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Re: Help with bob!



Tom Mcarthy asked:
There is a problem in Halliday, Resnick, and Walker that asks for the
deviation from vertical that a plumb bob experiences when it is hung at a
latitude of 40 degrees.

The correct answer to this problem is *zero*. The vertical direction at
any place is *defined* operationally as the direction that a plumb bob hangs.
This direction is also perpendicular to the horizontal which is defined as
the plane parallel to the surface of a free standing liquid such as a lake or
a puddle, and is quite close to the average orientation of the landscape
after averaging over the local inhomogeneities such as hills, valleys, etc.
On a large enough length scale the earth is shaped so that the local vertical
direction is perpendicular to the surface. The local direction of vertical
is parallel to the local effective gravitational field including both a
Newtonian gravitational contribution and a centrifugal contribution. To a
good approximation this means that the Earth's shape is that of an oblate
spheroid (ellipsiod of revolution) with an oblate eccentricity e = 0.0820.

I suspect that what Halliday, Resnick, and Walker wanted to ask was: 'What
is the deviation angle between the local vertical direction and the
direction pointing to the Earth's centroid?'. These two directions form
two different measuring schemes for measuring the latitude at a given
place. The angle between the local vertical and the vertical direction at
the equator at the same longitude measures the 'geographic' latitude. It
is also (to a *very* good approximation) the latitude measured by observing
the rotation of the celestial sphere, and is also sometimes called the
'astronomical' latitude. The angle between the local direction to the
Earth's centroid and the corresponding direction at the equator at the same
longitude is called the 'geodetic' latitude. It should be pointed out
that the Newtonian gravitational contribution to the local effective
gravitational field direction does *not* in general point to the Earth's
centroid. This is simply because the Earth does not have a spherically
symmetric distribution of mass.

It is a relatively simple exercise in trigonometry and analytic geometry
to relate the local geographic latitude to the local geodetic latitude
once it is assumed that the Earth's shape is spheroidal. If e (= 0.0820) is
the eccentricity we then have: tan(L') = (1 - e*e)*tan(L) where L is the
geographic latitude and L' is the geodetic latitude. Notice that the
geographic latitude is larger (in absolute value) than the geodetic
latitude. A formula for L - L' which avoids the subtraction roundoff
error of subtracting two nearly identical quantities is:

L - L' = arctan(e*e*tan(L)/(1 + (1 - e*e)*(tan(L))^2)) .

At L = 40 deg we have L - L' = 0.190 deg assuming that e = 0.0820.

The difference L - L' is a maximum at a geographic latitude
L = arctan(1/sqrt(1 - e*e)) and this has a corresponding geodetic latitude
L' = arctan(sqrt(1 - e*e)) = 90 deg - L. (This makes L' slightly less than
45 deg and L slightly greater than 45 deg.) For e = 0.0820 this maximum
difference is 0.193 deg.

David Bowman
dbowman@gtc.georgetown.ky.us