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The Universal Law of Gravitation

By a process of logic that I will not attempt to describe, Newton deduced that the force  F  between two objects with masses  m  and  M  separated by a distance  r  was given by

$$F = {GmM \over r^2}$$ (10.3)

where   $G = (6.67259 \pm 0.00085) \times 10^{-11}$ m$^3\cdot$kg $^{-1}\cdot$s^-2  is the Universal Gravitational Constant. Actually, Newton didn't know the value of G; he only postulated that it was universal - i.e. that it was the same constant of proportionality for every pair of masses in this universe. The actual determination of the value of G was first done by Cavendish in an experiment to be described below.

We should also express this equation in vector form to emphasize that the force on either mass acts in the direction of the other mass: if   $\vec{\mbox{\boldmath$\space F $\unboldmath }}_{12}$  denotes the force acting on mass  m₂  due to its gravitational attraction by mass  m₁  then

$$\vec{\mbox{\boldmath$ F $\unboldmath }}_{12} = - \, {G \, m . . . . . . er r_{12}^2} \; \hat{\mbox{\boldmath$ r $\unboldmath }}_{12}$$ (10.4)

where   $\hat{\mbox{\boldmath$\space r $\unboldmath }}_{12}$  is a unit vector in the direction of   $\vec{\mbox{\boldmath$\space r $\unboldmath }}_{12}$,  the vector distance from m₁ to m₂, and  r₁₂  is the scalar magnitude of $\vec{\mbox{\boldmath$\space r $\unboldmath }}_{12}$. Note that the reaction force   $\vec{\mbox{\boldmath$\space F $\unboldmath }}_{21}$  on m₁ due to m₂ is obtained by interchanging the labels "1" and "2" which ensures that it is equal and opposite because   $\vec{\mbox{\boldmath$\space r $\unboldmath }}_{21} \equiv - \vec{\mbox{\boldmath$\space r $\unboldmath }}_{12}$  by definition.