Exploring the "G" in Newton's Law of Universal Gravitation

By: Mark Mancini  | 
Newton's Law of Universal Gravitation
Isaac Newton proved that the force that causes an apple to fall to the ground is the same force that causes the moon to orbit the Earth. This is Newton's Law of Universal Gravitation, which he defined mathematically, using G as the gravitational constant. dmitro2009/Shutterstock

Here on the pale blue dot we call home, gravity is something we all experience every second of every day. And we're far more conscious of it thanks to Newton's Law of Universal Gravitation.

"Gravity is the glue that causes diffuse matter between the stars to slowly collapse and form new hydrogen-fusion machines (aka stars)," says University of Connecticut astrophysicist Cara Battersby. "It is the glue that binds galaxies together and it is responsible for our own Earth orbiting around the sun every year."


Gravity was also the key player in Sir Isaac Newton's famous "apple" story. One day, Newton was hanging out in Lincolnshire, England when he watched an apple fall out of a tree -- or so he claimed. Over the coming years, he'd tell many acquaintances — like Voltaire and biographer William Stukeley — that his great writings about the nature of gravity were inspired by this mundane little event.

Thus, the groundwork was laid for Newton's Law of Universal Gravitation -- central to which is a phenomenon called the universal gravitational constant, aka: "Big G" or just "G." In this article, we'll explore Newton's universal law, the conflicts proposed by Albert Einstein's theory, and gravitational force itself.


The Origins of Newton's Law of Universal Gravitation

Much has been said about that menacing apple in Newton's recollection. While the famous physicist told Stukeley this anecdote decades after it supposedly occurred, many academics have cast doubt on the tale. Regardless, the true intrigue of Newton's universal law is not whether or not the apple hit him, but that the force acting on the apple brought it straight down.

As Newton's assistant John Conduitt wrote:


[It] came into his thought that the same power of gravity (which made an apple fall from the tree to the ground) was not limited to a certain distance from the Earth but must extend much farther than was usually thought — Why not as high as the moon, said he to himself & if so that must influence her motion & perhaps retain her in her orbit.

Universal Gravitation Constant: The Equation

First thing's first. Before we tackle the Big G, we should step back and explain the universal gravitation equation of Newton's law. This equation continues to have huge ramifications on how we view the gravitational field that keeps us secured to Earth's surface.

As Katie Mack — an astrophysicist and the author of "The End of Everything (Astrophysically Speaking)" — says via email, gravity is "the mechanism by which things that have mass are attracted to each other". Newton was the first to make it clear that objects exert gravitational influences.


Newton realized the strength of this gravitational attraction between a given set of objects depends on (a) how massive they are and (b) how far apart they are. The inverse square law is a key principle here, whereby the gravitational force exerted is inversely proportional to the separation between objects. It's a dynamic his Law of Universal Gravitation puts into mathematical terms.

Here's the relevant equation:

F = (G x m1 x m2) / r2

The "F" stands for the "force of gravity"; "m1" means the mass of the first object; "m2" denotes the mass of the second object; and "r2" is shorthand for the distance squared between the centers of mass within object one and object two. And the "G"? Well friends, that's the Big G: The gravitational constant.


The Precise Value of Gravitational Force

"For any two masses, be they bowling balls or planets, the gravitational force between them is determined by their masses, their distance and the number G," says Mack. Thanks to experiments conducted by Henry Cavendish in the 1790s, we now know the gravitational constant has the numerical value of around 6.67 x 10 -11 Newtons (m2/kg2).

In this context, the term "Newtons" refers to a unit of measurement. One Newton is the amount of force you'll need to accelerate something with the mass of 2.2 pounds (1 kilogram) at 3.28 feet (1 meter) per second. (Like Anders Celsius and Charles F. Richter, Sir Isaac Newton earned a place on the vaunted list of scientists who've had units named in their honor.)


Cygnus X-1
Cygnus X-1 is a black hole about 15 times the mass of the sun in orbit with a massive blue companion star. Newton's description of gravity doesn't work for extremely strong gravity or very fast motion -- including black holes.

Newton vs. Einstein on Gravitational Force

Now, there's a layer of nuance we should acknowledge here. You see, the Law of Universal Gravitational Constant isn't quite as "universal" as the name implies. According to Battersby, "our classic picture of gravity" — which Newton articulated back in the 17th century — is "an accurate approximation of the reality of physics in most places in the entire universe (certainly on Earth)."

"However," she adds, "this theory has been superseded by Einstein's Theory of General Relativity, which is an improvement on 'Newtonian Gravity,' that posits that matter deforms space-time itself (like a heavy ball creating a dimple on a rubber sheet)."


That brings us to black holes. Capable of getting more than a million times more massive than our sun, black holes affect gravity in ways Newton's law just can't explain. General Relativity has been shown to make more accurate predictions about them.

"You have to start making corrections for the fact that Newton's description of gravity doesn't precisely work for extremely strong gravity or very fast motion," Mack says. "In those cases, we need to switch to Einstein's picture of gravity... But as long as you're not looking at one of those extreme cases, the equation that Isaac Newton wrote down in 1686 for what he called 'the Law of Universal Gravitation' truly is universal."

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Frequently Answered Questions

What is 9.8 m s2?
It is the acceleration due to gravity.