Chapter 1.7, Problem 37E

To determine

To find: The parametric equations for the cycloid when $\frac{\pi }{2}<\theta <\pi$ .

Answer to Problem 37E

The required parametric equations is $y=r\left(1-\cos \theta \right)$ is $x=r\left(\theta -\sin \theta \right)$ .

Explanation of Solution

Given:

The given range is $\frac{\pi }{2}<\theta <\pi$ .

Calculation:

The curve traced by $P$ point on the circumference of the circle as the circle rolls along the straight line known as cycloid.

Consider the radius of the circle is $r$ .

The parameter for the angle of rotation $\theta$ of the circle is $\theta =\frac{\pi }{2}$ for $P$ at the origin.

For the case when circle rotates $\theta$ radians from the angle $\theta =\frac{\pi }{2}$ , this is because when the circle is in contact with the line, then the distance it rolls from the origin is,

  $\begin{array}{c}\left|OT\right|=\text{arc}PT\\ =r\theta \end{array}$

The figure for the circle is shown in Figure 1

  

Figure

The centre of the circle is,

  $C\left(r\infty ,r\right)$

Consider the coordinates of $P$ is,

  $\left(x,y\right)$

Then, the expression for $x$ is,

  $\begin{array}{c}x=\left|OT\right|-\left|PQ\right|\\ =r\theta -r\sin \theta \\ =r\left(\theta -\sin \theta \right)\end{array}$

The expression for $y$ is,

  $\begin{array}{c}y=\left|TC\right|-\left|QC\right|\\ =r-r\cos \theta \\ =r\left(1-\cos \theta \right)\end{array}$