Pickering, Tracey – Homework 24 – Due: Mar 31 2006, noon – Inst: Drummond
1
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printout
should
have
15
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
The due time is Central
time.
001
(part 1 of 3) 10 points
Consider the two vectors
~
M
= (
a, b
) =
a
ˆ
ı
+
b
ˆ
and
~
N
= (
c, d
) =
c
ˆ
ı
+
d
ˆ
where,
a
and
c
represent the
x
displacement and
b
and
d
represent the
y
displacement in a Cartesian
xy
coordinate system.
Note:
ˆ
ı
and ˆ
represent unit vectors (
i.e.
vectors of length 1) in the
x
and
y
directions,
respectively.
What is the magnitude of the vector prod
uct
k
~
M
×
~
M
k
?
1.
k
~
M
×
~
M
k
=
a
2

2
a b
+
b
2
2.
k
~
M
×
~
M
k
= 2
a b
3.
k
~
M
×
~
M
k
=
a
2
+
b
2
4.
k
~
M
×
~
M
k
=
a
2

b
2
5.
k
~
M
×
~
M
k
=
a b
6.
k
~
M
×
~
M
k
=
p
a
2
+
b
2
7.
k
~
M
×
~
M
k
=
a
2
+ 2
a b
+
b
2
8.
k
~
M
×
~
M
k
=
a
+
b
9.
k
~
M
×
~
M
k
= 0
correct
10.
k
~
M
×
~
M
k
=

a b
Explanation:
The magnitude of the vector product of two
vectors
k
~
A
×
~
B
k
=
A B
sin
θ
, where
θ
is the
angle between the two vectors placed with
their tails at the same point.
When
~
A
and
~
B
are the same vector,
θ
= 0
◦
.
Since sin 0
◦
= 0, the vector product of a vector
with itself is zero.
002
(part 2 of 3) 10 points
What is the magnitude of the vector product
k
~
M
×
~
N
k
?
1.
k
~
M
×
~
N
k
=
a c
+
b d
2.
k
~
M
×
~
N
k
=
a
2
+
b
2
+
c
2
+
d
2
3.
k
~
M
×
~
N
k
=
a b c d
4.
k
~
M
×
~
N
k
=
a d

b c
correct
5.
k
~
M
×
~
N
k
=
a b

c d
6.
k
~
M
×
~
N
k
= 0
7.
k
~
M
×
~
N
k
=
a d
+
b c
8.
k
~
M
×
~
N
k
=
a b
+
c d
9.
k
~
M
×
~
N
k
=
a

b
10.
k
~
M
×
~
N
k
=
a c

b d
Explanation:
Take the vector products of the
x
 and
y

displacement of
~
M
and
~
N
individually
~
M
×
~
N
= (
a
ˆ
ı
+
b
ˆ
)
×
(
c
ˆ
ı
+
d
ˆ
)
=
a c
(ˆ
ı
×
ˆ
ı
) +
b c
(ˆ
×
ˆ
ı
)
+
a d
(ˆ
ı
×
ˆ
) +
b d
(ˆ
×
ˆ
)
=
a d

b c .
since ˆ
ı
⊥
ˆ
, we have ˆ
ı
×
ˆ
ı
= 0, ˆ
×
ˆ
ı
= +1,
ˆ
ı
×
ˆ
=

1, and ˆ
×
ˆ
= 0.
Note:
sin 0
◦
= 0 and sin 90
◦
= 1.
The
magnitude of
k
ˆ
ı
×
ˆ
k
= (1)(1) sin 90
◦
= 1
,
and
k
ˆ
×
ˆ
ı
k
= (1)(1) sin 90
◦
= 1
.
The vector product of two vectors, when
nonzero, is a vector. These two vector prod
ucts have direction given by
ˆ
ı
×
ˆ
=
ˆ
k
and
ˆ
×
ˆ
ı
=

ˆ
k ,
where
ˆ
k
is a unit vector pointing along the
positive
z
axis.
Thus the vectors ˆ
ı
×
ˆ
and
ˆ
×
ˆ
ı
point in opposite directions. The result
is
~
M
×
~
N
= (
a d

b c
)
ˆ
k .
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Pickering, Tracey – Homework 24 – Due: Mar 31 2006, noon – Inst: Drummond
2
003
(part 3 of 3) 10 points
What is the direction of the vector product
~
M
×
~
N
?
1.
along the
x
axis
2.
in the
yz
plane, but not along
y
or
z
3.
in the
xy
plane, but not along
x
or
y
4.
none of these since its direction cannot be
determined
5.
in the
xz
plane, but not along
x
or
z
6.
along the
y
axis
7.
along the
z
axis
correct
Explanation:
~
M
×
~
N
points along the
z
axis. It only has
a displacement in the
±
ˆ
k
direction.
004
(part 1 of 1) 10 points
A satellite of mass 3
m
moves on a plane in a
circular orbit of radius
r
3
m
i
about a spherical
planet of mass
M
and radius
R
. The magni
tude of the velocity is
v
3
m
i
.
M
is much larger
than 3
m
, so the center of mass of the system
can be regarded as being located at the center
of
M
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 Fall '08
 Turner
 Angular Momentum, Mass, Work, Pickering

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