The drive propeller of a ship starts from rest and accelerates at 2.95 10-3 rad/s2 for 2.05 103 s. For the next 1.90 103 s the propeller rotates at a constant angular speed. Then it decelerates at 2.30 10-3 rad/s2 until it slows (without reversing direction) to an angular speed of 4.00 rad/s. Find the total angular displacement of the propeller.
To find the total angular displacement of the propeller, we need to break down the motion into three phases: acceleration, constant speed, and deceleration. We will calculate the angular displacement for each phase and then sum them up.
Phase 1: Acceleration
We have the initial angular speed (ω0 = 0) and the angular acceleration (α = 2.95 × 10^(-3) rad/s^2) for a time duration (t = 2.05 × 10^3 s). To find the angular displacement (θ1) during this phase, we can use the equation:
θ1 = ω0*t + (1/2)α*t^2
Substituting the values, we get:
θ1 = 0*(2.05 × 10^3) + (1/2)*(2.95 × 10^-3)*(2.05 × 10^3)^2
θ1 = (1/2)*(2.95 × 10^-3)*(4.2025 × 10^6)
θ1 = 6.189 × 10^3 rad
Phase 2: Constant Speed
During this phase, the propeller rotates at a constant angular speed for a certain time duration (t2 = 1.90 × 10^3 s). Since the angular speed is constant, there is no change in angular displacement. Therefore, θ2 = ω*t2, where ω is the constant angular speed.
θ2 = (4.00 rad/s)*(1.90 × 10^3 s)
θ2 = 7.60 × 10^3 rad
Phase 3: Deceleration
We know the final angular speed (ωf = 4.00 rad/s) and the angular acceleration (α = -2.30 × 10^-3 rad/s^2) when the propeller decelerates until it slows down to ωf. We need to calculate the time duration (t3) during this phase first.
We can use the equation:
ωf = ω0 + α*t3
Rearranging the equation, we get:
t3 = (ωf - ω0) / α
Substituting the values, we get:
t3 = (4.00 rad/s - 0 rad/s) / (-2.30 × 10^-3 rad/s^2)
t3 = 1.739 × 10^3 s
Now, to calculate the angular displacement (θ3) during this phase:
θ3 = ω*t3 + (1/2)α*t3^2
Substituting the values, we get:
θ3 = (4.00 rad/s)*(1.739 × 10^3 s) + (1/2)*(-2.30 × 10^-3 rad/s^2)*(1.739 × 10^3 s)^2
θ3 = 6.956 × 10^3 rad
Total Angular Displacement:
Finally, we can find the total angular displacement by summing up the angular displacements of each phase:
Total Angular Displacement = θ1 + θ2 + θ3
Total Angular Displacement = 6.189 × 10^3 rad + 7.60 × 10^3 rad + 6.956 × 10^3 rad
Total Angular Displacement = 20.745 × 10^3 rad
Therefore, the total angular displacement of the propeller is 20.745 × 10^3 rad.