Final answer:
To find the angular velocity of a wheel that has undergone 12 revolutions, we apply the rotational kinematic equation and integrate the angular acceleration over the time it takes to complete those revolutions.
Step-by-step explanation:
The question is asking about the angular velocity of a wheel after it has turned through 12 revolutions, starting from rest with an angular acceleration given by α(t) = (6.0 rad/s´)t². To solve for angular velocity, we need to find the final angular velocity (ω) using the kinematic equation for rotational motion, which is ω = ω₀ + αt, where ω₀ is the initial angular velocity and α is the angular acceleration. However, since the angular acceleration is not constant in this case, we'll integrate the acceleration function over time to find the angular velocity.
To perform the calculation, we need to find the time it takes for the wheel to complete 12 revolutions. One revolution is 2π radians, so 12 revolutions is 12 × 2π = 24π radians. By using the rotational kinematic formula θ = ω₀t + 0.5αt², where θ is the angular displacement, ω₀ is the initial angular velocity (which is zero since the wheel starts from rest), and α is the angular acceleration, we can find the time by setting θ to 24π radians and solving for t. Once we have t, we can integrate the angular acceleration to find the angular velocity at that time.