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.