Answer: a) -333.8 rad/s² . b) 33.1 m c) 10.0 inch = 0.254 m
Explanation:
a) We can apply the definition of angular acceleration, as the rate of change of angular velocity over a time interval, as follows:
aθ = ωf - ωi / t
As ωf=0, we get the following value for aθ:
aθ = - ωi /t
Now, it is advisable to convert the ωi units from rpm to rad/sec, as follows:
3984 rev/min = 3984 rev. (2π /rev) . (60 s/1 min) = 417.2 rad/s
Replacing in the expression for aθ, we get:
aθ = -333.8 rad /s²
b) Assuming a constant angular acceleration, we can find the total angle travelled, Δθ, as follows:
ωf² - ωi² = 2 aθ Δθ ⇒ Δθ = -ωi² / 2 aθ = -(417.2)² /2 (-333.8) = 260.71 rad
By definition, an angle is the relationship between the arc subtended and the radius, so we can write the following expression:
Δs = r Δθ ⇒ Δs = 0.127 m. 260. 71 rad = 33.1 m
c) In order to get the net displacement, we need to know how many revolutions are equivalent to the distance travelled, in terms of circumferences, so we can write the following:
Nbr. of circumferences: 33.1 m / 2π r = 33.1 m / 2 π. 0.127 m = 41.5
So, taking any point on the rim of the blade, it travelled an entire number of circumferences during deceleration, plus half a circumference, which is equal to one diameter in net displacement.
So, the net displacement is equal to one diameter, i.e., 10 inch or 0.254 m.
