The drive propeller of a ship starts from rest and accelerates at 3.00 10-3 rad/s2 for 2.05 103 s. For the next 1.75 103 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 problem into smaller parts and then sum up the displacements.
Step 1: Calculating the initial angular velocity.
The propeller starts from rest and accelerates at 3.00 * 10^(-3) rad/s^2 for 2.05 * 10^3 s. We can use the formula for linear motion to calculate the initial angular velocity (ω0):
ω0 = ω0 + α * t
Here,
ω0 is the initial angular velocity,
α is the angular acceleration,
t is the time.
ω0 = 0 + (3.00 * 10^(-3)) * (2.05 * 10^3)
ω0 = 6.15 rad/s
Step 2: Calculating the angular displacement during constant angular speed.
The propeller rotates at a constant angular speed for 1.75 * 10^3 s. We can calculate the angular displacement (θ1) during this time using the formula:
θ1 = ωt
Here,
θ1 is the angular displacement,
ω is the angular velocity,
t is the time.
θ1 = (6.15) * (1.75 * 10^3)
θ1 ≈ 10762.5 rad
Step 3: Calculating the final angular velocity.
The propeller decelerates at -2.30 * 10^(-3) rad/s^2 until it reaches an angular speed of 4.00 rad/s. We can use the formula for linear motion to calculate the final angular velocity (ωf):
ωf^2 = ω0^2 + 2 * α * θ2
Here,
ωf is the final angular velocity,
ω0 is the initial angular velocity,
α is the angular acceleration,
θ2 is the angular displacement during deceleration.
θ2 = (4.00)^2 / (2 * (-2.30 * 10^(-3)))
θ2 ≈ 3478 rad
Step 4: Calculating the angular displacement during deceleration.
Using the formula, θ2 = ω0 * t + (1/2) * α * t^2, we can rearrange to solve for time (t):
t = (ωf - ω0) / α
Substituting the values, we can find t:
t = (4.00 - 6.15) / (-2.30 * 10^(-3))
t ≈ 1030.43 s
Finally, we can calculate the angular displacement (θ2) during deceleration:
θ2 = ω0 * t + (1/2) * α * t^2
θ2 = 6.15 * 1030.43 + (1/2) * (-2.30 * 10^(-3)) * (1030.43)^2
θ2 ≈ 3182.9 rad
Step 5: Summing up the angular displacements.
To find the total angular displacement, we add the angular displacements obtained in steps 2 and 4:
Total angular displacement = θ1 + θ2
Total angular displacement ≈ 10762.5 + 3182.9
Total angular displacement ≈ 13945.4 rad
Therefore, the total angular displacement of the propeller is approximately 13945.4 radians.