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What is the connection between Newton's laws of motion and his law of gravitation? Is there a connection where one suggests the other, or are they just two separate ideas that are assumed to be true?

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    $\begingroup$ They are completely separate, but neither was "assumed", not even at Newton's time. Some of the most important concepts of mechanics were already developed by Galileo and the motion of the planets suggested a law like the one that is named after Newton. There is a little bit of a controversy whether Newton was the first to notice that a 1/r potential could be used to explain planetary notion. $\endgroup$
    – CuriousOne
    Feb 16 '16 at 7:31
  • $\begingroup$ I meant assumed as in only proven experimentally and taken as laws of the universe without some mathematical derivation from another source. Like is the law of gravitation true just because we observe it without fail or can it be derived from Newton's motion laws. Or are the motion laws somehow related to gravity. $\endgroup$
    – BoddTaxter
    Feb 16 '16 at 7:37
  • $\begingroup$ They are different but there was also uncertainty as to whether inertial mass was different from gravitational mass. en.wikipedia.org/wiki/Mass#Inertial_vs._gravitational_mass $\endgroup$
    – Farcher
    Feb 16 '16 at 7:41
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    $\begingroup$ Laws in physics (or, more correctly, symmetries and equations of motion) are always derived from observation. They can not and are not being derived mathematically. $\endgroup$
    – CuriousOne
    Feb 16 '16 at 7:42
  • $\begingroup$ Well, you can apply newtons laws of motion while considering the gravitational forces. But , They are not actually derived from one another, if that was what you meant. Neither of them influences others existence! $\endgroup$
    – brainst
    Feb 16 '16 at 7:48
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The $F= \frac{GMm}{r^2}$ law (Or equivalently $U=-\frac{G M m}{r}$ potential law) can't be deduced from Newton's laws of motion. It's a little secret that parts of physics like this are, from a purely mathematical perspective, totally arbitrary.

For example, it's perfectly reasonable to play build-a-reality and plug in whatever interaction potentials you want. $U=\frac{A}{r^{12}}-\frac{B}{r^6}$ is a completely valid interaction in classical mechanics. It doesn't contradict anything.

In more modern physics the rules can be much more restrictive. (which is a good thing, because it means you can't just plug whatever you want into the theory!)

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Newton's Laws of Motion provide a framework for all types of forces, connecting observed motions to the forces that control them.

Newton's Universal Law of Gravitation tells you how to calculate the force of gravity. So if you want to solve a problem with gravity you use this law for the force, and the laws of motion to solve the problem.

Many practical problems include multiple forces, thus thre are force laws for friction, springs, etc. Some of these are just good approximations. Engineering uses all of them.

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