Final answer:
The period of the ball's motion with an angular velocity of 4π rad/s is 0.25 seconds, as the period is inversely proportional to the angular velocity.
Step-by-step explanation:
The student asked a question about uniform circular motion: specifically, they wanted to know the period of a ball's motion given its angular velocity of 4π rad/s. The period (T) of motion in uniform circular motion is the time it takes to complete one full revolution (2π radians). Given the angular velocity (ω), the period can be calculated using the formula T = ¹/⁷ω. In this scenario, T = 1/(4π rad/s) = 1/4 seconds per revolution or 0.25 s.
The angular velocity is defined as the change in angle (ΔΘ) per unit time (Δt). Since the given angular velocity is 4π rad/s, and one full revolution corresponds to an angle of 2π radians, it takes 0.25 seconds for the ball to complete one full cycle. Therefore, the period of the ball's motion is 0.25 seconds.