Geoscience Reference
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Fig. 9.2 A rotational system
of the Earth and the Moon
bound by the gravitational
force
F
f 2
F
A 2
O 2
F
x
r 2
F
f 2
f 1
CMI
x
x
O 2
A 2
r 1
O 1
x
A 1
Moon
Earth
inertia of the system is symmetric, similar to the center of mass in linear motion. It
can be represented mathematically as, for a system consisting of n bodies,
X
n
m i r i i D 0;
(9.1)
iD1
where i is a unit vector directing a body from CMI, r i is a radial distance from CMI
to the body, and m i is a mass of the body. The rotations of the Earth and the Moon
can be derived from the same fundamental physics, torque and reaction torque, as
in Fig. 9.1d .
Figure 9.2 shows a rotating system of the Earth and the Moon. The gravitational
force of the Earth acting on the Moon is highest at the point A 2 which is the nearest
point of the Moon to the Earth. The radial distance from A 2 to O 2 is largest due
to the different gravitational force. If the Moon does not spin, as the Moon moves
along its orbit, the point A 2 moves away and the new point comes in to the nearest
point. The highest point is not exactly on the line connecting two centers of the Earth
and the Moon, but it is shifted as shown in Fig. 9.2 . Here, it is assumed that Moon's
radial response to the tidal force is retarded by a certain time. The gravitational force
on the shifted highest point induces a force f 2 , which tends to make the spin of the
Moon synchronous to the period of orbital motion.
Since the force f 2 makes a torque with respect to CMI, there should be a
reaction torque on the Earth, r 1 0 f 1 D r 2 0 f 2 , as shown in Fig. 9.2 ,where r 1 0
and r 2 0 are radial vectors from CMI to the points A 1 and A 2 . The force f 1 on
the Earth is magnified by the ratio of distances r 2 0 / r 1 0 . As the force f 2 makes
the spin of the Moon synchronous to the period of orbital motion, the force f 1
will make the Earth rotate by about ten times stronger force than the force f 2 .
As soon as the Moon rotates synchronously to the period of orbital motion, the
forces f 1 and f 2 decrease to zero, and the rotation of the Earth will reach to
the speed accelerated by the torque and the reaction torque. To conserve total
angular momentum before and after torque and reaction torque were acted, orbital
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