Chapter 10.1, Problem 51E

$\text{(a)}$

To determine

To find: The parametric equations for the involute by expressing the coordinate $x$ and $y$ .

Answer to Problem 51E

The answer:

  $\begin{array}{c}x=\cos t+t\sin t\\ y=\sin t-t\cos t,\end{array}$

Where,

  $t\ge 0$

Explanation of Solution

Given information:

The figure is:

  

Calculation:

In accordance with the information provided in this exercise, the graph below.

  

In this exercise, the circle's radius is set to be equal to $\text{r=1}$ . The coordinates of the point $\text{Q}$ are $x_Q=\cos t$

and

  $y_Q=\sin t$ according to textbook's definitions of fundamental identities, which are presented in Appendix $1$ on page $579$ .

Know that the $\angle OQP$ is equal to $90$ degrees because the straight line overline $\overline{QP}$ lies on the tangent of the circle with centre at the origin and radius $\text{r=1}$ .

As a result of the above findings, infer that the coordinates $x$ and $y$ of the point $P$ in terms of $t$ for $t\ge 0$ is equal to using the textbook definitions of fundamental identities given in Appendix $1$ on page $579$ .

  $\begin{array}{c}x=x_Q+t\sin t\\ =\cos t+t\sin t\\ y=y_Q-t\cos t\\ =\sin t-t\cos t\end{array}$

Therefore, the required parametric equations for the involute by expressing the coordinate $x$ and $y$ are:

  $\begin{array}{c}x=\cos t+t\sin t\\ y=\sin t-t\cos t,\end{array}$

Where,

  $t\ge 0$

$\text{(b)}$

To determine

To find: The length.

Answer to Problem 51E

The answer: $L=2{\pi }^2$

Explanation of Solution

Given information:

The figure is:

  

Calculation:

Know that,

  

Hence part $(a)$ :

  $\begin{array}{c}x=\cos t+t\sin t\\ y=\sin t-t\cos t,\end{array}$

Then,

  $\begin{array}{c}\frac{dx}{dt}=-\sin t+\sin t+t\cdot \cos t\\ =t\cos t\\ \frac{dy}{dt}=\cos t-[\cos t+t\cdot (-\sin t)]\\ =t\sin t\end{array}$

Then to use the previous results that the length of the semicircle:

  

Therefore, the required length is $L=2{\pi }^2$ .