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angular momentum must decrease by increase of total spin angular momentum,
L
D
(
S
1
C
S
2
),
J
D
L
C
S
1
C
S
2
D
0.
The Moon will make the same kind of torque on the Earth and reaction torque
on the Moon. Since the tidal force of the Moon is much smaller, roughly, by the
mass ratio of the Earth and Moon, than the tidal force of the Earth, the torque and
reaction torque acted by the Moon are much smaller than those acted by the Earth
in Fig.
9.2
, and these torques will be neglected here. If the Earth rotates faster than
the period of orbital motion, the torque and the reaction torque acted by the Moon
are opposite in directions to the torques in Fig.
9.2
and may retard the speed of the
Earth's rotation slowly.
Torques and reaction torques acting on the points A
1
and A
2
with respect to the
CMI are
r
0
1
f
1
D
r
0
2
f
2
(9.2)
r
0
1
m
1
a
1
D
r
0
2
m
2
a
2
;
(9.3)
where the masses,
m
1
and
m
2
, are the mass increments in the points, A
1
and A
2
,
which have a relation
m
1
/
m
2
D
m
1
/
m
2
,where
m
1
and
m
2
are masses of the Earth
and the Moon, respectively. The forces,
f
1
and
f
2
, induce tangential velocities
v
1
and
v
2
in the points, as shown in Fig.
9.3
, which are transformed into spin velocities of
the Earth and the Moon, where the tangential velocities
v
1
and
v
2
are the relative
velocities of points A
1
and A
2
with respect to O
1
and O
2
, respectively. By integrating
Eq.
9.3
from initial velocities
v
10
,
v
20
to final velocities
v
1
,
v
2
, the velocities have a
relation:
r
0
1
m
1
.
v
1
v
10
/
D
r
0
2
m
2
.
v
2
v
20
/
(9.4)
It is assumed that initial velocities
v
10
,
v
20
which had not been driven by the forces
f
1
and
f
2
will decrease to zero by tidal friction or are negligibly small because of a
v
1
r
2
′
r
1
′
ʔ˕
A
2
O
1
O
2
x
A
1
CMI
v
2
ʔ˕
m
Earth
Moon
Fig. 9.3
Angles ' and
'
m
defined by the relative
velocity
v
2
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