The Law of Gravitation

Newton's Law of Universal Gravitation states that the gravitational force that one object exerts on another object is proportional to the masses of the objects and inversely proportional to the square of the distance between them.

You may ask why exactly it decreases with the square of the distance as opposed to simply the inverse. One way to think about it is in terms of particles called gravitons, which are hypothetical particles that carry the gravitational force. Gravitons spread themselves out uniformly in 3-dimensional space from a point of mass. The force of gravity is proportional to the density of the gravitational flux, or number of gravitons flowing through an object, so it decreases by the inverse square for reasons of geometry. As the radius of the spehere of gravitons emanating from a source increases, the surface area increases with the square of that, leaving fewer gravitons to occupy a specific point in space. (In fact, all of the fundamental forces decrease with the inverse of the square of distance for a similar reason.)

Moreover, the force vector points in the direction of the first object, and the force is always attractive. However, this is so well-known that most people do not bother to express gravity as a vector anyway.

More concisely, $$F\propto\frac{m_1m_2}{d^2}$$ where $F$ is the magnitude of the gravitational force, $m_1$ and $m_2$ are the two masses, and $d$ is the distance between the two objects. The symbol $\alpha$ means "is proportional to."

To calculate the force, we need to find the constant factor that links the two terms in the proportion. Many years of observation and calculation have provided us with the universal gravitational constant, commonly abbreviated $G$: $$G\approx 6.67384\times 10^{-11} \frac{m^3}{kg\cdot s^2}$$ Notice that this constant is incredibly small; this is why you don't feel yourself attracted to objects around you.

Also notice that this constant is not dimensionless; the units are required to make the equation dimensionally correct.

Now, we may write the complete equation for Newton's Law of Universal Gravitation. If $\hat{r}_{21}$ is the unit vector pointing toward object $1$ from object $2$, then the force that object $1$ exerts on object $2$, denoted $\vec{F}_{12}$, is: $$\vec{F}_{12}=\frac{Gm_1m_2}{d^2}\hat{r}_{21}$$

It is well-known that we can draw an analogue between gravitational and electrostatic fields, as the forces given are both inverse square laws as follows: $$F_g = G\frac{m_1m_2}{r^2} \text{and} F_E = \frac{1}{4\pi\epsilon_0} \frac{q_1q_2}{r^2}$$

A quick computation tells that the electrostatic force is much stronger than the gravitational force (just try substituting in the values of both forces for a proton and an electron), yet the gravitational force dominates on the macroscopic scale (think about planets, stars and galaxies).

Why is this so?

This is because electrostatic forces can be attractive or repulsive, whereas the gravitational force is purely attractive. On a large scale, the electrostatic forces "cancel" out whereas the gravitational forces do not and add up instead.