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19: The Cycloid

Last updated

Aug 8, 2022
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- 18.4: Area of a Catenoid
- 19.1: Introduction to Cycloids

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A cycloid generated by a rolling circle. (CC BY-SA 3.0; Zorgit).

19.1: Introduction to Cycloids
19.2: Tangent to the Cycloid
19.3: The Intrinsic Equation to the Cycloid
19.4: Variations
19.5: Motion on a Cycloid, Cusps Up
We shall imagine either a particle sliding down the inside of a smooth cycloidal bowl, or a bead sliding down a smooth cycloidal wire.
19.6: Motion on a Cycloid, Cusps Down
We imagine a particle sliding down the outside of an inverted smooth cycloidal bowl, or a bead sliding down a smooth cycloidal wire.
19.7: The Brachystochrone Property of the Cycloid
19.8: Contracted and Extended Cycloids
19.9: The Cycloidal Pendulum
If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the "string" is constrained between the adjacent arcs of the cycloid, and the pendulum's length is equal to that of half the arc length of the cycloid (i.e., twice the diameter of the generating circle), the bob of the pendulum also traces a cycloid path.
19.10: Examples of Cycloidal Motion in Physics
Several examples of cycloidal motion in physics come to mind.

Thumbnail: Schematic of a cycloidal pendulum. (Public Domain; 3piecesuits).

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