222 Chapter 8 Time and Time-Related Parameters
In Equation (8.13), vrepresents the average angular speed in radians per second, uis the
angular displacement (radians), and tis the time interval in seconds. Similar to the definition
of instantaneous velocity given earlier, the instantaneous angular speed is defined by making the
time increment smaller and smaller. Again, when we speak of angular velocity, we not only
refer to the speed of rotation but also to the direction of rotation. It is a common practice to
express the angular speed of rotating objects in revolutions per minute (rpm) instead of radi-
ans per second (rad /s). For example, the rotational speed of a pump impeller may be expressed
as 1600 rpm. To convert the angular speed value from rpm to rad /s, we make the appropriate
conversion substitutions.In practice, the angular speed of rotating objects is measured using a stroboscope or a
tachometer.
To get some idea how fast some common objects rotate, consider the following examples:
a dentist’s drill runs at 400,000 rpm; a current state-of-the-art computer hard drive runs at
7200 rpm; the earth goes through one complete revolution in 24 hours, thus the rotational
speed of earth is 15 degrees per hour or 1 degree every 4 minutes.
There exists a relationship between linear and angular velocities of objects that not only
rotate but also translate as well. For example, a car wheel, when not slipping, will not only
rotate but also translate. To establish the relationship between rotational speed and the trans-
lational speed we begin with(8.14)
dividing both sides by time increment t
(8.15)
and making use of the definitions of linear and angular velocities, we have
(8.16)
For example, the actual linear velocity of a particle located 0.1 m away from the center of a
shaft that is rotating at an angular speed of 1000 rpm (104.7 rad /s) is approximately 10.5 m/s.Example 8.5 Determine the rotational speed of a car wheel if the car is translating along at a speed of
55 mph. The radius of the wheel is 12.5 in.
Using Equation (8.16), we havev
V
r
a 55
miles
h
ba
1 h
3600 s
ba
5280 ft
1 mile
b
1 12.5 in.2a
1 ft
12 in.
b
77.4 rad/s739 rpm
Vrv
¢S
¢t
r
¢u
¢t
¢Sr¢u
1600 a
revolutions
minutes
ba
2 p radians
1 revolution
ba
1 minute
60 seconds
b167.5
rad
s
S
r
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