Answer:
Δθ = 15747.37 rad.
Explanation:
- The total angular displacement is the sum of three partial displacements: one while accelerating from rest to a certain angular speed, a second one rotating at this same angular speed, and a third one while decelerating to a final angular speed.
- Applying the definition of angular acceleration, we can find the final angular speed for this first part as follows:
- Since the angular acceleration is constant, and the propeller starts from rest, we can use the following kinematic equation in order to find the first angular displacement θ₁:
- Solving for Δθ in (2):
- The second displacement θ₂, (since along it the propeller rotates at a constant angular speed equal to (1), can be found just applying the definition of average angular velocity, as follows:
- Finally we can find the third displacement θ₃, applying the same kinematic equation as in (2), taking into account that the angular initial speed is not zero anymore:
- Replacing by the givens (α, ωf₂) and ω₀₂ from (1) we can solve for Δθ as follows:
- The total angular displacement is just the sum of (3), (4) and (6):
- Δθ = θ₁ + θ₂ + θ₃ = 5044.12 rad + 7252 rad + 3451.25 rad
- ⇒ Δθ = 15747.37 rad.