Angular Acceleration
Most of the cases we will consider in this chapter involve constant angular velocity, at least
during the time of the motion we will consider. But there are many examples of motion for which
the angular velocity is changing. For instance, in the above example of the roulette wheel, the ball
will eventually slow down and stop, meaning the angular velocity has decreased to zero.
Figure 6.1 A particle moves in a circle with changing angular velocity.
Figure 6.1 shows the motion of a particle with an increasing angular velocity. At time t i the
angular velocity is  i ; at the later time t f the angular velocity is  f . The change in angular
velocity during this time interval is
   f   i
The rate of change of velocity is acceleration; similarly, we can define the rate of change of
angular velocity as angular acceleration. In analogy with linear motion, we can define the
angular acceleration to be
change in angular velocity 
(0.1)

time interval
t
Angular acceleration for a particle in circular motion
The symbol for angular acceleration is the Greek letter  ; the units are rad/s2. changing angular
velocity implies an angular acceleration, for which we use the symbol . The sign of 
depends on the sign of the change in angular velocity—again in analogy with the onedimensional motion situation.
As we have seen, the definitions of angular velocity and angular acceleration can be easily
deduced by analogy with the related equations for one-dimensional motion, as we have seen.
Table 6.1 lists the variables for one-dimensional motion and angular motion and their definitions
side-by-side.

TABLE 6.1 Linear and angular motion variables
1D motion
variable
Postion
x
Velocity
v
Acceleration
a
Angular
motion
variable
1D motion definition
Angular motion definition
Angle
Displacement ∆x:
Angular displacement  :

x  xf  xi
   f   i
Angular
velocity
v
x
t


t
Angular
acceleration
a
v
t


t


All of the equations for one-dimensional motion have rotational analogs as well. We saw one
example above; Equation Error! Reference source not found. was obtained from a onedimensional motion equation by substituting angle for position and angular velocity for velocity.
Table 6. lists some one-dimensional motion equations and the analogous equations for angular
motion.
TABLE 6.2 Linear and angular motion equations
1D motion equation
Angular motion equation
Displacement of object moving
at constant speed:
Angular displacement of
object moving at constant
angular speed:
x  vt
Change in velocity of object
undergoing constant
acceleration:
  t
Change in angular velocity of
object undergoing constant
angular acceleration:
v  at
  t
Displacement of object
undergoing constant
acceleration:
Angular displacement of
object undergoing constant
angular acceleration:
1
x  v0 t  at 2
2
  0   0t   t 2
1
2
EXAMPLE 6.1 Spinning up a hard drive platter
The platter in a hard disk drive spins up to a steady 5400 rpm in a time of 2.0 seconds. What is
the angular acceleration? At the end of 2.0 seconds, how many revolutions has the platter made?
Prepare We convert 5400 rpm into angular velocity:
5400 rev 1 min 2 rad


 565 rad/s
min
60 s
1 rev
Solve Using the definition of angular acceleration in Table 6.1, we compute:
 565 rad/s

 283 rad/s2
t
2.0 s
During the time of this angular acceleration, we can compute the angular displacement using an
equation from Table 6.2:



1
1
2
   0 t   t 2  0 rad/s   283 rad/s2 2.0 s 
2
2
 566 rad
Each revolution corresponds to an angular displacement of 2π, so we compute:
566 rad
 90.1 revolutions
2 rad/revolution
The platter completes 90 complete revolutions during the first two seconds.
Assess As you might expect, the solution to this problem is exactly analogous to the solution of
a one-dimensional motion problem..
Number of revolutions 
In the above example, there is an angular acceleration; the angular velocity is changing. But, as
we have seen, there is an acceleration—a centripetal acceleration—even in cases in which the
angular velocity is constant. In the rest of the chapter, we will consider situations in which the
angular velocity is constant; when we speak of acceleration, we will mean the centripetal
acceleration.
A person swings a ball on the end of a string in a horizontal circle once
every second. For each of the following quantities, tell whether they are zero, constant (but not
zero) or changing.
A. Period
B. Velocity
C. Angular velocity
D. Acceleration
STOP TO THINK 6.1