1
Chapter 13
Gravitation
¾
Newton’s law of gravitation
¾
The acceleration of gravity on the surface of the earth, above it and
below it.
¾
Gravitational potential energy
¾
Kepler’s three laws of planetary motion
¾
Satellites (orbits, energy, escape velocity)
2
m
1
m
2
The tendency of objects with mass to move towards each other
Newton's Law of
is
known as gr
Gravit
avitati
ation
on.
12
has the following characteristics:
1.
The force acts along the line that connects the two particles.
2.
Its magnitude is given b
Newton's la
y the equat
w of gra
ion:
vitation
mm
FG
r
=
2
11
2
2
Here
and
are the masses of the two particles,
is the
separation between them and
is the gravitati
.
6.67
onal
10
constant.
N.m /
g
r
k
G
G
−
=×
2
ˆ
G
r
=
Fr
G
3
m
1
m
2
F
12
F
21
r
12
1
2
21
2
1
The two forces in a gravitational pair obey Newton's third law.
The gravitational
force
exerted on
by
is equal in magnitude and opposite in direction to the
force
exerted on
by
.
F
F
G
G
G
12
21
0
+=
FF
G
2
ˆ
G
r
=
G
4
The net gravitational force exerted by a group of particles is equal to the vecto
Gravita
r
sum o
tion and the Principle of S
f the contribution from eac
uperpos
h parti
ition
cle.
11
2
3
12
13
1
2 1
3
23
1
For example the net force
exerted on
by
and
is equal to:
Here
and
are the forces exerted on
by
and
, respectively.
In general the
force exerted on
mm m
m
m
=
+
F
F
G
GG
G
21
31
4
1
1
2
by n particles is given by the equation:
...
n
ni
i
=
=+++
∑
FF F F
F
F
GG G G
G
G
2
n
i
i
=
=
∑
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m
1
dm
r
d
F
G
1
The gravitation force exerted by a continuous extended object on a particle of mass
can be calculated using the principle of superposition.
The object is divided into
elements of mass
and the n
m
dm
1
1
1
1
2
et force on
is the vector sum of the forces
exerted by each element.
The sum takes the form of an integral:
Here
is the force exerted on
by
m
dm
d
m
Gm
d
dF
dm
r
=
=
∫
F
FF
G
G
G
1
d
=
∫
GG
6
12
1
2
Newton proved that a uniform shell attarcts a particle that is outside the shell as if
the mass of the shell were concetrated at the center of the shell:
mm
FG
r
=
m
1
m
2
r
F
1
m
2
m
1
If the particle is inside the shell, the net force is z
N
e
ote:
ro.
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 Spring '10
 Cohen
 Physics, Acceleration, Energy, Force, Gravity, Mass, Potential Energy, Kepler's laws of planetary motion

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