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Chapter 11
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)
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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 righthand 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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 Spring '11
 WANG
 Angular Momentum, Momentum

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