Chapter 10.1, Problem 52E

$\text{(a)}$

To determine

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

Answer to Problem 52E

The answer:

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

Where,

  $t\ge 0$

Explanation of Solution

Given information:

The figure is:

  

And,

The equation is: $x^2+y^2=a^2$

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 $r=a$ . 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 $r=a$ .

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 $579$ .

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

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

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

Where,

  $t\ge 0$

$\text{(b)}$

To determine

To find: The length.

Answer to Problem 52E

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

Explanation of Solution

Given information:

The figure is:

  

And,

The equation is:

  $x^2+y^2=a^2$

Calculation:

Know that,

  

Hence part $(a)$ :

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

Then,

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

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

  

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