Chapter 8.4, Problem 25E

To determine

To Describe: The path of the object and the time taken to complete one revolution

Answer to Problem 25E

The orientation of motion is anticlockwise.

The radius of the circle is 3 and the time required for one complete revolution around the circle is $2\pi$ .

Explanation of Solution

Given:

  $x=3\cos t$ and $y=3\sin t$ .

Calculation:

Find the radius r of a circle is given by

  $r=\sqrt{x^2+y^2}$ .

On substituting $x=3\cos t$ and $y=3\sin t$

  $\begin{array}{l}r=\sqrt{{(3\cos t)}^2+{(3\sin t)}^2}\\ r=\sqrt{9({\cos }^{2}t+{\sin }^{2}t)}\end{array}$

Using the identity ${\cos }^{2}x+{\sin }^{2}x=1$ .

  $\begin{array}{l}r=\sqrt{9(1)}\\ r=3\end{array}$

Hence, the radius of circle is $\boxed{3}$ .

On putting $t=0$ in $x=3\cos t$ ,

  $\begin{array}{l}x=3\cos (0)\\ x=3(1)\\ x=3\end{array}$

On putting $t=0$ in $y=3\sin t$

  $\begin{array}{l}y=3\sin (0)\\ y=3(0)\\ y=0\end{array}$

Thus, the position of the object at $t=0$ is $\boxed{x=3}$ and $\boxed{y=0}$ .

On putting $t=\frac{\pi }{2}$ in $x=3\cos t$ ,

  $\begin{array}{l}x=3\cos (\frac{\pi }{2})\\ x=3(0)\\ x=0\end{array}$

On putting $t=\pi$ in $x=3\cos t$

  $\begin{array}{l}x=3\cos (\pi )\\ x=3(-1)\\ x=-3\end{array}$

As the value of x decreases as t increases from 0 to $2\pi$ . The orientation of motion is anticlockwise.

Find the time t taken for one complete revolution around the circle.

On putting $t=0$ in $x=3\cos t$ , $y=3\sin t$ ,

  $\begin{array}{l}x=3\cos (0)\\ x=3(1)\\ x=3\end{array}$

  $\begin{array}{l}y=3\sin (0)\\ y=3(0)\\ y=0\end{array}$

On putting $t=2\pi$ in $x=3\cos t$ , $y=3\sin t$ ,

  $\begin{array}{l}x=3\cos (2\pi )\\ x=3(1)\\ x=3\end{array}$

  $\begin{array}{l}y=3\sin (2\pi )\\ y=3(0)\\ y=0\end{array}$

As the values of x and y are same at $t=0$ and $t=2\pi$ , therefore one complete revolution around the circle has been completed.

Hence, the time required for one complete revolution around the circle is $\boxed{2\pi }$ .

Conclusion:

Therefore, the radius of the circle is 3 and the time required for one complete revolution around the circle is $2\pi$ .

The orientation of motion is anticlockwise.