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19: The Cycloid

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    7057

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

    • 19.1: Introduction to Cycloids
      This page discusses the motion of a circle of diameter \(2a\) rolling counterclockwise along the line \(y = 2a\), creating a cycloid. It presents the coordinates of a point \(P\) on the circle after rolling \(2\theta\) using parametric equations \(x = a(2\theta + \sin 2\theta)\) and \(y = a(1 - \cos 2\theta)\). An alternative expression for \(y\) is also provided as \(y = 2a \sin^2 \theta\), along with an example for calculating the \(y\)-coordinate from the \(x\)-coordinate.
    • 19.2: Tangent to the Cycloid
      This page explores the characteristics of the tangent to a cycloid at point P, demonstrating that its slope is equal to tan \( \theta \), forming an angle \( \theta \) with the horizontal. It includes exercises for readers to validate this relationship and analyzes point A, the lowest point of the circle. The page concludes by challenging readers to prove that angle \( \psi \), which AP makes with the horizontal, is equal to \( \theta \), confirming that line AP serves as the tangent at point P.
    • 19.3: The Intrinsic Equation to the Cycloid
      This page explores the connection between arc length \(ds\), and differential elements \(dx\) and \(dy\) through the Pythagorean theorem, stating \(ds = \sqrt{(dx)^2 + (dy)^2}\). It introduces the intrinsic coordinate \(\psi\) as the angle of the tangent to the cycloid, equivalent to \(\theta\). The page includes an exercise to integrate \(ds\) to obtain the cycloid's intrinsic equation and to derive a relationship between arc length \(s\) and height \(y\) on the cycloid.
    • 19.4: Variations
      This page examines variations of the cycloid produced by a rolling circle on various lines and starting points, detailing four scenarios based on rotation direction and initial positions. It discusses angles \( \psi \) (tangent) and \( \theta \) (rolled), along with specific equations for \(x\), \(y\), and arc length \(s\), illustrating changes in these parameters with the circle's orientation. The interplay between angles and geometries is essential for comprehending cycloidal motion.
    • 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
      This page explores the brachistochrone problem, identifying the optimal path for a bead sliding under gravity from point O to point P as a cusps-up cycloid. It highlights the cycloid's isochronous property, provides the relevant parametric equations, and includes calculations to verify that this curve results in the shortest travel time for the bead. The findings confirm the initial hypothesis regarding the cycloid's efficiency.
    • 19.8: Contracted and Extended Cycloids
      This page explains the motion of a circle of radius a rolling along a line, particularly focusing on a point P at a distance r from the center. It distinguishes between contracted and extended cycloids based on the relationship between r and a. The section includes the relevant terminology and provides parametric equations for both cycloid types, supported by illustrative figures and examples.
    • 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).

    This page titled 19: The Cycloid is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.