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 assuming that it is a constant value.
We can use the formula for angular acceleration:
angular acceleration = (final angular velocity - initial angular velocity) / time
First, we need to convert the number of revolutions to radians. One revolution is equal to 2π radians, so 14 revolutions is equal to 14 x 2π = 28π radians.
Then, we can plug in the values:
angular acceleration = (1.3 rad/s - 14 rad/s) / (28π / 2π) = -12.7 rad/s²
Note that the negative sign indicates that the propeller is slowing down, which makes sense given that the initial angular velocity is greater than the final angular velocity.
To find the angular acceleration of the propeller, we can use the following formula:
angular acceleration (α) = (final angular velocity (ω2) - initial angular velocity (ω1)) / time
Given:
Initial angular velocity (ω1) = 14 rad/s
Final angular velocity (ω2) = 1.3 rad/s
Total revolutions (n) = 14
First, let's calculate the total time it takes for the propeller to complete 14 revolutions:
Total time (t) = (Total revolutions (n) * 2π) / Initial angular velocity (ω1)
t = (14 * 2π) / 14 rad/s
t = 2π s
Now, we can substitute the values into the formula to find the angular acceleration:
α = (1.3 rad/s - 14 rad/s) / (2π s)
α = (-12.7 rad/s) / (2π s)
This gives us the value of the angular acceleration of the propeller.