The C.G. of such a system is at \({C.G.}_{em}\) = \(\frac{{M}_{m}}{({M}_{e}+{M}_{m})}\).Orbs
\({C.G.}_{em}\) = 7.34767309*10^22/(7.34767309*10^22+5.97219*10^24)*(384400+6371+1737) = 4700.39 km.
from the center of Earth. This point is called the barycenter. It is about this center that the Moon is rotating, which explains the giant Moon phenomenon. The Moon orbit is both distorted and off center. It is largest when it is at the closest point in its orbit and, at the same time, closest to Earth on the same side as Earth about the system C.G.
The system C.G. is inside Earth (radius = 6371 km). Earth center and the Moon center and the system C.G. are all on the same plane. Earth rotates about the system C.G. perpendicular to this plane, with centripetal force of,
\({F}_{e}=\frac{{M}_{e}}{{M}_{e}+{M}_{m}}{F}_{ob}\)
\({F}_{ob}\) = \({g}_{m}\) = 0.000226 kms-2 adjusted Moon gravity based on earlier calculations.
\({F}_{e}\) = 5.97219*10^24/(7.34767309*10^22+5.97219*10^24)*0.000226 =0.000223 kms-2
\(\frac{{v}^{2}}{R}={F}_{e}\)
\(v^2 = {F}_{e}.R\)
Kinetic Energy in such a point mass system,
\(K.E. = 0.5mv^2 = 0.5m{F}_{e}.R\)
For a rigid body about an off center axis
\(K.E. = 0.5*I*w^2\)
where \(I = 0.4m*r^2 + m*R^2 = 0.4m6371^2 + m4700^2\),
\(I/m = 0.4*6371^2 + 4700^2\)
Assuming that in establishing a force \({F}_{e}\), the amount of K.E. imparted is the same whether it is a point mass or rigid body system,
\(0.5* I *w^2\) = \(0.5*m*{F}_{e}.R\)
Therefore,
\(w = (\frac{{F}_{e}.R}{(I/m)})^{0.5} = (\frac{{F}_{e}.R}{( 0.4*r^2 + R^2 )})^{0.5}\)
\(w\) = ( 0.000223*4700 / ( 0.4*6371^2 + 4700^2 ))^(0.5)
which gives a spin angular velocity of \(w\) =0.000165 rads-1
Which gives a period of 2*pi/(0.000165*60*60)=10.55 hrs. This is too fast, a 10 hr day!
It is possible to model Earth as an hollow sphere in which case,
\(w\) = ( 0.000223*4700 / ( 0.6667*6371^2 + 4700^2 ))^(0.5) and the period of this spin is 11.95 hrs.
Both a hollow sphere and a inner solid sphere,
And Earth modeled as a thin hollow sphere of crust, three thick hollow spheres of mantle from (r1=1221, r2=3480) the outer core, (r1=3480 to r2=5651) the lower mantle, (r1=5651,r2=6371) the upper mantle, and a solid inner core (r=1221), each acting independently under \({F}_{e}\). Link to details of Inner Earth.
\(w\) = ( 0.000223*4700 / (0.6667*6371^2 + 4700^2 +
0.4*((6371^5-5651^5)/(6371^3-5651^3)) + 4700^2+
0.4*((5651^5-3480^5)/(5651^3-3480^3)) + 4700^2+
0.4*((3480^5-1221^5)/(3480^3-1221^3)) + 4700^2+
0.4*1221^2 + 4700^2 )^(0.5)
= 0.0000775 rads-1.
and the period is 22.5 hrs. This is closer to the expected value of 23.9 hrs. Unfortunately, Moon gravity has been adjusted to 0.000223 kms-2 based on measured orbital speed and orbital radius. Based on the same model of Earth, but with a Moon gravity value of 0.00162 kms-2. The calculated period is 8.60 hrs.
