Physics 325
Lecture 9
Gravitational Force and Potential (cont.)
There are many similarities here between gravitation forces and electric forces.
In fact,
the only difference here is that the gravitational force is always attractive.
Otherwise we
just replace the masses by electric charges, and
G
by the appropriate electromagnetic
constant and we’re done.
In what follows, we carry
this analogy a bit further.
Given the vector
g
defined by Equation (8.4), we can define a gravitational
flux
due to a
mass
m
through an arbitrary surface
S
ˆ
m
S
S
g da
g nda
(9.1)
where
ˆ
n
is a unit vector normal to the surface
S
at
da
.
If the surface element
da
is a distance
r
away from the mass
m
(and outside of
m
), we can
write (9.1) as
2
cos
m
S
Gm
da
r
where
is the angle between
ˆ
and
g
n
.
For sim
plicity let’s take the surface to be a
sphere centered at
m
.
Then
0
and the integral just gives
4
.
In fact the integral will
give
4
regardless of the shape of the surface since the integral is over the solid angle of
the surface.
Nor does it matter where inside the surface the mass
m
is located.
The
gravitational flux due to mass
m
is then
4
m
Gm
(9.2)
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Going back to Equation (9.1)
, we can use Gauss’s theorem to write
ˆ
m
S
V
g nda
g dV
Therefore
4
4
g dV
Gdm
G
dV
which gives
4
g
G
Since
g
, we find
2
4
G
(9.3)
This is Poisson’s Equation.
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 Spring '08
 Staff
 mechanics, Force, Mass, General Relativity, Fundamental physics concepts, Central force field

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