Why do central forces, such as gravity, only act in one dimension?

[Werg22]

In a book of mine it is stated that the inversed square in the relationship of the radius and the intensity of the gravitational field is linked to the fact that the surface of a sphere is directly proportional to the square of its radius. How is this fact relevant?

 

[rbj]

In a book of mine it is stated that the inversed square in the relationship of the radius and the intensity of the gravitational field is linked to the fact that the surface of a sphere is directly proportional to the square of its radius. How is this fact relevant?

check out

http://en.wikipedia.org/wiki/Inverse_square_law

http://en.wikipedia.org/wiki/Flux

http://en.wikipedia.org/wiki/Gauss's_law

to get an idea why it is relevant for those who live in a 3 dimensional existence.

 

[El Skoro]

In a book of mine it is stated that the inversed square in the relationship of the radius and the intensity of the gravitational field is linked to the fact that the surface of a sphere is directly proportional to the square of its radius. How is this fact relevant?

Here's an analogy that helps me visualize inverse square relationships. If you have a source that gives off energy in all directions in 1 joule increments, that energy will move outward in all directions like the surface of an expanding sphere. As the sphere expands, its surface area increases in direct proportion to the square of its radius. On that surface, the total energy is still 1 joule. But as the sphere expands, the energy per unit of area is decreasing. Sphere expands, energy per area decreases, giving us the inverse relationship. And, as the sphere expands, itarea increase in direct proportion to the square of its radius.

Area of a sphere = 4 pi r squared

 

[daniel_i_l]

It is interesting to note that in 1-D, gravity would never weaken!

 

[rbj]

It is interesting to note that in 1-D, gravity would never weaken!

yup. one way to imagine this (in our 3-D existence) is with an infinite plane of mass in space (and nothing else except oneself). there would be some gravitational field from that plane of mass, but it would be the same whether you were 10 meters away from it, or a kilometer, or a lightyear. now imagine that this infinite plane of mass is featureless and perfectly smooth in surface appearance (sorta like the surface of that monolith in 2001: A Space Odyssey but extending infintitely in two dimensions). just by looking at it (now we'll pretend there is a sun somewhere illuminating it), you could not tell the difference between if you were 10 meters away from it, or a kilometer, or a lightyear distant.

now, for 2-D, let's imagine an infinite line of charge (or maybe better yet, a cylinder of charge that is infinitely long, but finite diameter). that has a 1/r gravitational field, and as you move away from it, it appears smaller only in one dimension, how thick it is.

for the point (or better yet, a little sphere) charge in 3-D, it's 1/r^2 and as you move away from it, it appears smaller in two dimensions.

i suppose, if we lived in a 4-D universe (4 spatial dimensions), instead of having these inverse-square laws for E&M, gravitation, power intensity of any radiation, etc., i think we would have inverse-cube laws.

 

[thepaqster]

hrm I'm not sure I understand when you say that gravity would never decrease in one dimension, can you explain a little more please? Because, the way I understand it, Central forces such as gravity actually are only in one dimension; that is directly towards the source of gravity. This is why the Earth rotates around the sun, because the force of gravity felt by the Earth is always directed towards the center of mass of the sun.
Also, there do exist 1/r^3 Force fields, they are just usually unstable and tend to diverge, so we don't see them as often as 1/r^2 force fields. For example, a neutron interacting with a proton in a Yukawa force field actually go by 1/r^3.

 

[rbj]

hrm I'm not sure I understand when you say that gravity would never decrease in one dimension, can you explain a little more please?

i don't really see what i could add. do you want me to dig out my old sophmore physics book and copy down the double integral that shows that the E field from an infinite plane of charge is constant with distance?

Because, the way I understand it, Central forces such as gravity actually are only in one dimension; that is directly towards the source of gravity. This is why the Earth rotates around the sun, because the force of gravity felt by the Earth is always directed towards the center of mass of the sun.

it's still at least a 2 dimensional scenario for anything to go around anything else. and since not all things orbiting the sun are in one plane, it must be a 3 dimenisional picture. just because you prefer to view all radial forces as acting in only one dimension doesn't mean they are. i think the right way to look at it is in a cartisian coodinate system.

Also, there do exist 1/r^3 Force fields, they are just usually unstable and tend to diverge, so we don't see them as often as 1/r^2 force fields. For example, a neutron interacting with a proton in a Yukawa force field actually go by 1/r^3.

dunno about 1/r^3 (i think they say it's 1/r^7), but i know from reading http://en.wikipedia.org/wiki/Fundamental_interaction that there are forces that are 1/r^n where n is not 2. it's the only way that it makes sense for the Strong Nuclear Force otherwise big atoms (like plutonium) would be just as stable as the small ones.