Classical Physics to Calculate Rotation Periods of Planets and the Sun - Planetary Exploration and Science: Recent Results and Advances - page 259

9.6 Experiment to Measure the Reaction Torque

As stated in Fig. 9.2 , the force f 1 in the reaction torque is a key to the Earth spin,
which is about ten times larger than the force f 2 on the Moon. Since the magnitude
and the direction of the force in the reaction torque are dependent on the position
along the radius and the latitude of the Earth, it is expected that the force will be
an important factor for a lot of Earth activities, flow of seawater, tides, air flows,
volcanoes, earthquakes, etc. The possibility to measure the force is discussed in
Fig. 9.12 . Since the force is largest at the point A 1 , a pendulum in a chamber as
shown in the magnification in Fig. 9.12 is located at the point A 1 , where the chamber
is just for blocking air flow to prevent any shift of pendulum by air.
If the Earth surface at the chamber is not accelerated by the force f 1 , then the
pendulum will move by a certain degree from the perpendicular direction to balance
with the sine component of gravitation. An accurate measurement of the pendulum
displacement induced by the libration of the Moon will provide information on the
reaction torque. Magnitude and direction of the force in the reaction torque might
be decided by measuring the displacement of pendulum, the position of the Moon
relative to the Earth, and eccentricity of the Moon.
For the libration of the Moon facing 58 % of its surface to the Earth, the largest
possible shift of pendulum at the equator is calculated to be 0.3043 nm with 1 m
pendulum which can be magnified to 152.2 m by a 5 km laser beam as a 5 m
pendulum and a 5 cm mirror are used as shown in Fig. 9.13 .FromEq. 9.3 and
Fig. 9.13a ,

r 1 m 1 a 1 D r 2 m 2 i 2 2Gm 1 R m
d 3

sin ı ı ;

(9.32)

where i 2 is a unit vector directing a 2 and the quantity in the parentheses is a
magnitude of a 2 by the Earth's tidal force on the surface of the Moon multiplied by
sin ı from 4 % (7 ı ) shift of the highest point with assumption that the highest point
moves by the same angle of 7 ı to calculate the largest possible shift of pendulum.

Earth
spin

f 1

f 1

A 2

O 1

O 2

A 1

A 1

f 2

Moon

Earth Fig. 9.12 A possible

experiment to measure the
reaction torque