To find the angular speed [tex]\(\omega\)[/tex] associated with rotating a central angle [tex]\(\theta = 200^\circ\)[/tex] in a time of [tex]\(t = 2\)[/tex] seconds, we need to follow these steps:
1. Convert the angle from degrees to radians.
- The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right)
\][/tex]
- Substituting in the given angle [tex]\(\theta = 200^\circ\)[/tex]:
[tex]\[
\theta_{\text{radians}} = 200 \times \left(\frac{\pi}{180}\right)
\][/tex]
Simplifying the expression:
[tex]\[
\theta_{\text{radians}} = \frac{200\pi}{180}
\][/tex]
Reducing the fraction:
[tex]\[
\theta_{\text{radians}} = \frac{10\pi}{9}
\][/tex]
2. Calculate the angular speed [tex]\(\omega\)[/tex].
- The angular speed [tex]\(\omega\)[/tex] is given by:
[tex]\[
\omega = \frac{\theta_{\text{radians}}}{t}
\][/tex]
- Given [tex]\(\theta_{\text{radians}} = \frac{10\pi}{9}\)[/tex] and [tex]\(t = 2\)[/tex] seconds, we substitute these values into the formula:
[tex]\[
\omega = \frac{\frac{10\pi}{9}}{2}
\][/tex]
Simplifying:
[tex]\[
\omega = \frac{10\pi}{9 \times 2} = \frac{10\pi}{18} = \frac{5\pi}{9}
\][/tex]
Therefore, the exact and fully simplified angular speed [tex]\(\omega\)[/tex] is:
[tex]\[
\omega = \frac{5\pi}{9} \text{ rad/second}
\][/tex]
