17.5 Angular Velocity Vector on Earth
So how do you find the angular velocity vector? One way is to use the formula shown
above. However, in important cases, this angular velocity vector can be determined from
simple geometric reasoning. An obvious example concerns motion on the surface of the
earth. Imagine you have a coordinate system fixed with the earth. Then it is
actually rotating through space because the earth is turning. However, to an
observer on the surface of the earth, these vectors are not moving and this observer
wants to understand motion in terms of these apparently fixed vectors. This is a
very interesting problem which can be understood relative to what was just
discussed.
Imagine a point on the surface of the earth which is not moving relative to the earth.
Now consider unit vectors, one pointing South, one pointing East and one pointing
directly away from the center of the earth.
Denote the first as i
, the second as
j, and the third as
k. If you are
standing on the earth you will consider these vectors as fixed, but of course
they are not. As the earth turns, they change direction and so each is in reality
a function of
t. What is the description of the angular velocity vector in this
situation?
Let i^{∗},j^{∗},k^{∗}, be the usual basis vectors fixed in space with k^{∗} pointing in the
direction of the north pole from the center of the earth and let i
,j,k be the unit
vectors described earlier with
i pointing South,
j pointing East, and
k pointing
away from the center of the earth at some point of the rotating earth’s surface
p.
(This means that the components of
p are constant with respect to the vectors fixed
with the earth. ) Letting
R be the position vector of the point
p, from the center
of the earth, observe that this is a typical vector having coordinates constant with respect
to
i,j,k. Also, since the earth rotates from West to East and the speed
of a point on the surface of the earth relative to an observer fixed in space is
ωsin
ϕ where
ω is the angular speed of the earth about an axis through the
poles and
ϕ is the polar angle measured from the positive
z axis down as in
spherical coordinates. It follows from the geometric definition of the cross product
that
Therefore, the vector of Theorem 17.4.2 is Ω
=
ωk^{∗} because it acts like it should for
vectors having components constant with respect to the vectors fixed with the earth. As
mentioned, you could let
θ,ρ,ϕ each be a function of
t and use the formula above along
with the chain rule to verify analytically that the angular velocity vector is what
is claimed above. That is, you would have
θ =
ωt and the other spherical
coordinates constant. See Problem
12 on Page
1090 below for a more analytical
explanation.