Environmental Engineering Reference
In-Depth Information
precisely understand the elliptical orbits of planets within our solar system and the
hyperbolic trajectories of comets.
Before launching into Newton's theory of gravity, it is perhaps worth recapping
that, following Einstein's work on the General Theory of Relativity, we now under-
stand that Newton's theory is only an approximation to Einstein's more accurate
theory. Nevertheless, it is an excellent approximation in almost all circumstances in
everyday life. For example, NASA's Apollo missions to the Moon were conducted
entirely using calculations based upon Newton's theory. That said, there is one
area of life where Newton's theory of gravity is inadequate. The GPS system uses
a network of satellites orbiting the Earth every 12 hours (or so). Accurate position
measurements require very accurate time keeping on the orbiting satellites and as
a result it is necessary to account not only for Special Relativistic corrections due
to the motion of the satellites but also the General Relativistic corrections which
correct Newton's theory of gravity. Without these corrections, the GPS system
would fail within minutes. At the end of this topic we shall discuss how Einstein's
General Relativity comes about and illustrate how it corrects Newton's theory, but
for now we satisfy ourselves with a detailed account of Newtonian gravity.
9.1 NEWTON'S LAW OF GRAVITY
Let us start by writing down Newton's Law of Gravity. Illustrated in Figure 9.1
are two point masses,
m
and
M
, separated by a distance,
r
. Newton's Law tells us
that a force
F
acts on the mass
m
such that
GMm
r
2
F
=−
e
r
,
(9.1)
where
e
r
is a unit vector pointing from the mass
M
towards the mass
m
,as
illustrated. Moreover, a force of equal magnitude but opposite direction also acts
upon the mass
M
. In short, the two masses attract each other with a strength
described by an inverse square law. Although we have taken care to specify the
law for point masses (i.e. idealized pointlike masses) we shall show in the next
section that the law also applies to extended spherical bodies provided that
r
is the
distance between the centres of the two masses.
As we shall very soon discover, the fact that the gravitational force acts along
the line joining the two bodies and depends only on the distance between them
m
F
M
e
r
Figure 9.1
The gravitational force on a mass
m
due to a second mass,
M
.
