The drive propeller of a ship starts from rest and accelerates at 2.90×10-3 rad/s2 for 2.10×103 s. For the next 1.40×103 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.
physics.nmu.edu/~ddonovan/classes/ph201/Homework/Chap08/CH08P29.html
To find the total angular displacement of the propeller, we need to calculate the displacement during each phase of its motion separately and then sum them up.
Phase 1: Acceleration
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In this phase, the propeller starts from rest and accelerates at a rate of 2.90×10-3 rad/s2 for 2.10×103 s.
We can use the formula for angular displacement during constant acceleration:
θ = ω0 * t + 1/2 * α * t^2
where:
θ is the angular displacement,
ω0 is the initial angular velocity,
α is the angular acceleration, and
t is the time.
In this case, ω0 = 0 (since the propeller starts from rest), α = 2.90×10-3 rad/s2, and t = 2.10×103 s.
θ1 = 0 * (2.10×103) + 1/2 * (2.90×10-3) * (2.10×103)^2
Simplifying the equation gives us the angular displacement during phase 1.
Phase 2: Constant Speed
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In this phase, the propeller rotates at a constant angular speed. The angular velocity is constant, so the angular displacement is simply given by:
θ2 = ω * t
where:
θ2 is the angular displacement,
ω is the constant angular velocity, and
t is the time.
In this case, ω is unknown, but the time t is given as 1.40×103 s, and we can solve for ω using the formula for angular velocity:
ω = ω0 + α * t
We know that the final angular velocity ω = 4.00 rad/s, and we already calculated the initial angular velocity ω0 = 0 from the previous phase.
Solve for ω using ω = ω0 + α * t, and substitute the values into θ2 = ω * t to calculate the angular displacement during phase 2.
Phase 3: Deceleration
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In this phase, the propeller decelerates at a rate of 2.30×10-3 rad/s2 until it slows (without reversing direction) to an angular speed of 4.00 rad/s.
We can use the same formula as in phase 1 for angular displacement during constant acceleration. However, since the propeller is decelerating, the angular acceleration α is negative.
θ3 = ω * t + 1/2 * α * t^2
In this case, we need to find the initial angular velocity ω, given that the final angular velocity ω = 4.00 rad/s, and the angular acceleration α = -2.30×10-3 rad/s2. The time is also given as 1.40×103 s.
Solve for ω using ω = ω0 + α * t, and substitute the values into θ3 = ω * t + 1/2 * α * t^2 to calculate the angular displacement during phase 3.
Total Angular Displacement
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To find the total angular displacement, we simply add up the angular displacements from all three phases:
Total θ = θ1 + θ2 + θ3
Substitute the calculated values for θ1, θ2, and θ3 into the equation to find the total angular displacement of the propeller.