an airplane propeller slows from an initial angular speed of 12.5rev/s to a final angular speed of 5rev/s during the process the propeller rotates through 21 rev. find the angular acceleration of the propeller if in radians per second squared, assuming its constant?
A 38.0-kg crate rests on a horizontal floor, and a 72.8-kg person is standing on the crate. Determine the magnitude of the normal force that (a) the floor exerts on the crate and (b) the crate exerts on the person.
To find the angular acceleration of the propeller, we can use the following formula:
angular acceleration (α) = (final angular speed (ωf) - initial angular speed (ωi)) / time
First, let's convert the values given to radians per second (rad/s):
Initial angular speed (ωi) = 12.5 rev/s = 12.5 * 2π rad/s
Final angular speed (ωf) = 5 rev/s = 5 * 2π rad/s
Next, let's calculate the time of rotation (t) in seconds. We can use the formula:
time (t) = angle (θ) / angular speed (ω)
Given that the propeller rotates through 21 rev, we can convert this to radians:
angle (θ) = 21 rev * 2π rad/rev
Now we can substitute the values into the formula for time:
time (t) = (21 rev * 2π rad/rev) / 12.5 * 2π rad/s
Simplifying, we get:
time (t) = 21 / 12.5 s
Finally, we can substitute the values into the formula for angular acceleration:
angular acceleration (α) = (5 * 2π rad/s - 12.5 * 2π rad/s) / (21 / 12.5 s)
Simplifying, we get:
angular acceleration (α) = (5 - 12.5) * (12.5 / 21) rad/s^2
Now, let's calculate the value:
angular acceleration (α) = -7.5 * (12.5 / 21) rad/s^2
To find the angular acceleration of the propeller, we can use the equation:
angular acceleration (α) = (final angular speed (ωf) - initial angular speed (ωi)) / time
In this case, the final angular speed (ωf) is 5 rev/s, the initial angular speed (ωi) is 12.5 rev/s, and the time is the rotation of the propeller through 21 rev.
First, we need to convert the units of angular speed from rev/s to rad/s. Since 1 revolution is equal to 2π radians, we can use the conversion factor:
1 rev = 2π rad
Therefore,
ωf = 5 rev/s = 5 * 2π rad/s = 10π rad/s
ωi = 12.5 rev/s = 12.5 * 2π rad/s = 25π rad/s
Now let's calculate the angular acceleration by substituting the values into the equation:
α = (ωf - ωi) / time = (10π rad/s - 25π rad/s) / 21 rev
Simplifying,
α = -15π rad/s / 21 rev
Now, to convert the angular acceleration from rev to rad per second squared, we need to multiply by the conversion factor:
1 rev = 2π rad
So,
α = -15π rad/s / 21 rev * 2π rad/1 rev
Simplifying,
α = -30π^2 rad/s² / 21
Therefore, the angular acceleration of the propeller is approximately -30π^2 / 21 rad/s².