homework 22 – BAUTISTA, ALDO – Due: Mar 24 2006, 4:00 am
1
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Question 1
Part 1 of 1.
10 points.
A uniform flat plate of metal is situated in
the reference frame shown in the figure below.
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
x
y
Calculate the
x
coordinate of the center of
mass of the metal plate.
Correct answer: 7
.
33333
(tolerance
±
1 %).
Explanation:
Basic Concept:
The center of mass coor
dinate is
x
≡
Z
x dm
m
,
where
m
≡
Z
dm ,
and
dm
=
σ y dx ,
where
σ
is the areal density
mass
area
of the plate.
Solution:
Let
(
x
1
, y
1
) = (6
,
0)
(
x
2
, y
2
) = (8
,
0)
(
x
3
, y
3
) = (8
,
4)
.
The equation for the hypotenuse is
y

y
1
x

x
1
=
y
3

y
1
x
3

x
1
.
The slope of the hypotenuse is
s
=
y
3

y
1
x
3

x
1
=
4

0
8

6
= +2
.
Rewriting the equation, we have
y
=
s
(
x

x
1
) +
y
1
= (+2) (
x

6) + 0
.
The
x
coordinate of the center of mass is
x
=
σ
Z
x
2
x
1
x y dx
σ
Z
x
2
x
1
y dx
=
Z
x
2
x
1
x s
(
x

x
1
)
dx
Z
x
2
x
1
s
(
x

x
1
)
dx
=
Z
x
2
x
1
x
(
x

x
1
)
dx
Z
x
2
x
1
(
x

x
1
)
dx
=
1
3
x
3

1
2
(
x
1
)
x
2
1
2
x
2

(
x
1
)
x
x
2
x
1
=
1
3
(
x
3
2

x
3
1
)

1
2
(
x
1
) (
x
2
2

x
2
1
)
1
2
(
x
2
2

x
2
1
)

(
x
1
) (
x
2

x
1
)
=
3
x
1
(
x
2
2

2
x
1
x
2
+
x
2
1
)
3 (
x
2
2

2
x
1
x
2
+
x
2
1
)
+
2 (
x
2

x
1
) (
x
2
2

2
x
1
x
2
+
x
2
1
)
3 (
x
2
2

2
x
1
x
2
+
x
2
1
)
=
x
1
+
2
3
(
x
2

x
1
)
(1)
= 6 +
2
3
(8

6)
= 7
.
33333
.
Alternate solution:
The center of mass
of a right triangle is
1
3
of the height or base
of the triangle measured from its right angle.
Therefore Eq. 1 is the
x
coordinate of the
center of mass of the metal plate.
The
y
coordinate of the center of mass of
the metal plate is
y
=
y
2
+
1
3
(
y
3

y
2
)
= 0 +
1
3
(4

0)
= 1
.
33333
.
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homework 22 – BAUTISTA, ALDO – Due: Mar 24 2006, 4:00 am
2
Note:
This problem has a different triangle
for each student.
Question 2
Part 1 of 1.
10 points.
A 2
.
63 kg particle has a velocity
v
2
.
63 kg
=
a
ˆ
ı
+
b
ˆ
,
where
a
=

59
.
5 m
/
s and
b
= 2
.
75 m
/
s
,
and
a 4
.
24 kg particle has a velocity
v
4
.
24 kg
=
c
ˆ
ı
+
d
ˆ
,
where
c
= 1
.
47 m
/
s and
d
= 9
.
45 m
/
s
.
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 Spring '08
 Turner
 Work, Correct Answer, kg, BAUTISTA

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