Chapter 8.4, Problem 26E

To determine

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

Answer to Problem 26E

The orientation of motion is clockwise.

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

Explanation of Solution

Given:

  $x=2\sin t$ and $y=2\cos t$ .

Calculation:

Find the radius r of a circle is given by

  $r=\sqrt{x^2+y^2}$ .

On substituting $x=2\sin t$ and $y=2\cos t$ ,

  $\begin{array}{l}r=\sqrt{{(2\sin t)}^2+{(2\cos t)}^2}\\ r=\sqrt{4({\cos }^{2}t+{\sin }^{2}t)}\end{array}$

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

  $\begin{array}{l}r=\sqrt{4(1)}\\ r=2\end{array}$

Hence, the radius of circle is $\boxed{2}$ .

On putting $t=0$ in $x=2\sin t$ and $y=2\cos t$ ,

  $\begin{array}{l}x=2\sin (0)\\ x=2(0)\\ x=0\end{array}$

  $\begin{array}{l}y=2\cos (0)\\ y=2(1)\\ y=2\end{array}$

Thus, the position of the object at $t=0$ is $\boxed{x=0}$ and $\boxed{y=2}$ .

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

  $\begin{array}{l}x=2\sin (\frac{\pi }{2})\\ x=2(1)\\ x=2\end{array}$

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

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

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

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

On putting $t=0$ in $x=2\sin t$ and $y=2\cos t$ ,

  $\begin{array}{l}x=2\sin (0)\\ x=2(0)\\ x=0\end{array}$

  $\begin{array}{l}y=2\cos (0)\\ y=2(1)\\ y=2\end{array}$

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

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

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

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

The orientation of motion is clockwise.