Editing Cycloid (section)

== Cycloidal pendulum ==
[[File:CyloidPendulum.png|right|thumb|Schematic of a cycloidal pendulum.]]
If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the string is constrained to be tangent to one of its arches, and the pendulum's length ''L'' is equal to that of half the arc length of the cycloid (i.e., twice the diameter of the generating circle, ''L = 4r''), the bob of the [[pendulum]] also traces a cycloid path. Such a pendulum is [[Tautochrone curve|isochronous]], with equal-time swings regardless of amplitude. Introducing a coordinate system centred in the position of the cusp, the equation of motion is given by:
<math display="block">\begin{align}
  x &= r[2\theta(t) + \sin 2\theta (t)] \\
  y &= r[-3-\cos2\theta (t)],
\end{align}</math>
where <math>\theta</math> is the angle that the straight part of the string makes with the vertical axis, and is given by
<math display="block">\sin\theta (t) = A \cos(\omega t),\qquad \omega^2 = \frac{g}{L}=\frac{g}{4r},</math>
where {{math|''A'' < 1}} is the "amplitude", <math>\omega</math> is the radian frequency of the pendulum and ''g'' the gravitational acceleration.

[[File:Isochronous cycloidal pendula.gif|thumb|Five isochronous cycloidal pendula with different amplitudes.]]
The 17th-century Dutch mathematician [[Christiaan Huygens#Horology|Christiaan Huygens]] discovered and proved these properties of the cycloid while searching for more accurate pendulum clock designs to be [[History of longitude|used in navigation]].<ref>C. Huygens, "The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula (sic) as Applied to Clocks," Translated by R. J. Blackwell, Iowa State University Press (Ames, Iowa, USA, 1986).</ref>