B Gravity Inside a Hollow Sphere

[jayromano]

TL;DR Summary

Gravitional forces at the centre of the earth's core

Hello all,

Suppose you created a 20x20 foot hollow sphere at the most central point of the earth’s core. This sphere is protected from the heat and oxygenated, in fact it’s a perfect 22 degree C inhabitable environment for a human.

What kind of gravitational forces would one experience in this sphere? Given the Earth's mass surrounds the human and is relatively balanced across all outward vectors, would one experience something approximating weightlessness?

Thanks,
An insomnaic who is kept awake by these thoughts.

 

[anuttarasammyak]

No gravitational forces there.

 

[berkeman]

Welcome to PhysicsForums!

would one experience something approximating weightlessness?

Yep!

Do you know why? We can show you using calculus -- would you be comfortable with that?

 

[jayromano]

Welcome to PhysicsForums!

Yep!

Do you know why? We can show you using calculus -- would you be comfortable with that?

If you can spare the time to explain using calculus I would be very grateful! But I cannot guarantee I'll understand :)

 

[anuttarasammyak]

Gauss's law explains it.
ref. https://en.wikipedia.org/wiki/Gauss's_law_for_gravity

 

[Keith_McClary]

using calculus

I think if you assume force is proportional to mass, you might be able to prove it with a symmetry argument (no calculus).

 

[Nugatory]

I think if you assume force is proportional to mass, you might be able to prove it with a symmetry argument (no calculus).

You can, and that’s the way I learned it.

@jayromano consider the gravitational force on us from any small chunk of the earth. Because we’re at the center, there will be a piece the exact same size and distance away on the opposite side creating the same force but in the opposite direction. These cancel, so there’s no net force on us.

There’s a followup question that goes will keep your insomnia going for a bit longer: why do we experience no net force as weightlessness instead of being pulled in opposite directions?

 

[jayromano]

There’s a followup question that goes will keep your insomnia going for a bit longer: why do we experience no net force as weightlessness instead of being pulled in opposite directions?

Thanks for the reply Nugatory,

My thinking was that if you are in the sphere, then you will have mass all around you generating an outward force vector, essentially pulling you away from the center of the sphere. However, because the mass around you is relatively balanced in all directions, there would be a state of balance or equilibrium were the sum of all outwards vectors results in no movement in anyone direction. I visualize it a bit like standing on a circular platform, there are many ropes tied to the platform, all heading in different directions, but each rope is perfectly opposite to another rope, there are people pulling on all the ropes with the same amount of force, the net result is no platform movement (like a stalemate tug-of-war).

The way you describe, it sounds like I might be wrong? It sounds like there are somehow inwards and outwards forces which cancel out resulting in no force, rather than just outward forces balancing each other?

 

[Nugatory]

I visualize it a bit like standing on a circular platform, there are many ropes tied to the platform, all heading in different directions, but each rope is perfectly opposite to another rope, there are people pulling on all the ropes with the same amount of force, the net result is no platform movement (like a stalemate tug-of-war).

That's why the platform doesn't move in the tug of war game... but if the people pulling on the ropes are strong enough the platform will be torn apart, breaking into separate pieces each attached to one rope and moving away from one another.

That doesn't happen when the force is generated by gravity between the platform and the surrounding spherical shell, and this is because of the difference between the forces acting on the platform as a whole and the forces acting on any single small piece of the platform. Try working out the forces at the attachment points for the ropes in the tug of war case and the gravity case and you'll see the difference.
(or you could follow @anuttarasammyak's hint above and google for "Gauss's theorem" - the wikipedia page he linked may be more calculus than you want to take on, but Google will throw up some more intuitive explanations).