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CHAPTER
10
Conservation of Angular Momentum
1* ·
True or false: (
a
) If two vectors are parallel, their cross product must be zero.
(
b
) When a disk rotates about its
symmetry axis,
w
is along the axis.
(
c
) The torque exerted by a force is always perpendicular to the force.
(
a
) True
(
b
) True
(
c
) True
2
·
Two vectors
A
and
B
have equal magnitude. Their cross product has the greatest magnitude if
A
and
B
are
(
a
) parallel.
(
b
) equal.
(
c
) perpendicular.
(
d
) antiparallel. (
e
) at an angle of 45
o
to each other.
(
c
)
3
·
A force of magnitude
F
is applied horizontally in the negative
x
direction to the rim of a disk of radius
R
as
shown in Figure 1029. Write
F
and
r
in terms of the unit vectors
i
,
j
, and
k
, and compute the torque produced by the
force about the origin at the center of the disk.
F
= 
F
i
;
r
=
R
j
;
t
=
r
·
F
=
FR
j
·

i
=
FR
i
·
j
=
FR
k
.
4
·
Compute the torque about the origin for the force
F
= 
mg
j
acting on a particle at
r
=
x
i
+
y
j
, and show that this
torque is independent of the
y
coordinate.
Use Equs. 101 and 107
t
= 
mgx
i
·
j

mgy
j
·
j
= 
mgx
k
5* ·
Find
A
·
B
for (
a
)
A
= 4
i
and
B
= 6
i
+ 6
j
,
(
b
)
A
= 4
i
and
B
= 6
i
+ 6
k
, and
(
c
)
A
= 2
i
+ 3
j
and
B
= 3
i
+ 2
j
.
Use Equ. 107; Note that
i
·
i
=
j
·
j
=
k
·
k
= 0
(
a
)
A
·
B
= 24
i
·
j
= 24
k
.
(
b
)
A
·
B
= 24
i
·
k
= 24
j
.
(
c
)
A
·
B
= 4
i
·
j

9
j
·
i
= 13
k
.
6
·
Under what conditions is the magnitude of
A
×
B
equal to
A B
?
A
·
B
=
AB
sin
q
=
A B
=
AB
cos
q
if
sin
q
=
cos
q
or tan
q
= ±1;
q
= ±45
o
or
q
= ±135
o
.
7
·
A particle moves in a circle of radius
r
with an angular velocity
w
. (
a
) Show that its velocity is
v
=
w
·
r
.
(
b
)
Show that its centripetal acceleration is
a
c
=
w
·
v
=
w
·
(
w
·
r
)
.
(a)
Let
r
be in the
xy
plane. Then if
w
points in the positive
z
direction, i.e.,
w
=
w
k
, the particle’s velocity is in
the
j
direction when
r
=
r
i
(see Figure) and has the magnitude
r
w
. Thus,
v
=
w
·
r
=
r
w
j
.
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View Full Document Chapter 10
Conservation of Angular Momentum
(
b
)
a
=
d
v
/
dt
= (
d
w
/
dt
)
×
r
+
w
·
(
d
r
/
dt
) = (
d
w
/
dt
)
×
r
+
w
·
v
=
a
t
+
w
·
(
w
·
r
) =
a
t
+
a
c
,
where
a
t
and
a
c
are the tangential and centripetal accelerations, respectively.
8
··
If
A
=
4
i
,
B
z
= 0,
B
= 5, and
A
·
B
=
12
k
, determine
B
.
B
=
B
x
i
+
B
y
j
(
B
z
= 0); write
A
·
B
B
x
2
+
B
y
2
=
B
2
; solve for
B
x
12
k
= 4
B
y
i
·
j
= 4
B
y
k
;
B
y
= 3
B
x
= 4 ;
B
= 4
i
+ 3
j
9* ·
If
A
=
3
j
,
A
·
B
= 9
i
, and
A B
= 12, find
B
.
Let
B
=
B
x
i
+
B
y
j
+
B
z
k
; write
A B
and find
B
y
Write
A
·
B
and determine
B
x
and
B
y
A B
= 3
B
y
= 12;
B
y
= 4
9
i
= 3
B
x
j
·
i
+
3
B
z
j
·
k
= 3
B
x
k
+ 3
B
z
i
;
B
x
= 0,
B
z
= 3
B
= 4
j
+ 3
k
10 ·
What is the angle between a particle’s linear momentum
p
and its angular momentum
L
?
From Equ. 108 it follows that
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This homework help was uploaded on 04/09/2008 for the course PHYS 161,260,27 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff
 Angular Momentum, Force, Momentum

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