The attraction between every two objects

[Khadar Mariano]

There is an attraction between every two objects,the strength of the the attraction is directly proportional with the masses of the to objects and indirectly proportional with the distance between the two objects squared, so where this is square came from, why not only distance.

 

[256bits]

Khadar,
You can blame at least three individuals for that. Kepler, with his laws of planetary motion, Galileo, who changed the world view on heliocentrism and of falling objects, and Newton who came up with the law of universal gravitation.

As you have said, the attraction is proportional to the masses of the objects. Double the mass and the force of attraction also doubles, no problem there.

But double the distance. Well, we logically assume it should vary inversely, but how - inversely to distance, to distance squared, maybe distance cubed, or even some other power. To find out we can do some experiment and vary the distance and measure the force. It turns out the force is indirectly proportional to distance squared.

In Newton's words “I deduced that the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly." ref http://en.wikipedia.org/wiki/Gravitation

Mathematically, if we take a certain size sphere1 of diameter r, and then double the radius to obtain another sphere2 with radius 2r, the surface area of sphere2 will be found to be 4 times the surface area of sphere1. Since we live in a 3-D world or universe, gravity also behaves the same way - the strength of gravity will also follow the inverse square law.

 

[Bandersnatch]

IMO, the last paragraph in 256bits' post is the most important for getting an intuitive understanding as to the why. Let me repeat my elaboration on this topic from an earlier thread(https://www.physicsforums.com/showthread.php?t=697273):


As far as the inverse sqare law goes, that's the result of the world being (spatially) three-dimensional.
It's easy to see why, when you begin by considering a source of whatever you want(e.g., radiation, force) in one-dimensional space.

So, we've got a line(1d space) with a point-like source located somewhere on it. The source is causing some sort of interaction to propagate from it in all directions. In 1d space, that means all the output is divided equally between the two directions. Whichever point on the line you choose, no matter how distant from the source, the strength of the interaction measured there will always be the same as at any other point, and equal to half the source strength. There's nowhere else for the interaction to dissipate but being split in half.
So in such a space, e.g. the gravitational force would not have \frac{1}{R^2} in it. It would have no R at all, as the strenght is distance independent.

Now let's take a 2d space(a plane). In this case, the interaction propagates in the form of expanding circles centered around the source. If you pick one point somewhere on the plane, you'll find out that the interaction has been spread thinner, as the same amount of it needs to cover more space. At any given distance R, there are 2\pi R points that equally share the original interaction between them. Since the interaction has to be divided between all the points on the circle with size dependent on an R variable, the force of gravity would have the factor \frac{1}{R} in it.

Three-dimensional space adds another dimension into which the interaction must spread, so that at any given distance R there is 4\pi R^2(concentric spheres) points sharing what the source had produced. Hence the \frac{1}{R^2} factor in the equations.

 

[Khadar Mariano]

thank you guys