Chapter 1.7, Problem 38E

To determine

To Prove: The parametric equations of the trochoid are $x=r\theta -d\sin \theta$ and $y=r-d\cos \theta$ , then sketch for the trochoid for the case $d<r$ and for $d>r$ .

Answer to Problem 38E

The parametric equations are $x=r\theta -d\sin \theta$ and $y=r-d\cos \theta$ . The required sketch for $d<r$ is shown in Figure 2 and the for $d>r$ is shown in Figure 3

Explanation of Solution

Given:

The given equations are $x=r\theta -d\sin \theta$ and $y=r-d\cos \theta$ .

The given condition are $d<r$ and for $d>r$ .

Calculation:

Consider the case for $d<r$ .

Consider the point $P$ at the distance $d$ from the centre of the circle $c$ of the radius $r$ , where $d<r$ . Point P makes an angle $\theta$ with the centre $c$ .

The required diagram is shown in Figure 1

Figure 1

The distance between the centre of the circle and origin is $r\theta$ .

Here, the distances are,

  $\begin{array}{l}\left|PC\right|=d\sin \theta \\ \left|CR\right|=d\cos \theta \end{array}$

Consider the coordinates of $P$ is $x,y$ then,

  $\begin{array}{c}x=r\theta -\left|PR\right|\\ =r\theta -d\sin \theta \end{array}$

Also,

  $\begin{array}{c}y=r-\left|CR\right|\\ =r-d\cos \theta \end{array}$

Thus, the parametric equations are,

  $\begin{array}{c}x=r\theta -d\sin \theta \\ y=r-d\cos \theta \end{array}$

Consider the case for $d>r$ .

The required sketch for $d<r$ is shown in Figure 2

Figure 2

The sketch for $d>r$ is shown in Figure 3

Figure 3