# 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.