Environmental Engineering Reference
In-Depth Information
Eq. (2.15). One newton of force (N) causes a mass of one kilogram to accelerate
at one m s
−
2
,i.e.1N
kg m s
−
2
.
The attentive reader may well have noticed that we have introduced two different
types of mass. There is the mass
m
that appeared in our definition of the force due
to gravity and then there is the mass
m
that is the constant of proportionality in
Newton's Second Law. A priori they are two different quantities and we were quite
right to keep them distinct. But careful experiments reveal something remarkable:
all bodies fall with the same acceleration in the vicinity of the surface of the Earth.
We shall use the symbol
g
to denote that special acceleration. Now we know the
gravitational force acting on any body close to the Earth (Eq. (2.5)) and that can
be inserted into Newton's Second Law to give
≡
m
g
=
mg.
(2.17)
The fact that
g
is a constant leads (since
g
is also a constant) to the conclusion
that
m
∝
m
. Since we haven't yet defined the scale for gravitational masses we
are perfectly at liberty to fix the constant of proportionality to unity (i.e. to choose
g
=
g
) and henceforth
m
=
m
and we need not distinguish between the two
different types of mass, although we ought to be impressed that Nature has arranged
for their equivalence. This all may seem like pedantry but it is not. Einstein took
very seriously the equivalence of gravitational and inertial mass and it played a
crucial role in his development of the General Theory of Relativity. The General
Theory is our modern theory of gravity and we shall introduce it in Section 14.2.
It is characterized by the fact that it offers an explanation for the equivalence
of inertial and gravitational mass - in fact gravitational mass never appears in
Einstein's theory.
So far we have only been talking about the effect of gravity on objects close to
the surface of the Earth using
F
mg
. Actually, this is a special case of the more
general result discovered by Newton, building on the earlier studies of Johannes
Kepler following observations of the planets within our Solar System. The more
general result states that the gravitational force acts between any two massive
bodies according to
=
G
Mm
r
2
F
=−
e
r
,
(2.18)
where
M
and
m
are the two masses,
r
is the separation of their centres and
e
r
is
a unit vector pointing from the centre
3
of the mass
M
to the centre of the mass
m
.
G
is a constant of proportionality to be fixed by the data (it is usually called
Newton's gravitational constant). The force
F
is then the force acting upon mass
m
(the force on the mass
M
is equal in magnitude but opposite in direction). We have
taken care to completely specify the force, making an appropriate use of vectors,
but the maths should not obscure the simple fact that this is an inverse square law
of attraction (i.e. the forces act to pull the bodies towards each other), which grows
3
Gravitation will be studied in much more detail in Chapter 9 but for now it suffices to consider only
the gravitational forces between spherical bodies.
