Chapter 11.1, Problem 51E

(a)

To determine

To find: The parametric equation for the involute by expressing the coordinates $x$ and $y$ of $p$ in terms of $t$ for $t\ge 0$ .

Answer to Problem 51E

The required parametric equation is $x=\cos t+t\sin t$ , $y=\sin t-t\cos t$ for $t\ge 0$ .

Explanation of Solution

Given Data:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider the initial point of the tracing is $\left(1,0\right)$ on the x axis. The unwound portion of the string is the tangent to the circle and $t$ is the radian measure of the angle from the positive x axis of the segment OQ.

Then, from the above data and the geometry of the figure shows that the parametric equation of the involute is,

  $\begin{array}{l}x=\cos t+t\sin t\\ y=\sin t-t\cos t\\ t\ge 0\end{array}$

(b)

To determine

To find: The length of the involute for $0\le t\le 2\pi$ .

Answer to Problem 51E

The required length is $L=2{\pi }^2$ .

Explanation of Solution

Consider the parametric equation are,

  $\begin{array}{l}x=\cos t+t\sin t\\ y=\sin t-t\cos t\\ t\ge 0\end{array}$

The derivative with respect to t is,

  $\begin{array}{c}\frac{dx}{dt}=-\sin t+t\cos t+\sin t\\ =t\cos t\end{array}$

Then, the derivative of $y$ is,

  $\begin{array}{c}\frac{dy}{dt}=\cos t+t\sin t-\cos t\\ =t\sin t\end{array}$

Consider the length of the parametric curve is obtained as,

  $L=\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$

Then, for the interval $0\le t\le 2\pi$ the length is,

  $\begin{array}{c}L={\displaystyle {\int }_0^{2\pi }\sqrt{{\left(t\cos t\right)}^2+{\left(t\sin t\right)}^2}dt}\\ ={\displaystyle {\int }_0^{2\pi }tdt}\\ ={\left(\frac{t^2}{2}\right)|}_0^{2t}\\ =\frac{{\left(2\pi \right)}^2}{2}\end{array}$

Solve further as,

  $L=2{\pi }^2$