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.
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).
To find the angular acceleration of the propeller when it's at a constant value, we can use the formula:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given:
Initial angular velocity (ω₁) = 14 rad/s
Final angular velocity (ω₂) = 1.3 rad/s
Number of revolutions (n) = 14
First, we need to convert the number of revolutions to radians:
Number of revolutions (n_rad) = 14 * 2π = 28π rad
To find the time, we can use the formula:
Total displacement (θ) = (angular velocity initial + angular velocity final) / 2 * time
Rearranging the formula, we get:
time = (2 * θ) / (ω₁ + ω₂)
Substituting the given values, we have:
time = (2 * 28π) / (14 + 1.3)
Simplifying, we get:
time = 56π / 15.3
Now, we can calculate the angular acceleration:
angular acceleration (α) = (ω₂ - ω₁) / time
= (1.3 - 14) / (56π / 15.3)
= (-12.7) / (56π / 15.3)
≈ -0.716 rad/s²
Therefore, the angular acceleration of the propeller when it's at a constant value is approximately -0.716 rad/s².