Would you explain this for me, please?

Question

F=Gmm/r^2
Newton's law of gravitation.
My book says "if you double the distance between two objects, the attraction between them becomes four times weaker."
I'm trying to understand how formulas are created from natural observation. So, why is it weaker by 4 times? I understand it's weaker because it's in the denominator but what if the distance is 3? Then 3^2=9, which is tripled. I have a math and physics book with two problems like this that explain it as an even decrease. Why is this? Because it is squared? But only 2^2 and 4^2 produce answers that are multiplied by four-all others do not-what am I missing? Thanks!

Answer 1

I'm afraid that what you are missing is the mathematical consequence of having an "inverse square law of attraction," which is how gravity behaves, or is described.

Incidentally, the force F_12 between two bodies of masses m1 and m2 at separation (or distance between them), ' r ,' is:

F_12 = G m1 m2 / r^2.

(I like to separate terms out in a formula like this, for greater reading clarity.)

To see the consequences of changing r, just CONCENTRATE on the (1/r^2) factor, since the rest of the formula (G m1 m2) is UNCHANGING.


So how does 1/r^2 behave, as r is increased? :

r .......... 1/r^2

1 .......... 1
2 .......... 1/4
3 .......... 1/9
4 .......... 1/16
5 .......... 1/25 etc.

So, as you move the masses away from each other in successive integer steps so that they are successively 1, 2, 3, 4, 5, ... times further away than at the start, the force decreases by the SQUARES of the current distance, so that it is INVERSELY PROPORTIONAL to 1 (that's the starting value), 4, 9, 16, 25, ... , that is DIRECTLY PROPORTIONAL to 1, 1/4. 1/9. 1/16, 1/25, ... etc.

The question about how this formula was "created from natural observation" is a fascinating historical detective story. It involves probably the greatest scientist of all time, Isaac Newton, with whom only Albert Einstein is on a par.

(Einstein himself was in awe of Newton's scientific accomplishments. They not only laid the basis for much of modern science, but could also be said to be the basis for today's technological society.) Newton was trying to understand the fundamental logical and mechanical explanation for observations and empirical "Laws" of planetary motion made by Johannes Kepler. He was the first person to declare and show that the same laws of motion operated both on the Earth, and in "the Heavens." (Until then, these were treated as distinct spheres --- almost literally --- operating on different "rules." Even Galileo thought that there was a "linear law of inertia" here on Earth, but a "circular law of inertia" out there in space.)

This fascinating story played a very large role in different introductory university courses I taught for both non-science and science students during the past thirty years. I hope that you have the chance to learn about this some day from an equally enthusistic teacher !

Answer 2

Think of gravity as going out in waves (just for illustration). It covers the surface of a sphere which radiates out. Since the surface of a sphere is 4πr² as the radius increases the surface increases by r². Since the same gravity covers more surface it is relatively weaker by r².