Question

The force of gravity on a space craft increases as it gets (closer to/further) from the Earth. Which one would be correct?

Answer 1

Answer:

Gravitational field strength: Decreases as you get closer to the center. [1]

Gravitational potential: Increases as you get closer to the center. [2]

Force of gravity: Undetermined until some other mass is specified that interacts with the Earth’s gravitational field, but would decrease in direct proportion to the gravitational field strength. [3]

Notes

[1] The gravitational field strength, g , for a point mass is g=GMr2 and if take the Earth to be a ball with constant density which it isn’t in either case, but good enough for our purposes, we can express the mass as M(r)=ρ(r)⋅V(r)=ρ⋅43πr3 which yields the expression for the gravitational field strength as a function of distance:

g=GMr2=G⋅ρ⋅43πr3r2=43πGρr

From this we can see that the gravitational field strength is zero at the center and is linear with respect to depth, or g=κ⋅r

[2] The derivation for the interior gravitational potential is simple enough but a bit lengthy, but it is simply:

ϕ(r)=−GM2R3(3R2−r2)

where R is the radius of the sphere and r is the radial coordinate. Here we see that the gravitational potential is a maximum at the center and a minimum at the surface. From the surface outwards, the exterior solution, the gravitational potential decreases to zero at infinity in the usual form ϕ(r)=−GMr−1 .

[3] To have a force there needs to be an interaction with the Earth and some other mass, so we have for the force: F(r)=−m∇ϕ(r)=m⋅g(r) , but no other mass is mentioned. Still, the force scales proportionally to g(r) and so decreases until zero at the center.

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Answer 2

The answer is closer to

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