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 is a constant value.
We can use the formula for angular acceleration:
α = (ωf - ωi) / t
where ωi is the initial angular velocity, ωf is the final angular velocity, and t is the time taken to change between the two velocities.
In this case, ωi = 14 rad/s, ωf = 1.3 rad/s, and the propeller completes 14 revolutions, or 28π radians, during this time. So we have:
t = 28π / (ωi - ωf)
t = 28π / (14 rad/s - 1.3 rad/s)
t ≈ 6.39 s
Now we can use the formula to find the angular acceleration when the propeller is at a constant velocity:
α = (ωf - ωi) / t
α = (1.3 rad/s - 14 rad/s) / 6.39 s
α ≈ -2.04 rad/s²
We get a negative value for the angular acceleration because the propeller is slowing down. If it were speeding up, we would get a positive value.
To find the angular acceleration of the propeller when it is at a constant value, we can use the formula:
Angular acceleration (α) = Change in angular velocity (Δω) / Change in time (Δt)
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 calculate the change in angular velocity:
Δω = ω2 - ω1
= 1.3 rad/s - 14 rad/s
= -12.7 rad/s
Next, we need to convert the number of revolutions to radians:
Number of radians (θ) = 2π * number of revolutions
= 2π * 14
= 28π rad
Now, we can calculate the change in time:
Δt = θ / Δω
= (28π rad) / (-12.7 rad/s)
Finally, we can calculate the angular acceleration:
Angular acceleration (α) = Δω / Δt
= (-12.7 rad/s) / [(28π rad) / (-12.7 rad/s)]
Simplifying this equation will give you the value of the angular acceleration.