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:

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

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:

$|\overrightarrow{𝐑}|=r$

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

${\overrightarrow{F}}_g=G\frac{M_1{M}_2}{r^2}\cdot \frac{\overrightarrow{𝐑}}{r}$

And this gives us our final vector equation:

${\overrightarrow{F}}_g=G\frac{M_1{M}_2\overrightarrow{𝐑}}{r^3}$

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.