Final answer:
The total angular displacement of the propeller is 16.10 x 10³ rad.
Step-by-step explanation:
To find the total angular displacement of the propeller, we need to find the angular displacements for each phase and then add them up.
In the first phase, the propeller starts from rest and accelerates at 2.50 x 10⁻³ rad/s² for 2.50 x 10³ s. Using the equation α = (ωf - ωi) / t, we can find the final angular velocity (ωf) in this phase:
ωf = ωi + αt = 0 + (2.50 x 10⁻³ rad/s²)(2.50 x 10³ s) = 6.25 rad/s
The angular displacement (Δθ) in this phase can be found using the equation Δθ = ωi*t + 1/2 αt²:
Δθ = 0 + 1/2 (2.50 x 10⁻³ rad/s²)(2.50 x 10³ s)² = 3.13 x 10³ rad
In the second phase, the propeller rotates at a constant angular speed. The angular displacement in this phase can be found using the equation Δθ = ωt:
Δθ = (6.25 rad/s)(1.38 x 10³ s) = 8.63 x 10³ rad
In the third phase, the propeller decelerates at 2.70 x 10⁻³ rad/s² until it slows down to an angular speed of 2.35 rad/s. Using the equation ωf = ωi + αt and solving for t, we can find the time it takes to reach the final angular velocity:
t = (ωf - ωi) / α = (2.35 rad/s - 6.25 rad/s) / (-2.70 x 10⁻³ rad/s²) ≈ 1.852 x 10³ s
The angular displacement in this phase can be found using the equation Δθ = ωit + 1/2 αt²:
Δθ = (2.35 rad/s)(1.852 x 10³ s) + 1/2 (-2.70 x 10⁻³ rad/s²)(1.852 x 10³ s)² ≈ 4.34 x 10³ rad
Finally, we can find the total angular displacement by adding up the angular displacements from each phase:
Total angular displacement = 3.13 x 10³ rad + 8.63 x 10³ rad + 4.34 x 10³ rad = 16.10 x 10³ rad