Chapter 1, Problem 32RE

(a)

To determine

To find: The parametric equation for the particle that moves counter clockwise halfway around the circle from the top and bottom.

Answer to Problem 32RE

The parametric equation is $x=2+2\cos t$ , $y=2\sin t$ for the parameter range of $\frac{\pi }{2}\le t\le \frac{3\pi }{2}$ .

Explanation of Solution

Given:

The given equation is ${\left(x-2\right)}^2+y^2=4$ .

Calculation:

The general form for the parametric equation of the circle with the centre at $\left(h,k\right)$ and radius $r$ is,

  $\begin{array}{l}x=h+r\cos t\\ y=k+r\sin t\end{array}$

The range of $t$ is,

  $0\le t\le 2\pi$

Consider the given equation of the circle is,

  ${\left(x-2\right)}^2+y^2=4$

Then, the parametric equation is,

  $\begin{array}{l}x=2+2\cos t\\ y=2\sin t\end{array}$

As the path of the particle is in counter clockwise direction from the top to the bottom, the value of $t$ varies from $\frac{\pi }{2}$ to $\frac{3\pi }{2}$ . Thus, the parameter range is,

  $\frac{\pi }{2}\le t\le \frac{3\pi }{2}$

The parametric equation is $x=2+2\cos t$ , $y=2\sin t$ for the parameter range of $\frac{\pi }{2}\le t\le \frac{3\pi }{2}$ .

(b)

To determine

To find: The graph for the semi-circular path.

Explanation of Solution

Given:

The given equation is ${\left(x-2\right)}^2+y^2=4$ .

Calculation:

Consider the parametric equation are,

  $\begin{array}{l}x=2+2\cos t\\ y=2\sin t\end{array}$

Consider the parameter range is,

  $\frac{\pi }{2}\le t\le \frac{3\pi }{2}$

The table for the value of $x$ and $y$ for different values of the parameter is shown in Table 1

Table 1

$t$	$x$	$y$
$\frac{\pi }{2}$	$2$	$2$
$\frac{3\pi }{4}$	0.5858	1.414
$\pi$	0	0
$\frac{5\pi }{4}$	0.5858	-1.414
$\frac{3\pi }{2}$	2	-2

The graph for the parametric equations is shown in Figure 1

  

Figure 1