Chapter 11.1, Problem 52E

(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 52E

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 radius of the circle is a and cantered at the origin with the equation of the circle is $x^2+y^2=a^2$

Calculation:

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

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

(b)

To determine

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

Answer to Problem 52E

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

Explanation of Solution

Consider the parametric equation are,

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

The derivative with respect to t is,

  $\frac{dx}{dt}=at\cos t$

Then, the derivative of $y$ is,

  $\frac{dy}{dt}=at\sin t$

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,

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