Suppose that point $P$ is on a circle with radius $r$, and ray $O P$ is rotating with angular speed $\omega$. Complete parts (a) through (c).
\[
r=8 \mathrm{~cm}, \omega=\frac{\pi}{12} \text { radian per } \mathrm{sec}, \mathrm{t}=8 \mathrm{sec}
\]
(a) What is the angle generated by $P$ in time t?
\[
\theta=\square \text { radian }
\]
(Type an exact answer, using $\pi$. Use integers or fractions for any numbers in the expression.)
Answer
Final Answer: The angle generated by \(P\) in time \(t\) is \(\boxed{\frac{2\pi}{3}}\) radians.
Step 1 ：Given that the angular speed \(\omega = \frac{\pi}{12}\) radian per sec and time \(t = 8\) sec.
Step 2 ：The angle generated by a point on a circle with radius \(r\) and angular speed \(\omega\) in time \(t\) is given by the formula \(\theta = \omega \cdot t\).
Step 3 ：Substitute the given values into the formula to find the angle \(\theta\).
Step 4 ：So, \(\theta = \frac{\pi}{12} \cdot 8 = \frac{2\pi}{3}\) radians.
Step 5 ：Final Answer: The angle generated by \(P\) in time \(t\) is \(\boxed{\frac{2\pi}{3}}\) radians.
