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Gauss's Law of Universal Gravity
--------------------------------
In a nutshell: The gravitational flux through
any closed surface is proportional to the
enclosed mass. Consider:
_ _ _
∫∇.F dV = ∫(F.n) dσ ... The Divergence Theorem
F = -GM/R^{2} for a unit mass
∫(F.n) dσ = -(GM/R^{2})4πR^{2}
= -4πGM
Where,
dσ is a vector with magnitude equal to a tiny
piece of the surface area. It is outward
pointing and normal to the surface.
F is a vector field (gravitational field).
M is the mass contained within the surface.
V is the volume contained within the surface
Now, M =∫ρ dV where ρ is the mass density (ΔM/ΔV)
∫(F.n) dσ = -4πG∫ρ dV
Using the Divergence Theorem:
_
∫∇.F dV = -4πG∫ρ dV
_ _
∇.F = -4πGρ
=> -4πG∫ρ dV = F4πR^{2}
_
=> F = -MG/R^{2}
Drill hole through center of earth. At any
point in the fall, the only mass exerting a
net force on the test body is that contained
inside the current radius because the shell
of matter outside the radius has no effect.
So the force at point R is:
F4πR^{2} = (-4/3)πR^{3}ρG
=> F = (-4/3)ρGR
This has the same form Hooke's law and would
result in Harmonic oscillation about the center
of the earth.