Physics 325
Lecture 24
Motion near the Earth’s surface
In this lecture, we describe the motion of particles near the surface of the earth. The
dominant noninertial effect is due to the rotation of the earth about its axis, as the effects
of other motion (e.g. revolution of the earth around the sun and the motion of the solar
system in our galaxy) are comparatively negligible.
Consider the coordinate system shown below used to describe the motion of a particle
near the surface of the earth:
We have a fixed inertial frame
,,
x y z
with origin at the center of the earth and the
moving frame
x y z
on the Earth’s surface. The effective force
eff
F
as measured in the
moving system becomes (recall Equation 23.3)
0
(
) 2
eff
f
r
F
mg
mR
m
r
m
r
m
v
(24.1)
where
0
g
is the Earth’s gravitational field vector given by
0
2
E
R
M
g
G
e
R
(24.2)
where
E
M
is the Earth’s mass and
R
is the Earth’s radius. We choose Earth’s angular
velocity
to lie along the inertia system’s
z
direction and recall that
5
7.3 10
rad/sec which is a pretty slow rotation (why the Earth’s frame is a decent approximation
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View Full Documentof an inertial frame). Since
is very nearly constant in time, we can safely ignore the
mr
term in Equation (24.1).
According to Equation 22.10, we have
()
ff
RR
R
(24.3)
Equation (24.1) then becomes
0
(
)
2
eff
r
F
mg
m
r
R
m
v
(24.4)
The first and second terms in Equation (24.4) are what we experience (and measure) on
the surface of Earth as the effective
g
, and we will henceforth denote it as
g
. Due to the
rotation of the earth, the acceleration due to gravity (i.e. the acceleration of a freely
falling body in the rotating frame of Earth) appears to be modified by the centrifugal
acceleration and the magnitude is reduced, as shown in below
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 Spring '08
 Staff
 mechanics, Inertia, Rotation, Feff, Focault pendulum

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