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Angular Momentum
CHAPTER OUTLINE
11.1
The Vector Product and
Torque
11.2
Angular Momentum
11.3
Angular Momentum of a
Rotating Rigid Object
11.4
Conservation of Angular
Momentum
11.5
The Motion of Gyroscopes
and Tops
11.6
Angular Momentum as a
Fundamental Quantity
336
±
Mark Ruiz undergoes a rotation during a dive at the U.S. Olympic trials in June 2000. He
spins at a higher rate when he curls up and grabs his ankles due to the principle of conser-
vation of angular momentum, as discussed in this chapter. (Otto Greule/Allsport/Getty)

T
he central topic of this chapter is angular momentum, a quantity that plays a key
role in rotational dynamics. In analogy to the principle of conservation of linear mo-
mentum, we ﬁnd that the angular momentum of a system is conserved if no external
torques act on the system. Like the law of conservation of linear momentum, the law of
conservation of angular momentum is a fundamental law of physics, equally valid for
relativistic and quantum systems.
11.1
The Vector Product and Torque
An important consideration in deﬁning angular momentum is the process of multiply-
ing two vectors by means of the operation called the
vector product
. We will introduce
the vector product by considering torque as introduced in the preceding chapter.
Consider a force
F
acting on a rigid object at the vector position
r
(Fig. 11.1). As we
saw in Section 10.6, the
magnitude
of the torque due to this force relative to the origin
is
rF
sin
±
, where
is the angle between
r
and
F
. The axis about which
F
tends to pro-
duce rotation is perpendicular to the plane formed by
r
and
F
.
The torque vector
is related to the two vectors
r
and
F
. We can establish a mathe-
matical relationship between
,
r
, and
F
using a mathematical operation called the
vector product,
or
cross product:
±
r
²
F
(11.1)
We now give a formal deﬁnition of the vector product. Given any two vectors
A
and
B
, the
vector product A
²
B
is deﬁned as a third vector
C
, which has a magnitude of
AB
sin
²
, where
is the angle between
A
and
B
. That is, if
C
is given by
C
³
A
²
B
(11.2)
then its magnitude is
C
±
AB
sin
(11.3)
The quantity
AB
sin
is equal to the area of the parallelogram formed by
A
and
B
, as
shown in Figure 11.2. The
direction
of
C
is perpendicular to the plane formed by
A
and
B
, and the best way to determine this direction is to use the right-hand rule illustrated
in Figure 11.2. The four ﬁngers of the right hand are pointed along
A
and then
“wrapped” into
B
through the angle
. The direction of the upright thumb is the direc-
tion of
A
²
B
³
C
. Because of the notation,
A
²
B
is often read “
A
cross
B
”; hence,
the term
cross product.
Some properties of the vector product that follow from its deﬁnition are as follows:
1.
Unlike the scalar product, the vector product is
not
commutative. Instead, the or-
der in which the two vectors are multiplied in a cross product is important:
A
²
B
³´
B
²
A
(11.4)
337
O
r
P
φ
x
F
y
τ
=
r
×
F
z
τ
Active Figure 11.1
The torque
vector
lies in a direction perpen-
dicular to the plane formed by the
position vector
r

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