565
Chapter 13
1. The gravitational force between the two parts is
2
2
2
=
=
Gm M
m
G
F
mM
m
r
r
which we differentiate with respect to
m
and set equal to zero:
2
= 0 =
2
= 2
dF
G
M
m
M
m
dm
r
.
This leads to the result
m
/
M
= 1/2.
2. The gravitational force between you and the moon at its initial position (directly
opposite of Earth from you) is
0
2
(
)
m
ME
E
GM m
F
R
R
where
m
M
is the mass of the moon,
ME
R
is the distance between the moon and the Earth,
and
E
R
is the radius of the Earth. At its final position
(directly above you), the
gravitational force between you and the moon is
1
2
(
)
m
ME
E
GM m
F
R
R
.
(a) The ratio of the moon’s gravitational pulls at the two different positions is
2
2
2
8
6
1
2
8
6
0
/(
)
3.82 10 m 6.37 10 m
1.06898.
/(
)
3.82 10 m 6.37 10 m
m
ME
E
ME
E
m
ME
E
ME
E
GM m R
R
F
R
R
F
GM m R
R
R
R
Therefore, the increase is 0.06898, or approximately 6.9%.
(b) The change of the gravitational pull may be approximated as
1
0
2
2
2
2
3
1 2
1 2
(
)
(
)
4
.
m
m
m
m
E
E
ME
E
ME
E
ME
ME
ME
ME
m
E
ME
GM m
GM m
GM m
GM m
R
R
F
F
R
R
R
R
R
R
R
R
GM mR
R

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*Sign up*CHAPTER 13
566
On the other hand, your weight, as measured on a scale on Earth, is
2
E
g
E
E
GM m
F
mg
R
.
Since the moon pulls you “up,” the percentage decrease of weight is
3
3
22
6
7
5
1
0
24
8
7.36 10
kg
6.37 10 m
4
4
2.27 10
(2.3 10 )%.
5.98 10
kg
3.82 10 m
m
E
g
E
ME
F
F
M
R
F
M
R
3. The magnitude of the force of one particle on the other is given by
F
=
Gm
1
m
2
/
r
2
,
where
m
1
and
m
2
are the masses,
r
is their separation, and
G
is the universal gravitational
constant. We solve for
r
:
11
2
2
1
2
12
6.67 10
N m / kg
5.2kg
2.4kg
19m
2.3 10
N
Gm m
r
F
.
4. We use subscripts
s
,
e
, and
m
for the Sun, Earth and Moon, respectively. Plugging in
the numerical values (say, from Appendix C) we find
2
2
2
30
8
2
24
11
/
1.99 10
kg 3.82 10 m
2.16.
/
5.98 10
kg 1.50 10 m
sm
s
m
sm
s
em
em
e
m
em
e
sm
F
Gm m
r
m
r
F
Gm m
r
m
r
5. The gravitational force from Earth on you (with mass
m
) is
2
E
g
E
GM m
F
mg
R
where
2
2
/
9.8 m/s .
E
E
g
GM
R
If
r
is the distance between you and a tiny black hole of
mass
11
1 10 kg
b
M
that has the same gravitational pull on you as the Earth, then
2
.
b

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