Environmental Engineering Reference
In-Depth Information
Thus
GMm
r
2
x
i
r
.
F
i
=−
(9.13)
Since
r
=
x
i
e
i
we can write this equation as
GMm
r
2
r
r
,
F
=−
(9.14)
which is none other than Newton's Law since
r
/r
e
r
. Eq. (9.10) therefore
provides us with a potential energy function which we can use to compute the
work done on our particle of mass
m
as it moves in the field of the mass
M
.
The existence of this function implies that the gravitational force is conser-
vative.
In choosing Eq. (9.10) to define the gravitational potential energy of our mass
m
we have also specified that the zero of potential energy is at
r
=
=∞
. This choice
has a nice physical interpretation. Let us consider the case where the mass
m
starts
from rest infinitely far away from the mass
M
, then the conservation of energy
tells us that
1
2
mv
2
.
U(
∞
)
+
0
=
U(r)
+
(9.15)
The zero on the left hand side expresses the fact that the particle starts out at rest.
Thus we see that
U(r)
is equal to the kinetic energy that a particle of mass
m
would have after falling to a distance
r
from the mass
M
starting from an initial
speed of zero at infinity. Equivalently,
−
−
U(r)
is the work done by gravity on the
particle as it falls from infinity to
r
.
9.2 THE GRAVITATIONAL POTENTIAL
So far we have talked about the force due to gravity acting on and the gravita-
tional potential energy of a particle of mass
m
due to the field associated with a
mass
M
. There is nothing wrong with such an approach however the idea that the
mass
M
produces a gravitational field independent of whether there is another mass
present or not is a very intuitive one and one that is conveniently expressed when
we think in terms of the gravitational field strength and the gravitational potential,
which are defined as follows.
The gravitational field strength,
g
, is simply defined to be the force which would
act on a unit mass placed in the field whilst the gravitational potential,
,isthe
gravitational potential energy of a unit mass placed in the field, i.e.
F
m
,
g
≡
U
m
≡
(9.16)
