122
Chapter 12
12.5.
I
DENTIFY
:
Use Eq.(12.1) to calculate
g
F
exerted by the earth and by the sun and add these forces as vectors.
(a) S
ET
U
P
:
The forces and distances are shown in Figure 12.5.
Let
E
F
!
and
S
F
!
be the
gravitational forces exerted
on the spaceship by the
earth and by the sun.
Figure 12.5
E
XECUTE
:
The distance from the earth to the sun is
11
1.50
10
m.
r
=
×
Let the ship be a distance
x
from the
earth; it is then a distance
r
x
−
from the sun.
E
S
F
F
=
says that
2
2
E
S
/
/(
)
Gmm
x
Gmm
r
x
=
−
(
)
2
2
E
/
/
m
x
ms
r
x
=
−
and
2
2
S
E
(
)
(
/
)
r
x
x
m
m
−
=
S
E
/
r
x
x
m m
−
=
and
S
E
(1
/
)
r
x
m m
=
+
11
8
30
24
S
E
1.50
10
m
2.59
10
m
1
/
1+
1.99
10
kg/5.97
10
kg
r
x
m m
×
=
=
=
×
+
×
×
(from center of earth)
(b) E
VALUATE
:
At the instant when the spaceship passes through this point its acceleration is zero. Since
S
E
m
m
"
this equalforce point is much closer to the earth than to the sun.
12.6.
I
DENTIFY
:
Apply Eq.(12.1) to calculate the magnitude of the gravitational force exerted by each sphere.
Each
force is attractive.
The net force is the vector sum of the individual forces.
S
ET
U
P
:
Let +
x
be to the right.
E
XECUTE
:
(a)
(
)
(
)
(
)
(
)
(
)
(
)
11
2
2
11
g
2
2
5.00 kg
10.0 kg
6.673
10
N
m
/kg
0.100 kg
2.32
10
0.400 m
0.600 m
x
F
−
−
⎡
⎤
=
×
⋅
−
+
= −
×
Ν
⎢
⎥
⎢
⎥
⎣
⎦
, with the
minus sign indicating a net force to the left.
(b)
No, the force found in part (a) is the
net
force due to the other two spheres.
E
VALUATE
:
The force from the 5.00 kg sphere is greater than for the 10.0 kg sphere even though its mass is less,
because
r
is smaller for this mass.
12.7.
I
DENTIFY
:
The force exerted by the moon is the gravitational force,
M
g
2
Gm m
F
r
=
. The force exerted on the
person by the earth is
w
mg
=
.
S
ET
U
P
:
The mass of the moon is
22
M
7.35
10
kg
m
=
×
.
11
2
2
6.67
10
N m /kg
G
−
=
×
⋅
.
E
XECUTE
:
(a)
22
11
2
2
3
moon
g
8
2
(7.35
10
kg)(70 kg)
(6.67
10
N
m
/kg
)
2.4
10
N
(3.78
10
m)
F
F
−
−
×
=
=
×
⋅
=
×
×
.
(b)
2
earth
(70 kg)(9.80 m/s
)
690 N
F
w
=
=
=
.
6
moon
earth
/
3.5
10
F
F
−
=
×
.
E
VALUATE
:
The force exerted by the earth is much greater than the force exerted by the moon.
The mass of the
moon is less than the mass of the earth and the center of the earth is much closer to the person than is the center of
the moon.
12.8.
I
DENTIFY
:
Use Eq.(12.2) to find the force each point mass exerts on the particle, find the net force, and use
Newton°s second law to calculate the acceleration.
S
ET
U
P
:
Each force is attractive.
The particle (mass
m
) is a distance
1
0.200 m
r
=
from
1
8.00 kg
m
=
and
therefore a distance
2
0.300 m
r
=
from
2
15.0 kg
m
=
.
Let +
x
be toward the 15.0 kg mass.
E
XECUTE
:
11
2
2
8
1
1
2
2
1
(8.00 kg)
(6.67
10
N
m
/kg
)
(1.334
10
N/kg)
(0.200 m)
Gm m
m
F
m
r
−
−
=
=
×
⋅
=
×
, in the
x
−
direction.
11
2
2
8
2
2
2
2
2
(15.0 kg)
(6.67
10
N
m
/kg
)
(1.112
10
N/kg)
(0.300 m)
Gm m
m
F
m
r
−
−
=
=
×
⋅
=
×
, in the
x
+
direction.
The net force is
8
8
9
1
2
(
1.334
10
N/kg
1.112
10
N/kg)
(
2.2
10
N/kg)
x
x
x
F
F
F
m
m
−
−
−
=
+
= −
×
+
×
= −
×
.
9
2
2.2
10
m/s
x
x
F
a
m
−
=
= −
×
.
The
acceleration is
9
2
2.2
10
m/s
−
×
, toward the 8.00 kg mass.
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 Spring '06
 Buchler
 Physics, Force, Mass, Gravitational forces, Gravitational constant

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