Chapter 11.1, Problem 42E

To determine

To determine: The parametric equations for the motion of a point on the exterior of a wheel radius a as the wheel rolls along the horizontal line without slipping.

Answer to Problem 42E

The required parametric equations a Prolate Cycloid are $x=at-b\sin t$ and $y=a-b\cos t$ .

Explanation of Solution

Consider the center of the wheel is at $\left(at,0\right)$ .

Also, the coordinates of the point is:

  $x=at+b\cos \theta$ , $y=a+b\sin \theta$ ; $a<b<2a$

To write the expression as $\theta$ in terms of t , consider some integer be k .

So, $\begin{array}{c}t+\theta =\frac{3}{2}\pi +2k\pi \\ \theta =\frac{3}{2}\pi +2k\pi -t\end{array}$

Substitute $\frac{3}{2}\pi +2k\pi -t$ for $\theta$ in the coordinates.

  $x=at+b\cos \left(\frac{3}{2}\pi +2k\pi -t\right)$ , $y=a+b\sin \left(\frac{3}{2}\pi +2k\pi -t\right)$

  $x=at-b\sin t$ and $y=a-b\cos t$ ; $a<b<2a$

The above equations are the parametric equations of a Prolate Cycloid.

Hence, the required parametric equations of a Prolate Cycloid are $x=at-b\sin t$ and $y=a-b\cos t$ .