19.1: Introduction to Cycloids

Let us set up a coordinate system $Oxy$, and a horizontal straight line $y = 2a$. We imagine a circle of diameter $2a$ between the $x$-axis and the line $y = 2a$, and initially the lowest point on the circle, P, coincides with the origin of coordinates O. We now allow the circle to roll counterclockwise without slipping on the line $y = 2a$, so that the centre of the circle moves to the right. As the circle rolls on the line, the point P describes a curve, which is known as a cycloid.

When the circle has rolled through an angle $2\theta$, the centre of the circle has moved to the right by a horizontal distance $2a \theta$, while the horizontal distance of the point P from the centre of the circle is $a \sin 2 \theta$ and the vertical distance of the point $P$ below the centre of the circle is $a \cos 2 \theta$. Thus the coordinates of the point $P$ are

$$x = a(2 \theta + \sin 2 \theta) \label{19.1.1} \tag{19.1.1}$$

and

$$y = a (1 - \cos 2 \theta ). \label{19.1.2}\tag{19.1.2}$$

Equations $\ref{19.1.1}$ and $\ref{19.1.2}$ are the parametric equations of the cycloid. Using a simple trigonometric identity, Equation $\ref{19.1.2}$ can also be written

$$y = 2a \sin^2 \theta . \label{19.1.3}\tag{19.1.3}$$

Example $\PageIndex{1}$

When the $x$-coordinate of P is 2.500$a$, what (to four significant figures) is its $y$-coordinate?

Solution

We have to find $2 \theta$ by solution of $2 \theta +\sin 2 \theta$. By Newton-Raphson iteration or otherwise, we find $2 \theta$ = 0.931 599 201 radians, and hence y = 0.9316$a$.