9.4.
A fan blade rotates with angular velocity given by 𝜔𝑧 (𝑡) = 𝛾 − 𝛽𝑡 2 , where  = 5.00 rad/s and 
= 0.800 rad/s3. (a) Calculate the instantaneous angular acceleration z at t = 3.00 s and the
average angular acceleration av-z for the time interval t = 0 to t = 3.00 s. How do these two
quantities compare? If they are different, why are they different?
z
 z  dz / dt .  av-z  t .
Identify:
d 2
(t )  2t
dt
Set Up:
Execute:
(a)
αz (t ) 
dωz
 2 βt  (  1.60 rad s3 )t.
dt
3
2
(b) αz (3.0 s)  (  1.60 rad s )(3.0 s)  4.80 rad s .
αav-z 
ωz (3.0 s)  ωz (0) 2.20 rad s  5.00 rad s

 2.40 rad s 2. ,
3.0 s
3.0 s
which is half as large (in magnitude) as the acceleration at t  3.0 s.
Evaluate:
 z (t ) increases linearly with time, so  av-z 
 z (0)   z (3.0 s)
2
.  z (0)  0 .