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366 CHAPTER 12 Rotation of a Rigid Body

body is rigid if the particles in the body do not move relative to one another. Thus, the
rigid body
Abody has a fixed shape, and all its parts have a fixed position relative to one another.
A hammer is a rigid body, and so is a baseball bat. A baseball is not rigid—when struck
a blow by the bat, the ball suffers a substantial deformation; that is, different parts of
the ball move relative to one another. However, the baseball can be regarded as a rigid
body while it flies through the air—the air resistance is not sufficiently large to pro-
duce an appreciable deformation of the ball. This example indicates that whether a
body can be regarded as rigid depends on the circumstances. No body is absolutely
rigid; when subjected to a sufficiently large force, any body will suffer some deforma-
tion or perhaps even break into several pieces. In this chapter, we will ignore such
deformations produced by the forces acting on bodies. We will examine the motion
of bodies under the assumption that rigidity is a good approximation.

12.1 MOTION OF A RIGID BODY

A rigid body can simultaneously have two kinds of motion: it can change its position
in space, and it can change its orientation in space. Change of position is translational
motion; as we saw in Chapter 10, this motion can be conveniently described as motion
Hammer rotates of the center of mass. Change in orientation is rotational motion; that is, it is rotation
about its center
of mass. about some axis.
As an example, consider the motion of a hammer thrown upward (see Fig. 12.1).
The orientation of the hammer changes relative to fixed coordinates attached to the
ground. Instantaneously, the hammer rotates about a horizontal axis, say, a horizontal
axis that passes through the center of mass. In Fig. 12.1, this horizontal axis sticks out
of the plane of the page and moves upward with the center of mass. The complete
motion can then be described as a rotation of the hammer about this axis and a simul-
taneous translation of the axis along a parabolic path.
In this example of the thrown hammer, the axis of rotation always remains hori-
zontal, out of the plane of the page. In the general case of motion of a rigid body, the
axis of rotation can have any direction and can also change its direction. To describe
such complicated motion, it is convenient to separate the rotation into three compo-
nents along three perpendicular axes.The three components of rotation are illustrated
by the motion of an aircraft (see Fig. 12.2): the aircraft can turn left or right (yaw), it
can tilt to the left or the right (roll), and it can tilt its nose up or down (pitch). However,
FIGURE 12.1 A hammer in free fall
under the influence of gravity. The center of in the following sections we will usually not deal with this general case of rotation
mass of the hammer moves with constant with three components; we will mostly deal only with the simple case of rotation about
vertical acceleration g, just like a particle in a fixed axis, such as the rotational motion of a fan, a roulette wheel, a compact disc, a
free fall. swinging door, or a merry-go-round (see Fig. 12.3).

Axes of rotation for the
three motions are all
mutually perpendicular.
z

pitch

roll

FIGURE 12.2 Pitch, roll, and yaw y
x
motions of an aircraft. yaw