Geography Reference
In-Depth Information
2.2
THE VECTORIAL FORM OF THE MOMENTUM EQUATION IN
ROTATING COORDINATES
In an inertial reference frame, Newton's second law of motion may be written
symbolically as
F
D
a
U
a
Dt
=
(2.3)
The left-hand side represents the rate of change of the absolute velocity
U
a
, fol-
lowing the motion as viewed in an inertial system. The right-hand side represents
the sum of the real forces acting per unit mass. In Section 1.5 we found through
simple physical reasoning that when the motion is viewed in a rotating coordinate
system certain additional apparent forces must be included if Newton's second
law is to be valid. The same result may be obtained by a formal transformation of
coordinates in (2.3).
In order to transform this expression to rotating coordinates, we must first find a
relationship between
U
a
and the velocity relative to the rotating system, which we
will designate by
U
. This relationship is obtained by applying (2.2) to the position
vector
r
for an air parcel on the rotating earth:
D
a
r
Dt
=
D
r
Dt
+
×
r
(2.4)
but D
a
r
/
Dt
≡
U
a
and D
r
/
Dt
≡
U
; therefore (2.4) may be written as
U
a
=
U
+
×
r
(2.5)
which states simply that the absolute velocity of an object on the rotating earth is
equal to its velocity relative to the earth plus the velocity due to the rotation of the
earth.
Now we apply (2.2) to the velocity vector
U
a
and obtain
D
a
U
a
Dt
D
U
a
Dt
=
+
×
U
a
(2.6)
Substituting from (2.5) into the right-hand side of (2.6) gives
D
a
U
a
Dt
D
Dt
(
U
=
+
×
r
)
+
×
(
U
+
×
r
)
(2.7)
D
U
Dt
+
2
R
=
2
×
U
−
where
is assumed to be constant. Here
R
is a vector perpendicular to the axis of
rotation, with magnitude equal to the distance to the axis of rotation, so that with
the aid of a vector identity,
2
R
×
(
×
r
)
=
×
(
×
R
)
=−
