Geoscience Reference
In-Depth Information
with components
G
v
x − ξ
l
3
g
x
=
∂W
∂x
dv
+
ω
2
x,
=
−
G
v
y
g
y
=
∂W
∂y
−
η
dv
+
ω
2
y,
=
−
(2-11)
l
3
G
v
z
g
z
=
∂W
∂z
−
ζ
−
dv,
=
l
3
is called the
gravity vector
; it is the total force (gravitational force plus
centrifugal force) acting on a unit mass. As a vector, it has
magnitude
and
direction
.
The magnitude
g
is called gravity in the narrower sense. It has the phys-
ical dimension of an acceleration and is measured in gal (1 gal = 1 cm s
−
2
),
the unit being named in honor of Galileo Galilei. The numerical value of
g
is about 978 gal at the equator, and 983 gal at the poles. In geodesy, another
unit is often convenient - the milligal, abbreviated mgal (1 mgal = 10
−
3
gal).
In SI units, we have
1 gal = 0
.
01 m s
−
2
,
1 mgal = 10
µ
ms
−
2
.
(2-12)
The direction of the gravity vector is the direction of the
plumb line
,orthe
vertical; its basic significance for geodetic and astronomical measurements
is well known.
In addition to the centrifugal force, another force called the
Coriolis force
acts on a moving body. It is proportional to the velocity with respect to the
earth, so that it is zero for bodies resting on the earth. Since in classical
geodesy (i.e., not considering navigation) we usually deal with instruments
at rest relative to the earth, the Coriolis force plays no role here and need
not be considered.
Gravitational and inertial mass
The reader may have noticed that the mass
m
has been used in two con-
ceptually completely different senses: as
inertial mass
in Newton's law of
inertia,
force
=
mass
acceleration
and as
gravitational mass
in Newton's
law of gravitation (1-1). Thus,
m
in gravitation, which is a “true” force, is
the gravitational mass, but
m
in the centrifugal “force”, which is an accel-
eration, is the inertial mass. The Hungarian physicist Roland Eotvos had
shown experimentally already around 1890 that both kinds of masses are
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