12.
Picture the Problem
: The 2.0kg mass is gravitationally
attracted to the other three masses.
Strategy:
Add the gravitational forces using the component
method of vector addition.
Use equation 121 and the
geometry of the problem to determine the magnitudes of the
forces. Let
1
1.0 kg,
m
=
2
2.0 kg,
m
=
3
3.0 kg,
m
=
and
4
4.0 kg.
m
=
Solution:
1. (a)
Find the
x
com
ponent of the force on
2
m
:
(
29
(
29
(
29
(
29
(
(
29
(
1
2
2
4
2
2
12
24
1
2
2
4
12
1
4 12
2
2
2
2
3
24
12
24
12
24
11
2
2
2
2
8
cos
4.0 kg
0
1.0 kg
6.67
10
N
m
/kg
2.0 kg
0.20 m
0.20 m
0
1.3
10
N
x
x
m m
m m
F
G
G
r
r
m m
m m
r
m
m r
G
G
Gm
r
r
r
r
r
F
θ


=
+
=
+
=
+
÷
÷
=
×
×
+
+
=
×
2.
Find the
y
component
of the force on
2
m
:
(
29
(
29
(
29
(
29
(
(
29
(
2
3
2
4
2
2
23
24
2
3
3
2
4
14
4 14
2
2
2
2
3
24
23
24
23
24
11
2
2
2
2
8
sin
4.0 kg
0
3.0 kg
6.67
10
N
m
/kg
2.0 kg
0.10 m
0.20 m
0
4.5
10
N
y
y
m m
m m
F
G
G
r
r
m m
m
m m
r
m r
G
G
Gm
r
r
r
r
r
F
θ


=
+
=
+
=
+
÷
÷
=
×
×
+
+
=
×
3.
Use the components of
F
r
to
find its magnitude and
direction.
(
29
(
29
2
2
2
2
8
8
8
8
1
1
8
1.3
10
N
4.5
10
N
4.7
10
N
4.5
10
N
tan
tan
74
below horizontal and to
1.3
10
N
x
y
y
x
F
F
F
F
F
θ







=
+
=
×
+
×
=
×
×
=
=
=
°
÷
÷
×
4.
(b)
If the sides of the rectangle are all doubled, all forces are reduced by a factor of
2
2
4;
=
the
directions of the forces are unchanged.
Insight:
Doubling all the masses will quadruple the gravitational force, because the gravitational
force depends upon the product of the masses of each object.
Therefore, the force would stay
exactly the same if we doubled all the masses and doubled the length of the sides of the rectangle.
21.
Picture the Problem
: The volcano on Io spews material at high speed straight upward.
The mass
slows down, rising to a height of 5.00 km before coming to rest momentarily under the influence
of Io’s gravitation.
Strategy:
Use conservation of energy to relate the initial kinetic energy of the ejected material to
its potential energy at the maximum altitude.
This relation will allow the calculation of the surface
gravity of Io.
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 Spring '11
 BEAN
 Mass, Potential Energy, Gravitational forces

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