Question

An airplane propeller slows from 14 rad/s to 1.3 rad/s as the propeller completes 14 revolutions. Find the angular acceleration of the propeller when it's a constant value.

Answer 1

We can use the formula:

angular acceleration = (final angular velocity - initial angular velocity) / time

We are given the initial and final angular velocity:

initial angular velocity = 14 rad/s
final angular velocity = 1.3 rad/s

We are also given the number of revolutions completed during the time interval:

revolutions = 14

We need to first convert revolutions to radians. One revolution is equal to 2π radians, so 14 revolutions is:

14 revolutions * 2π radians/revolution = 28π radians

Now we can use the formula:

time = (final angle - initial angle) / average angular velocity

The average angular velocity is the average of the initial and final angular velocity:

average angular velocity = (14 rad/s + 1.3 rad/s) / 2 = 7.65 rad/s

The final and initial angles are:

final angle = 28π radians
initial angle = 0 radians (assuming the propeller starts at rest)

Plugging in these values, we get:

time = (28π radians - 0 radians) / 7.65 rad/s = 3.66 seconds

Now we can find the angular acceleration:

angular acceleration = (1.3 rad/s - 14 rad/s) / 3.66 seconds = -3.36 rad/s^2

Note that the negative sign indicates that the angular acceleration is in the opposite direction to the initial velocity (i.e. it's slowing down).