The propeller blades of an airplane are 2.4 m long. The plane is getting ready for takeoff, and the propeller starts turning from rest at a constant angular acceleration. The propeller blades go through two revolutions between the fifth and the seventh second of the rotation. Find the angular speed at the end of 8.9 s.

To solve this problem, we can use the equations of rotational motion.

First, let's find the initial angular velocity (ω_0) of the propeller when it starts turning from rest. The problem states that the propeller goes through two revolutions between the fifth and seventh second of rotation. Therefore, the time it takes for two revolutions is 7 - 5 = 2 seconds.

We know that the angular displacement (θ) is given by θ = ω_0t + (1/2)αt^2, where ω_0 is the initial angular velocity, α is the angular acceleration, and t is the time.

Since the propeller makes two complete revolutions, the angular displacement is 2π radians.

Substituting the given values, we get:
2π = ω_0(2) + (1/2)α(2)^2.

Simplifying the equation, we have:
2π = 2ω_0 + 2α.

Now, let's find the angular speed (ω) at the end of 8.9 seconds. We can use the equation ω = ω_0 + αt.

Substituting the given values, we get:
ω = ω_0 + α(8.9).

To solve for ω, we need to find the values of ω_0 and α.

From the first equation, 2π = 2ω_0 + 2α, we can isolate ω_0 in terms of α:
ω_0 = π - α.

Now, substitute this value of ω_0 into the second equation:
ω = (π - α) + α(8.9).

Simplifying further, we have:
ω = π + α(8.9).

To find α, we can use the formula for average angular acceleration (α_avg = Δω / Δt), where Δω is the change in angular velocity and Δt is the change in time.

From the given information, we know that the propeller goes from rest (ω_0 = 0) to some final angular velocity (ω) in a time of 8.9 seconds.

Thus, Δω = ω - ω_0 = ω.

Substituting the values, we get:
α_avg = ω / Δt = ω / 8.9.

Finally, we solve for α:
α = α_avg * 8.9 = (ω / 8.9) * 8.9 = ω.

Now that we have the value of ω, we can substitute it back into the equation ω = π + α(8.9) to find the angular speed at the end of 8.9 seconds.

Note: Since α = ω, the equation becomes:
ω = π + ω(8.9).

Simplifying further, we find:
ω - ω(8.9) = π
ω(1 - 8.9) = π
ω(-7.9) = π
ω = π / (-7.9).

Therefore, the angular speed at the end of 8.9 seconds is ω = π / (-7.9).