Cycloidal Paths in Physics

TL;DR

This paper derives parametric equations for cycloidal paths in various physical contexts, including rolling disks, celestial orbits, and electron-spin resonance, revealing the underlying cycloidal motion in these phenomena.

Contribution

It provides a unified derivation of cycloidal paths in rolling motion, celestial orbits, and spin resonance, extending classical cycloid analysis to new physical situations.

Findings

Derived equations for cycloid, curtate, and prolate cycloid paths.

Showed cycloidal paths occur in celestial and quantum systems.

Modified cycloid equations for slipping rolling and spin resonance.

Abstract

A popular classroom demonstration is to draw a cycloid on a blackboard with a piece of chalk inserted through a hole at a point P with radius r = R from the center of a wood disk of radius R that is rolling without slipping along the chalk tray of the blackboard. Here the parametric equations versus time are derived for the path of P from the superposition of the translational motion of the center of mass (cm) of the disk and the rotational motion of P about this cm for r = R (cycloid), r < R (curtate cycloid) and r > R (prolate cycloid). It is further shown that the path of P is still a cycloidal function for rolling with frictionless slipping, but where the time dependence of the sinusoidal Cartesian coordinates of the position of P is modified. In a similar way the parametric equations versus time for the orbit with respect to a star of a moon in a circular orbit about a planet that is in a circular orbit about a star are derived, where the orbits are coplanar. Finally, the general parametric equations versus time for the path of the magnetization vector during undamped electron-spin resonance are found, which show that cycloidal paths can occur under certain conditions.