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12.2 Rotation about a Fixed Axis 369

When a rigid body rotates, the position angle changes in time. The body then
has an angular velocity (the Greek letter omega).The definition of the angular veloc-
ity for rotational motion is mathematically analogous to the definition of velocity for
translational motion (see Sections 2.2 and 2.3). The average angular velocity is
defined as
¢f
(12.2) average angular velocity
¢t

where is the change in the angular position and t the corresponding change in time.
The instantaneous angular velocity is defined as

df
(12.3) instantaneous angular velocity
dt
According to these definitions, the angular velocity is the rate of change of the
angle with time.The unit of angular velocity is the radian per second (1 radian/s).The
radian is the ratio of two lengths [compare Eq. (12.1)], and hence it is a pure number;
thus, 1 radian/s is the same thing as 1/s. However, to prevent confusion, it is often
useful to retain the vacuous label radian as a reminder that angular motion is involved.
Table 12.1 gives some examples of angular velocities.
If the body rotates with constant angular velocity, then we can also measure the
rate of rotation in terms of the ordinary frequency f, or the number of revolutions per
second. Since each complete revolution involves a change of by 2 radians, the fre-
quency of revolution is smaller than the angular velocity by a factor of 2 :

f (12.4) frequency
2p

This expresses the frequency in terms of the angular velocity. The unit of rotational
frequency is the revolution per second (1 rev/s). Like the radian, the revolution is a
pure number, and hence 1 rev/s is the same thing as 1/s. But we will keep the label rev
to prevent confusion between rev/s and radian/s.
As in the case of planetary motion, the time per revolution is called the period of
the motion. If the number of revolutions per second is f, then the time per revolution
is 1/f, that is,
1
T (12.5) period of motion
f

TABLE 12.1 SOME ANGULAR VELOCITIES

2
Computer hard disk 8 10 radians/s Helicopter rotor 40 radians/s
2
Circular saw 7 10 radians/s Compact disc (outer track) 22 radians/s
2
Electric blender blades 5 10 radians/s Phonograph turntable 3.5 radians/s
2
Jet engine 4 10 radians/s Neutron star (pulsar) rotation 0.1 radian/s
2
Airplane propeller 3 10 radians/s Earth rotation 7.3 10 5 radian/s
2
Automobile engine 2 10 radians/s Earth revolution about Sun 2.0 10 7 radian/s
Small fan 60 radians/s