A unit mass at rest relative to the earth's surface
experiences not only the attractive force 
but is also
subject to the centrifugal force associated with the
earth's rotation. It is this net force which is actually
measured in pendulum experiments.
Measurements of gravity at different latitudes on the earth's
surface reveal a slight increase from the equator to the
poles. The mean values of gravity at sea level for
different latitudes are given in Table 2.02-I. At least part
of this variation is attributed to the variation in the
centrifugal force, which is a maximum at the equator and
zero at the poles.
Let 
represent the gravity vector at given latitude
and let 
represent the centrifugal
force experienced by a unit mass which is fixed on the
earth's surface at the latitude , then
The gravitational attraction vector 
for spherical
earth is directed towards the exact center of the earth, while
the force is directed outwards and normal
to the axis of the earth. These forces are shown in Fig. 2.02-1.
The direction of the resultant of 
and
is that assumed by a plumb line at the
latitude .
The magnitude of , if we again regard the earth as
a sphere, can be expressed by
where 
is the angular speed of the earth expressed in
radians per unit time.
The value of is given in Table 2.02-II. It should be
noted that the rotation relative to the stars is 366/355 revolutions
per day, there being one extract revolution peryear of which we are
not aware if we use the sun as a reference.
Referring to Fig. 2.02-2, we note that
and
The maximum value of 
according to Eq. (17) however is only
which is that at the equator. This is only about 0.3 of one per cent of
and, consequently, Eq.s (18) and (19) can be written as
with no sensible loss of accuracy. According to Eq. (22), the maximum
angular deviation of the plumb line from a radial line to the earth's
center occurs at latitude 
(N or S) and has the
magnitude
or 5.9 minutes of arc. The actual deviation is somewhat different
from this because of the influence of the nonspherical shape of
the earth in the direction of at intermediate latitudes.
The direction of the plumb line at the equator and poles is
directed towards the exact center of the earth.
At latitude (N or S) the value of g is, according
to Eq. (21),
Elimination of between Eqs. (21) and (24) leads to the
relation
Taking 
as the observed mean value of 980.6, this would
indicate that g has the values 978.9 and 982.3 cm/sec 
at
the equator and poles, respectively. The indicated range of
3.4 cm/sec falls short of the actual observed range
of 5.2 cm/sec (Table 2.02-I). The discrepancy must be
attributed to the ellipsoidal shape of the earth.
The effect of the greater radius at the equator than that at the
poles i to bring about an increase in from the equator to
the poles. The variation can be expressed by a relation
similar to Eq. (25).
The final variation of g with latiatude and elevation can be expressed by the relation
where z is elevation above sea level and 
is the
value of g at latitude and sea level. The value
of 
which leads to a fit of the measured values at sea
level is 0.00259.