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Chapter 10
Angular Momentum
Conceptual Problems
1
•
True or false:
(
a
) If two vectors are exactly opposite in direction, their cross product must be
zero.
(
b
) The magnitude of the cross product of 2 vectors is at a minimum when the two
vectors are perpendicular.
(
c
) Knowing the magnitude of the cross product of two nonzero vectors and their
individual magnitudes uniquely determines the angle between them.
Determine the Concept
The cross product of vectors
A
G
and
B
G
is defined to be
n
AB
B
A
ˆ
sin
φ
=
×
G
G
where
n
ˆ is a unit vector normal to the plane defined by
A
G
and
B
G
.
(
a
) True. If
A
G
and
B
G
are in opposite direction, then sin
=
sin(180
°
) = 0
.
(
b
) False. If
A
G
and
B
G
are perpendicular, then sin
=
sin(90
°
) = 1 and the cross
product of
A
G
and
B
G
is a maximum.
(
c
) False.
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
×
=
−
AB
B
A
G
G
1
sin
, because of the magnitude of
B
A
G
G
×
, gives the
reference angle associated with
B
A
G
G
×
.
2
•
Consider two nonzero vectors
G
A
and
G
B
. Their cross product has the
greatest magnitude if
G
A
and
G
B
are (
a
) parallel, (
b
) perpendicular, (
c
) antiparallel,
(
d
) at an angle of 45° to each other.
Determine the Concept
The cross product of the vectors
A
G
and
B
G
is defined to
be
n
AB
B
A
ˆ
sin
=
×
G
G
where
n
ˆ is a unit vector normal to the plane defined by
A
G
and
B
G
. Hence, the cross product is a maximum when sin
=
1. This condition
is satisfied provided
A
G
and
B
G
are
perpendicular
.
)
(
b
is correct.
3
•
What is the angle between a force
F
G
and a torque vector
τ
G
produced
by
F
G
?
Determine the Concept
Because
n
rF
F
r
ˆ
sin
=
×
=
G
G
G
, where
n
ˆ is a unit vector
normal to the plane defined by
r
G
and
F
G
, the angle between
F
G
and
G
is
.
90
°

Chapter 10
962
4
•
A particle of mass
m
is moving with a constant speed
v
along a straight
line that passes through point
P
. What can you say about the angular momentum
of the particle relative to point
P
? (
a
) Its magnitude is
mv
. (
b
) Its magnitude is
zero. (
c
) Its magnitude changes sign as the particle passes through point
P
. (
d
) It
varies in magnitude as the particle approaches point
P
.
Determine the Concept
L
G
and
p
G
are related according to
.
p
r
L
G
G
G
×
=
Because the
motion is along a line that passes through point
P
,
r
= 0 and so is
L
.
)
(
b
is
correct.
5
•
[SSM]
A particle travels in a circular path and point
P
is at the
center of the circle. (
a
) If the particle’s linear momentum
G
p
is doubled without
changing the radius of the circle, how is the magnitude of its angular momentum
about
P
affected? (
b
) If the radius of the circle is doubled but the speed of the
particle is unchanged, how is the magnitude of its angular momentum about
P
affected?
Determine the Concept
L
G
and
p
G
are related according to
.
p
r
L
G
G
G
×
=
(
a
) Because
L
G
is directly proportional to
p
G
,
L
is doubled.

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