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19.3: The Intrinsic Equation to the Cycloid

  • Page ID
    7053
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    An element \(ds\) of arc length, in terms of \(dx\) and \(dy\), is given by the theorem of Pythagoras: \( ds = ((dx)^2 + (dy)^2))^{1/2} \) or, since \(x\) and \(y\) are given by the parametric Equations 19.1.1 and 19.1.2, by And of course we have just shown that the intrinsic coordinate \( \psi \) (i.e. the angle that the tangent to the cycloid makes with the horizontal) is equal to \( \theta \).

    Exercise \(\PageIndex{1}\)

    Integrate \(ds\) (with initial condition \(s\) = 0, \( \theta \) = 0) to show that the intrinsic equation to the cycloid is

    \[ s = 4 a \sin \psi \label{19.3.1}\tag{19.3.1} \]

    Also, eliminate \( \psi \) (or \( \theta \)) from Equations \( \ref{19.3.1}\) and 19.1.2 to show that the following relation holds between arc length and height on the cycloid:

    \[ s^2 = 4 ay. \label{19.3.2}\tag{19.3.2} \]


    This page titled 19.3: The Intrinsic Equation to 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; a detailed edit history is available upon request.

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