. In this work we report one such analysis. We augment our analysis to study the sensitivity of the aforementioned metrics with the inclusion of kinetic friction. Furthermore, with a major caveat we link this work to our recently reported analysis [2]; applying Mathematica, we evaluate sets of parameters to form a cycloid and a catenary of equal arc length between two points on a vertical plane—something that never has been done before. We then compare the runtime and kinematics of two beads sliding along these two curves. The analysis embodies cases with and without kinetic friction. Among various interesting observations, we report the fact that the runtime of the cycloid even with friction is shorter than the corresponding catenary. [1] Weisstein, Eric W., MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BrachistochroneProblem.html (See the paper for further references) [2] Sarafian, H., aInternational Mathematica Symposium, IMS06, Avignon, France.