We kicked off our countdown to Newton’s birthday with his equations of motion, so it seems fitting to close out the section on classical mechanics with another of Newton’s equations, this time the Law of Universal Gravitation:

Like all the other equations to this point, I’m cribbing this from the formula sheet for my just-completed intro mechanics class, which means it’s in the notation used by Matter and Interactions. This is sub-optimal in some ways– I prefer to have subscripts on the *r* to remind you which way it points, but I don’t care enough to re-do the equation.

So, this is the famous inverse-square law of gravity, saying that every object with mass in the universe attracts every other object with mass in the universe with a force that is equal to some constant times the product of their masses divided by the square of the distance between them. The r-hat at the end is a unit vector telling you that the force is along the line between the two objects.

Why is this important?

This is important because it was this law, perhaps even more than the laws of motion, that made Newton’s name, and launched physics as a mathematical science. Newton hit upon this equation for the gravitational force in the 1660’s, and published it in the Principia Mathematica in 1686. There’s some argument as to whether he was the first to think of the inverse-square nature of the law (Robert Hooke even claimed that Newton stole the idea from him), but he unquestionably put it on a solid mathematical footing, showing that it was reasonable to use it not only for tiny particles, but also for planet-sized spherical masses.

This last bit was the killer app of the late seventeenth century, because this law of gravitation allowed Newton to explain and correctly predict the elliptical orbits of the planets. No other theory to that point had been able to do as much, and having a solid explanation of planetary motion was a huge milestone in the path to modern science.

(Popular legend, of course, has Newton coming up with this law after being struck by a falling apple in his family’s orchard. This is almost certainly apocryphal, but Newton *did* do a lot of productive thinking in the 1660’s while staying with family in the country, an outbreak of plague having closed the universities. In some sense, then, the birth of physics can be credited to the Black Death, which is either ironic or appropriate, depending on your feelings about the subject. )

We now know that this law of gravity is a good description of reality over a huge range of sizes– masses ranging from a few kilograms to billions of times the mass of the Sun, and separations of less than the thickness of a human hair up to millions of light-years. This mathematical form also provides the paradigm for thinking about forces and interactions in physics: it depends on intrinsic properties of the two interacting objects, and the distance between them, and that’s it. It’s a *universal* law, that does not depend on the composition or history of the objects, and that’s a big part of what makes it powerful.

This is also the first equation I’ve put on this advent calendar that is not fully correct. In fact, we now have a more general theory of gravity, Einstein’s general relativity, which gives a more accurate description of reality in certain circumstances, and also provides an explanation of *why* gravity his this form, something Newton was unable to do. (Pressed to explain the mechanism by which gravity acted, he huffily replied (in Latin) “hypotheses non fingo,” meaning “I feign no hypotheses.”) The math involved in general relativity is fiendishly complex, though, and for the vast majority of situations in which you might want to know the gravitational force between two objects, Newton’s law is more than sufficient.

We’ll return to this question again later, though. For now, just take a moment to appreciate Sir Isaac’s greatest achievement, and come back later for the next equation of the season.

(I intended this to be Sunday’s advent calendar post, but getting and decorating (with SteelyKid’s help) our Christmas tree was more important, so instead you get two equations today…))

Chad Orzel wrote (December 12, 2011 10:10 AM):

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F_{grav, on 2 by 1}= […]>

does not depend on the composition or history of the objectsNoting that a while back

Chad Orzel wrote (December 1, 2011 10:26 AM):

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d/dt[p] =F_{[…]}there seems to be some, perhaps implicit, dependency on the histories of the objects (separately), as parametrized by “t“, after all …“Newton hit upon this equation for the gravitational force in the 1660’s, and published it in the Principia Mathematica in 1686.”

You might want to rephrase this statement because Newton did not use equations in the Principia. Newton worked with ratios and equality of ratios only and did not use branded units.

Frank Wappler@1 –

there seems to be some, perhaps implicit, dependency on the histories of the objectsCertainly the gravitational force depends on the history of an object. The reason there’s less gravitational force between the Earth and a Voyager spacecraft than the Earth and a 1977 Plymouth Voyager is that some point in their history NASA launched the spacecraft towards interstellar space. But the influence of that history is entirely contained by the effect on the variables on the right hand side. Once you’ve chosen your system and the instant you’re observing it (and thus what you mean by m1, m2, and r), the gravitational force is completely defined by the equation.

That’s what Chad is getting at – the functional form of the equation is the same regardless of the composition or history of the objects, even though the parameters you feed into might change.

RM@3 –

Once you’ve chosen your system and the instant you’re observing it (and thus what you mean by m1, m2, and r) […]Can values of m1 and m2 be measured (or “be appropriately

chosen“) without “considering history” of the corresponding objects(even if these obtained values are taken as referring only to one particular instant, or trial, or application in the formula) ?

As an analogy: The derivative of a function, even only at one particular point, cannot be calculated or be considered without taking account of the function in an entire neighborhood of that point.

Apart from this I agree:

Obviously the values of m1, m2 and r may (or may not) be found to vary, from trial to trial

(the derivative of a given function may or may not vary from point to point, too);

and obviously the formula stated above is applied equally in each trial, and it doesn’t show any explicit “history dependence”.

And it doesn’t explicitly show quite “what the quantities mean” or how their values might be determined either.

The apple story was given to us by Newton biographer William Stukeley, who wrote that Newton was reminded of it when the two were having tea in the shade on an apple tree. “…he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind…” Of course, the apple hadn’t hit Newton, but, â ‘Why should that apple always descend perpendicularly to the ground, ‘ thought he to himself, occasioned by the fall of an apple, as he sat in a contemplative moodâ¦”

There’s a photocopy here: http://ttp.royalsociety.org/silverlight/?id=1807da00-909a-4abf-b9c1-0279a08e4bf2

… check page 42.

That gravity should drop off as the square of the distance was actually first proposed by neither Newton nor Hooke, but by Ismael Boulliau, as detailed here: https://thonyc.wordpress.com/2011/09/28/the-man-who-inverted-and-squared-gravity/

I agree that Newton’s law is sufficient. We don’t need to make physics more complicated than it already is. Relativity is actually more complicated than its presented in the textbooks. Einstein had to simplify and approximate long series to make the equations found in the textbooks today.