Chapter 1.7, Problem 17E

To determine

To find: The motion of the particle with the position $\left(x,y\right)$ as it varies in the in the given interval.

Answer to Problem 17E

The particle follows the circular motion.

Explanation of Solution

Given:

The value of $x$ and $y$ is given by,

  $\begin{array}{l}x=3+2\cos \\ y=1+2\sin t\end{array}$

The value of the range is,

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

Calculation:

Consider the given interval,

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

Consider the given expressions,

  $\begin{array}{l}x=3+2\cos \\ y=1+2\sin t\end{array}$

The variation in the variable $x$ and $y$ is shown in Table 1

Table 1

$t$	$x$	$y$
$\frac{\pi }{2}=90°$	3	3
$120°$	2	2.732
$150°$	$1.27$	2
$180°$	$1$	$1$
$210°$	$1.27$	0
$240°$	$2$	$-1.732$
$270°$	3	$-1$

The required graph is shown in Figure 1

  

Moving counter clock wise along the circle the equation is,

  ${\left(x-3\right)}^2+{\left(y-1\right)}^2=4$

The above equation ranges from $\left(3,3\right)$ to $\left(3,-1\right)$ .

This gives that the $x$ and $y$ particles moves in the circle.