Exploring the Ratio of Gravity to Curvature of the Earth

[Kingarthur]

If I dig 2 holes a little space apart, and then they intersect straight down, because of the curvature of the earth, then there is more than one up direction from the center. The closer 2 points at an equal distance down from the surface in separate holes get to the center, the less distance there is between the 2 points. I think that ratio has got to be related to gravity! Can we discuss the ratio and how it's related to gravity? By the way, is gravity different near the center? At the center?

 

[bp_psy]

If I dig 2 holes a little space apart, and then they intersect straight down, because of the curvature of the earth, then there is more than one up direction from the center. The closer 2 points at an equal distance down from the surface in separate holes get to the center, the less distance there is between the 2 points. I think that ratio has got to be related to gravity! Can we discuss the ratio and how it's related to gravity? By the way, is gravity different near the center? At the center?

If we consider the Earth to a perfect sphere having a constant density(or varying only as function of radius).then the gravitational pull of the Earth on an object at any height or depth would be constant on the surface of a sphere with radius [tex]\vec{r}=r(earth)+height[/tex] or [tex]\vec{r}=r(earth)-depth[/tex].

The gravitational force between 2 force on the surface of such sphere would not depend on the gravitational force due to the earth.It would depend only on the linear distance between them and their masses.

[tex]\ F={\frac {Gm_{{1}}m_{{2}}}{{r}^{2}}}[/tex]

So yes the gravitational force between them would increase as you go down because the distance between the objects decreases.Not because the Earth does something to them.

By the way, is gravity different near the center? At the center?

The the gravitational force due to Earth on each of the 2 objects approaches 0 as you get close to the center. The gravitational force between the 2 objects is still depend only on their distance

 

[Kingarthur]

Thank you for your response. I was sort of getting at whether or not the ratio has anything to do with the shape of the earth. The world could be square, but it is not. I'm thinking that mass causes gravity, and that things fall to the core, but then stop, or possibly some things rise as vapor. The Earth's shape causes things that fall in separate holes to get closer to one another while they go "straight down". So, I wonder if that ratio of getting closer as they go down is proof of the cause of the Earth's shape; if the Earth was forced to be a ball by gravity. I'm new to science, and maybe I don't know how to express this idea. Obviously the objects’ gravitational attraction to each other is a separate issue, related to their mass.