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Last modified: November 20, 2021 ✓

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/R2 for a unit mass

∫(F.n) dσ = -(GM/R2)4πR2

          = -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 = F4πR2
   _
=> F = -MG/R2

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:

F4πR2 = (-4/3)πR3ρG

=> F = (-4/3)ρGR  

This has the same form Hooke's law and would 
result in Harmonic oscillation about the center 
of the earth.

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