To find the angular speed [tex]\(\omega\)[/tex] (in radians per second) of a rotating object, you can use the formula for angular speed:
[tex]\[
\omega = \frac{\theta}{t}
\][/tex]
where:
- [tex]\(\theta\)[/tex] is the angular displacement in radians,
- [tex]\(t\)[/tex] is the time in seconds.
In this problem, we are given:
- [tex]\(\theta = 27 \pi\)[/tex] radians,
- [tex]\(t = 12\)[/tex] seconds.
Substitute the given values into the formula:
[tex]\[
\omega = \frac{27 \pi}{12}
\][/tex]
Now, simplify the fraction:
[tex]\[
\omega = \frac{27 \pi}{12} = \frac{27}{12} \pi = \frac{27 \div 3}{12 \div 3} \pi = \frac{9}{4} \pi
\][/tex]
Thus, the angular speed [tex]\(\omega\)[/tex] is:
[tex]\[
\omega = \frac{9 \pi}{4} \text{ rad/sec}
\][/tex]
So the exact, fully simplified angular speed is:
[tex]\[
\boxed{\frac{9 \pi}{4} \text{ rad/sec}}
\][/tex]
