# Newton's Law of Gravity

Newton's universal law of gravitation was first published by Isaac Newton in his Principia in 1687 and it states that

$$\vec{F}_g=G\frac{m_1m_2}{r^2}\hat{r}_{1,2},\tag{1}$$

where $$\vec{F}_g$$ is the gravitational force exerted by the mass $$m_1$$ on the mass $$m_2$$, $$r$$ is the separation distance between the two masses, $$G$$ is a constant called the gravitational constant, and $$\hat{r}_{1,2}$$ is a unit vector whose tail coincides with $$m_1$$ and whose arrow points at $$m_2$$. The gravitational constant was experimentally determined to be

$$G=6.67408(31)×10^{−11}m^3⋅kg^−1⋅s^−2.\tag{2}$$

According to Equation (1) and Newton's third law of motion, any pair of two objects that have masses exert the forces $$\vec{F}_g$$ and $$-\vec{F}_g$$ on each other. This is true for any pair of two objects, no matter how light they are and no matter how far apart they are. Two grains of dust on the opposite sides of the universe would have very small masses $$m_1$$ and $$m_2$$ and their separation distance $$r$$ would be vast; despite this, if you plugged those values into Equation (1), you would still get some nonezero value (albeit, it would be very tiny). Thus, according to Newton's law of gravity and third law, two grains of dust on the opposite sides of the universe are pulling on one another and attracted to one another. Since all of the galaxies, stars, planets, life, and grains of dust in the universe have mass, according to Newton's laws all of those things are pulling on each other. This must have been an astonishing realization for Newton: that everything in the universe is attracted to everthing else.

Shortly after the birth of the universe, all of the matter in the universe was distributed very smoothly. But there were slight non-uniformities in this distribution of matter: certain regions were denser than others. According to Newton's law of gravity, since the regions of space with denser matter distributions have more mass packed into them than the surround regions of space with comparitively rareified distributions of matter, it follows that the denser mass distributions will exert greater gravitational forces than the rarefied distributions of mass. Thus, the rareified distributions of matter will tend to be drawn to those denser regions. As time rolls forward, the dense regions will become denser and denser and the matter in the rareified regions will become more and more sparse. After many eons, the first generations of stars and galaxies were formed through this process. And smaller conglomerates of matter grew bigger and bigger—pulling in more and more matter—until eventually the first planets were formed.

Not only has Newton's law of gravity taught us a great deal about the processes which lead to the formation of galaxies, stars, and planets but it also had, together with Newton's laws of motion, taught us the fundamental nature of motion in the universe. These laws attempted to answer the question: why does everything move the way it does? Why, for example, if I threw a rock, it would trace out a parabolic trajectory and eventually fall to the ground? Why does the Moon revolve around the Earth? According to Newton's law of gravity and laws of motion, the rock falls towards the ground—but, counterintuitively, the Moon is also falling towards the ground. Both the rock and the Moon fall towards the ground for the same reason and due to the same cause: the mass of the Earth is very enormous and exerts a force on both the rock and the Moon which causes them to fall. But why, unlike the rock, does the Moon never actually hit the ground? The answer is because the Earth is so massive that only objects traveling at fantastically high speeds could move fast enough for the Earth to curve underneath them as they are falling thereby allowing them to avoid hitting the ground. But the strength of the gravitational force $$F_g$$ exerted by a world (like the Earth) depends on how massive the world is. Humans will likely one day live on Mars' moon Phobos. This world is very unmassive and thus exerts a much smaller gravitational force than does the Earth. Thus, an object falling towards Phobos could avoid hitting the ground by traveling at a much slower speed. According to Newton's laws, if you through a small rock in a straight line parallel to Phobos's surface, that tiny velocity would be sufficient for the rock to travel in a circle all the way around Phobos and hit you in the back of the head.

Why do the planets revolve around the Sun? Or more precisely, what causes them to go around the Sun? Rocks or even a big rock like the Moon can fall to the Earth because, according to the law of gravity, the mass of the Earth is so big that it exerts a vast force on rocks and the Moon causing them to fall towards the Earth. The Sun is millions of times more massive than the Earth; so massive that, according to the law of gravity, it causes all of the planets to fall towards it. And because the planets are moving at the right speed, they fall towards the Sun in the shape of a circle.

Newton's law of gravity and laws of motion have taught us some very surprising and unexpected things about the nature of motion and the universe. It is therefore little doubt that they are viewed as one of the greatest achievmenets in human thought of all time.