Why is distance squared in Newton's gravitational formula

[sheenktk]

Can anyone tell me that why we take r square instead of r?

 

[xts]

Because if we took r instead of r^2 then predictions of such theory would not fit to experimental results. Especially plredicted planet movements would not fit to observations. If we take r^2 - they fit perfectly well.

 

[homeomorphic]

The original reason why Newton did that is that the only central force law (in terms of powers of r--and this might be valid for any function of r, but I'm not sure) that produces elliptical orbits for the planets with the Sun at the focus of the ellipse. So, it's basically the only option, to the extent that the orbits really are elliptical (which is really only a good approximation).

A linear force law also produces elliptical orbits, but then the Sun would have to be at the center of the ellipses, which is not what we see.

Newton's law of gravity was one of his greatest achievements because he was able to demonstrate these facts using Euclidean geometry. His proofs are complicated, but they are described in a somewhat readable form in Brackenridge's book, The Key to Newton's Dynamics (I say it's complicated, but it's possible to give a fairly simple outline of the main points, which is done in the first few chapters--unfortunately, I am too rusty on it to describe it here, and I never found the time to finish the rest of the book).There may also be other explanations for the r^2 (I saw one in a book about quantum field theory, and you could derive it as some sort of limiting case in general relativity).

 

[mrspeedybob]

Imagine we are talking about light instead of gravity for a moment...

A bulb emits light at a rate of 1 watt. If you construct a sphere around the bulb with a radius of 1 meter then the total interior surface area of the sphere is 4pi and the intensity of the light striking the interior is 1/4pi watts per square meter.

If the sphere were 2 meters in diameter then the surface area would be 16pi and the intensity of the light striking the interior is 1/16pi watts per square meter.

As you can see, the intensity of light striking the interior of the sphere is inversely proportional to the square of it's radius.

Now if we imagine that a mass is a source radiating gravitons just like a light bulb radiates photons the reason for the inverse square law becomes obvious.

 

[A.T.]

A linear force law also produces elliptical orbits, but then the Sun would have to be at the center of the ellipses,

Yeah, like shown here:

[URL]http://bluelyon.files.wordpress.com/2011/02/isaac-Newton-pound-note.jpg[/URL]

Trajectories for different laws of gravity:

http://megaswf.com/serve/1161536