Question

A fan blade rotates with angular velocity given by ω_z(t)= γ − β t², where γ = 5.00 rad/s and β = 0.800 rad/s³.

a- Calculate the angular acceleration as a function of time.

b- Calculate the instantaneous angular acceleration α_z at t = 3.00 s and the average angular acceleration a_av-z for the time interval t=0 to t₀ = 3.00 s.

How do these two quantities compare? If they are different, why are they different?

Answer 1

a) The angular acceleration is the first derivative of velocity by time:

$$\alpha (t)=0-2\beta t=-1.6t.$$

b) The instantaneous value of acceleration:

$$\alpha (3)=-4.8\text{ rad/s}^2.$$

The average angular acceleration is the sum of angular accelerations divided by time it took:

$$\alpha_\text{avg}=\dfrac{\alpha (3)+\alpha (0)}{t}=-1.6\text{ rad/s}^2.$$

The values are different because the average value is averaged along a certain time, while the instantaneous value is the value at a certain moment.