Geography Reference
In-Depth Information
a coordinate system fixed in space will remain in uniform motion in the absence of
any forces. Such motion is referred to as
inertial motion
; and the fixed reference
frame is an inertial, or absolute, frame of reference. It is clear, however, that an
object at rest or in uniform motion with respect to the rotating earth is not at rest or
in uniform motion relative to a coordinate system fixed in space. Therefore, motion
that appears to be inertial motion to an observer in a geocentric reference frame
is really accelerated motion. Hence, a geocentric reference frame is a
noninertial
reference frame. Newton's laws of motion can only be applied in such a frame if the
acceleration of the coordinates is taken into account. The most satisfactory way of
including the effects of coordinate acceleration is to introduce “apparent” forces
in the statement of Newton's second law. These apparent forces are the inertial
reaction terms that arise because of the coordinate acceleration. For a coordinate
system in uniform rotation, two such apparent forces are required: the centrifugal
force and the Coriolis force.
1.5.1
Centripetal Acceleration and Centrifugal Force
A ball of mass
m
is attached to a string and whirled through a circle of radius
r
at a
constant angular velocity ω. From the point of view of an observer in inertial space
the speed of the ball is constant, but its direction of travel is continuously changing
so that its velocity is not constant. To compute the acceleration we consider the
change in velocity δ
V
that occurs for a time increment δt during which the ball
rotates through an angle δθ as shown in Fig. 1.5. Because δθ is also the angle
between the vectors
V
and
V
+
δ
V
, the magnitude of δ
V
is just
|
δ
V
|=|
V
|
δθ.If
we divide by δt and note that in the limit δt
→
0,δ
V
is directed toward the axis
of rotation, we obtain
D
V
Dt
=|
Dθ
Dt
r
r
V
|
−
Fig. 1.5
Centripetal acceleration is given by the rate of change of the direction of the velocity vector,
which is directed toward the axis of rotation, as illustrated here by δ
V
.
