Introduction - An Introduction to Dynamic Meteorology - page 12

However, | V |= ωr and Dθ/Dt = ω, so that

D V
Dt =−

ω 2 r (1.6)

Therefore, viewed from fixed coordinates the motion is one of uniform accel-
eration directed toward the axis of rotation and equal to the square of the angular
velocity times the distance from the axis of rotation. This acceleration is called
centripetal acceleration . It is caused by the force of the string pulling the ball.
Now suppose that we observe the motion in a coordinate system rotating with
the ball. In this rotating system the ball is stationary, but there is still a force acting
on the ball, namely the pull of the string. Therefore, in order to apply Newton's
second law to describe the motion relative to this rotating coordinate system,
we must include an additional apparent force, the centrifugal force , which just
balances the force of the string on the ball. Thus, the centrifugal force is equivalent
to the inertial reaction of the ball on the string and just equal and opposite to the
centripetal acceleration.
To summarize, observed from a fixed system the rotating ball undergoes a
uniform centripetal acceleration in response to the force exerted by the string.
Observed from a system rotating along with it, the ball is stationary and the force
exerted by the string is balanced by a centrifugal force.

1.5.2 Gravity Force

An object at rest on the surface of the earth is not at rest or in uniform motion relative
to an inertial reference frame except at the poles. Rather, an object of unit mass
at rest on the surface of the earth is subject to a centripetal acceleration directed
toward the axis of rotation of the earth given by

2 R , where R is the position −

10 − 5 rad s − 1 is the
angular speed of rotation of the earth. 3 Since except at the equator and poles the
centripetal acceleration has a component directed poleward along the horizontal
surface of the earth (i.e., along a surface of constant geopotential ), there must be a
net horizontal force directed poleward along the horizontal to sustain the horizontal
component of the centripetal acceleration. This force arises because the rotating
earth is not a sphere, but has assumed the shape of an oblate spheroid in which there
is a poleward component of gravitation along a constant geopotential surface just
sufficient to account for the poleward component of the centripetal acceleration
at each latitude for an object at rest on the surface of the earth. In other words,
from the point of view of an observer in an inertial reference frame, geopotential

vector from the axis of rotation to the object and = 7.292 ×

3 The earth revolves around its axis once every sidereal day, which is equal to 23 h 56 min 4 s
(86,164 s). Thus, = 2π/(86, 164 s) = 7.292 × 10 − 5 rad s − 1 .