Environmental Engineering Reference
In-Depth Information
and hence
g
=−
∇
.
(9.17)
Being a scalar function, the gravitational potential is usually much easier to
deal with than the field strength, which is a vector quantity. Moreover, all of
the information we need is contained in the potential for all we need to do is
differentiate it in order the compute the field strength. Thus, the conservative nature
of the gravitational force has led to a big calculational simplification. Our general
task will therefore be to compute the gravitational potential for whichever problem
we are faced with. Given this we can compute the other interesting quantities.
Let us next show how to compute the gravitational potential for a general dis-
tribution of mass which is described by a density
ρ(
x
)
(this is the mass per unit
volume at position
x
). Figure 9.2 illustrates the general situation and our goal is
to determine the potential at a general point P. To do this we must compute and
sum up the potential at P arising from each and every tiny element of mass, like
the one illustrated in the figure. The potential at P arising from a volume element
d
V
located at position
x
which has a mass equal to
ρ(
x
)
d
V
is given by
Gρ (
x
)
d
V
r
d
(
x
)
=−
.
(9.18)
Hence the potential at P arising from a general distribution of mass is the sum over
all such volume elements, i.e. it is the triple integral
G
ρ(
x
)
r
(
x
)
=−
d
V.
(9.19)
V
P
r
d
V
x
′
x
O
Figure 9.2 To compute the potential at P due to a general distribution of matter we sum
over the infinity of volume elements d
V
.
