# Modern Physics/The Law of Gravitation

## Law of Gravitation

Of Newton's accomplishments, the discovery of the universal law of gravitation ranks as one of his greatest. Imagine two masses, M1 and M2, separated by a distance r. The gravitational force has the magnitude

$F_g = G \frac{M_1 M_2}{r^2}$

where G is the gravitational constant:

$G \approx 6.67 \times 10^{-11} \frac{m^3}{kg \cdot s^2}$

The force is always attractive, and acts along the line joining the centre of the two masses.

## Vector Notation

Let's say that we have two masses, M and m, separated by a distance r, and a distance vector R. The relationship between R and r is given by:

$|\vec{\mathbf{R}}| = r$

We will also change our force into a force vector, acting in the direction of R:

$\vec{F}_g = G \frac{M_1 M_2}{r^2} \cdot \frac{\vec{\mathbf{R}}}{r}$

And this gives us our final vector equation:

$\vec{F}_g = G \frac{M_1 M_2 \vec{\mathbf{R}}}{r^3}$

Notice that since the ratio between R and r is normalized, the addition of these terms does not alter the equation, only the direction in which the force is acting.