Linear Algebra, Theory And Applications, 2012a

66 MATRICES AND LINEAR TRANSFORMATIONS
and consequently,
v = R ′ + x ′ i + y ′ j + z ′ k + Ω × r B = R ′ + x ′ i + y ′ j + z ′ k + Ω× (xi + yj + zk) .
Now consider the acceleration. Quantities which are relative to the moving coordinate
system and quantities which are relative to a fixed coordinate system are distinguished by
using the subscript B on those relative to the moving coordinate system.
Ω×v B
{ }} {
a = v ′ = R ′′ + x ′′ i + y ′′ j + z ′′ k+ x ′ i ′ + y ′ j ′ + z ′ k ′ + Ω ′ × r B
⎛
⎞
v B
Ω×r B (t)
{ }} { { }} {
⎜
+Ω× ⎝x ′ i + y ′ j + z ′ k+ xi ′ + yj ′ + zk ′ ⎟
⎠
= R ′′ + a B + Ω ′ × r B +2Ω × v B + Ω× (Ω × r B ) .
The acceleration a B is that perceived by an observer who is moving with the moving coordinate
system and for whom the moving coordinate system is fixed. The term Ω× (Ω × r B )
is called the centripetal acceleration. Solving for a B ,
a B = a − R ′′ − Ω ′ × r B − 2Ω × v B − Ω× (Ω × r B ) . (2.28)
Here the term − (Ω× (Ω × r B )) is called the centrifugal acceleration, it being an acceleration
felt by the observer relative to the moving coordinate system which he regards as fixed, and
the term −2Ω × v B is called the Coriolis acceleration, an acceleration experienced by the
observer as he moves relative to the moving coordinate system. The mass multiplied by the
Coriolis acceleration defines the Coriolis force.
There is a ride found in some amusement parks in which the victims stand next to
a circular wall covered with a carpet or some rough material. Then the whole circular
room begins to revolve faster and faster. At some point, the bottom drops out and the
victims are held in place by friction. The force they feel is called centrifugal force and it
causes centrifugal acceleration. It is not necessary to move relative to coordinates fixed with
the revolving wall in order to feel this force and it is pretty predictable. However, if the
nauseated victim moves relative to the rotating wall, he will feel the effects of the Coriolis
force and this force is really strange. The difference between these forces is that the Coriolis
force is caused by movement relative to the moving coordinate system and the centrifugal
forceisnot.
2.6.2 The Coriolis Acceleration On The Rotating Earth
Now consider the earth. 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. Letting R (t) be
the position vector of the point p, from the center of the earth, observe the coordinates of
R (t) are constant with respect to i (t) , j (t) , k (t) . 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 ω |R| sin 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
R ′ = ωk ∗ × R