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 find the angular speed at the end of 8.9 seconds, we need to calculate the angular acceleration first.

Given that the propeller blades go through two revolutions between the fifth and the seventh second of rotation, we can determine the time taken for each revolution:

Time for two revolutions = 7 seconds - 5 seconds = 2 seconds

The time taken for one revolution is half of the time for two revolutions:

Time for one revolution = 2 seconds / 2 = 1 second

Next, let's calculate the angular acceleration. We know that the initial angular velocity is zero since the propeller starts turning from rest.

The formula to calculate angular acceleration is:

Angular acceleration (α) = (Final angular velocity (ω) - Initial angular velocity (ω0)) / Time (t)

Since the initial angular velocity (ω0) is zero, the formula simplifies to:

Angular acceleration (α) = Final angular velocity (ω) / Time (t)

Now, let's calculate the angular acceleration using the values we know:

Angular acceleration (α) = 2 revolutions / 1 second = 2 radians/second^2

We have the angular acceleration (α) and the time (t) to find the angular speed at the end of 8.9 seconds using the formula:

Angular speed (ω) = Initial angular velocity (ω0) + Angular acceleration (α) * Time (t)

Since the initial angular velocity (ω0) is zero, the formula simplifies to:

Angular speed (ω) = Angular acceleration (α) * Time (t)

Now, plug in the values:

Angular speed (ω) = 2 radians/second^2 * 8.9 seconds

Calculating this gives us:

Angular speed (ω) = 17.8 radians/second

Therefore, the angular speed at the end of 8.9 seconds is 17.8 radians/second.