Why does gravity cancel out for all points inside a sphere?

[rbj]

Hi rbj. To put this in context, this is what I posted in answer to a post by Chaos;

i had conceded that my original response was wrong. i was meaning to dispel the notion of a sort of "Faraday cage" for gravity.

what you said is true, but that shell won't stop the gravitational effect if another large planet swung around that spherical shell. objects inside the spherical shell would start to move in the direction of that object that came around to one side of the shell (on the outside). but, i recognized that this objection of mine was out of context to what was asked and i hope that i set the record straight. now i look at the topic as one of Gauss's Law and, you're right, any objects inside that uniform spherical shell experience no net gravitational force.

 

[Rach3]

For the inverse-cubed hypothetical, here's an intuitive reasoning:

The force an an off-center test particle inside a spherical uniform mass shell, under inverse-cubed gravity, is instantaneously equal to the force due to an inverse-squared gravity, for some particular non-uniform spherical mass shell. This modified shell, would have mass density inversely proportional to the distance from the test particle at that instant. Clearly it will accelerate in the direction of the nearest wall.

 

[rcgldr]

Another interesting property of inverse square law is that the strength from an infinitely large plane is constant, no matter how far you are from the plane.

 

[Chaos' lil bro Order]

Nice answers, thank you.
Guess my 2 posts are gone forever.
Moving on...

What predictions does MOND make on gravity's propogation and weakening with distance?

 

[jasc15]

Well that would be petty cool, and gravity doesn’t need to be so small as Dave said depending on how you design this world. Make the diameter 1/4 the size of Earth - we would need only 1/16 the mass of Earth to create a1 G gravity. (Likely already to dense to be real)
Now if we concentrate that mass into a 10% thick shell (An even denser configuration) with a familiar surface. Plus a few openings with no leaks below a certain altitude so the surface could hold a couple oceans of water. Now just add enough air till gravity brings it up to a pressure that we like. It would take a lot of air as it would of course leak into the inside where it would experience no significant gravity increase thus the pressure inside would rise to a uniform level matching the surface with a small gravitation bias to the center due to the mass of the air.

All the making for a breathable relatively weightless environment.
Could be fun - But what of the weather inside?
Interior rain would collect at the center for a low pressure / gravity blob of an ocean, could get very weird.
I expect possible only within the imagination of a Sci-Fi man made planet.

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Also related to the OP, as to “checking” the inside and outside gravity mathematically you can also do an approximate manual calculation in a flat circle. Use four points each with a mass of 1/4 that of Earth in a circular orbit to replace the earth. The orbit of the Moon would still be the same. Calculations at and near the center would be zero. But if you move out to close to the inside edge of the circle, or measure the outside force to near the circle would have some geometry aberrations biased to a nearby point. That distortion would be reduced by increasing the mass points to 16 or more but then the number of calculations start to get out of hand.

i can't imagine that air would fill the sphere, or water fot that matter. what would the pressure be? the only reason there is any air pressure at the surface is because of gravity. but then, would there just be an air-vacuum interface at the surface of a hole in the sphere? neither of these possibilities holds up to much scrutiny, yet i can't think of another.

 

[Hazzattack]

Hi, was just reading this topic and was wondering if anyone could give me a link to a practice problem/solution for this type of question. I was trying to prove it through Gauss' law but get slightly confused on some of it.. and find it awful hard to follow when it isn't written in proper form (i.e /frac[r] (always find this hard to follow).

Would really like to see how it works out, i assume you do something similar to when you calculate a surface integral, with r varying between r < R.

Thanks,

Harry

 

[Hazzattack]

also perhaps you could show it through treating it like a point mass( in EM a point charge).
thanks again.

 

[my_wan]

Another interesting GR feature of this effect is that even though the gravitational acceleration goes the zero everywhere inside such a sphere the gravitational time dilatation remains slowed to the same as on the surface. So you have what is essentially flat space-time inside and far removed from the mass, yet two observers with no relative motion between them them in these respective flat regions of space-time and yet their clocks will still have different relative rates.

This also means that the Gravitational constant (big G) associated with masses in the flat regions of space can 'apparently' differ if the difference in depth of field is not properly accounted for. Or conversely they can merely disagree on each others total mass which is the approach used by GR for good reason. Keeping the physical constants constant in this way keeps the laws of physics consistent in all cases, even though relativistically speaking you could get the same observational effects even if you assumed they varied.

 

[Redbelly98]

... find it awful hard to follow when it isn't written in proper form (i.e /frac[r] (always find this hard to follow).

FYI, I have edited those posts (Post #'s 7 and 25) to make the equations readable. We have recently switched to new equation processor software, and some older posts do not display properly using the newer software.