The propeller blades of an airplane are 2.7 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 ninth second of the rotation. Find the angular speed at the end of 8.7 s.

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To find the angular speed at the end of 8.7 seconds, we first need to determine the number of revolutions the propeller blades have completed in that time interval.

Given:
Length of the propeller blades (l) = 2.7 m
Time interval (t) = 8.7 s
Number of revolutions between the fifth and ninth second of rotation = 2 revolutions

First, we need to find the total angle covered by the propeller blades in the time interval.

Since there are 2 revolutions between the fifth and ninth second, we can calculate the total angle covered using the formula:

Total angle (θ) = 2 * 2π

Next, we need to calculate the average angular velocity during the time interval.

Average angular velocity (ω_avg) = θ / t

Finally, to find the angular speed at the end of 8.7 seconds, we need to find the final angular velocity.

Using the formula for uniformly accelerated angular motion:

Final angular velocity (ω_final) = Initial angular velocity (ω_initial) + (angular acceleration (α) * t)

Since the propeller starts from rest, the initial angular velocity (ω_initial) is 0.

We need to find the angular acceleration (α) to calculate the final angular velocity.

Since the propeller starts from rest and undergoes constant angular acceleration, we can use the formula:

θ = ω_initial * t + 0.5 * α * t^2 (where θ is the total angle)

Since ω_initial is 0, the formula simplifies to:

θ = 0.5 * α * t^2

Rearranging the equation, we can solve for α:

α = (2 * θ) / t^2

With the value of α determined, we can calculate the final angular velocity using the formula:

ω_final = α * t

By following this step-by-step process, we can find the angular speed at the end of 8.7 seconds based on the given information.