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I am having trouble comprehending how anyone could come up with this formula:

$$F = \frac{GMm}{d^2}.$$

Could someone walk me through this?

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  • $\begingroup$ Guess and check! $\endgroup$ – Andrew Oct 8 '14 at 2:20
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    $\begingroup$ Newton never did express his universal law of gravitation in that form. He reasoned in terms of proportionalities and worked to avoid needing to know the constant of proportionality. The form in which you wrote Newton's law of gravitation with the constant of proportionality included (the constant G) didn't appear until late in the 19th century. $\endgroup$ – David Hammen Oct 8 '14 at 4:36
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Well, there are 4 parts to the right-hand side; let's look at each in turn.

The first $M$ is the mass of the gravitating body. You would expect that the force it exerts should be larger if it has a larger mass. Moreover, it's not unreasonable to think the force should be directly proportional to this mass: twice as much mass should grab things with twice as much force.

By symmetry, the dependence on $m$ should be the same as on $M$. This symmetry is really that of Newton's Third Law: for every action (force that $M$ exerts on $m$) there is an equal and opposite reaction (force that $m$ exerts on $M$).

How about the $d^2$ in the denominator? Well, we expect that the force with which one thing attracts another decreases with distance: get far enough away and you should feel something's effect on you diminish to arbitrarily small magnitude. Moreover, the power of 2 is an effect of living in 3D space. Consider a point source of light. Any sphere of radius $r$ centered on the source will capture the same total power, so the power per unit area (how bright the light appears) is proportional to $1/r^2$, since the area of a sphere is proportional to $r^2$.

In fact, this $1/r^2$ dependence was known for certain things and even suggested for gravity. Hooke in particular felt Newton got too much credit because he (Hooke) had suggested an inverse square law earlier (though he didn't really have rigorous math or predictions relating to it). Needless to say, this soured their relations, prompting Hooke to join the Leibniz camp in the calculus debate.

Finally, there's the $G$, which is just a reflection of the fact that our arguments are all about proportions and so they leave an overall proportionality constant undetermined. The value of $G$ is in fact notoriously difficult to measure, and this wasn't done in Newton's time. Instead, one would make do with taking ratios of quantities such that $G$ dropped out.

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    $\begingroup$ So newton had a sense of gravity flux, and field strength? $\endgroup$ – ja72 May 26 '15 at 12:42
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Newton actually calculated, using a lot of obscure geometry and limiting concepts, the orbits that various force forms would generate. One of those forms was the inverse square force. If you want to know what the others were, find a copy of the Principia and wade through it.

The result wasn't published until Edmund Halley asked Newton if he knew the nature of the force (our modern wording) that would cause something to have a repeating orbit. Halley was investigating the possibility that sightings of a bright comet every 76 years could possibly be the same comet. Newton immediately responded that it was an inverse square force. Halley was shocked with the quick answer. Newton had worked out the math for various forces out of curiosity and never told anyone. Halley forced him (with his financial backing) to publish the results. Halley was also impressed that the orbits from the inverse square force would mathematically produce the same results as Kepler had extracted from Tycho's planetary data.

Newton's reasoning was that the same force which gave an object weight (or gravitas, from the Latin) was the same force which kept the moon in orbit around the Earth and the Earth around the Sun, etc.

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