Newton's Law of Gravitation

We already know about Newton's three laws of motion, the second of which is . These three laws describe motion in general. They never refer to a specific force. Newton however did also study in detail a specific force, namely gravity. He conjectured that the gravitational force between two bodies of mass m₁ and m₂ whose centers are separated by a distance of r has a magnitude of

The minus sign tells us that the force points inwards. The value of G was determined later in 1798 by Cavendish. It's value measured today is

G = 6.67 × 10^-11Nm² kg^-2

However, it is interesting that today the gravitational constant is the least accurately known of all the fundamental constants. For instance, its most accurately known value is actually G = (6:67259±0.00085)×10^-11 Nm²kg^-2 whereas for example the charge of the electron is (1:60217733 ± 0.00000049) × 10^-19 Coulomb or the speed of light is 299 792 458 m sec^-1 which are known much more accurately than G. Another example is the strength of the electrical force, called the fine structure constant, α^-1 = 137:0359895 ± 0.0000061.

Note: Now the vector form of Newton's Law is

where  is a unit vector point from one mass out to the other. The gravitational force is an inward force and that's why the minus sign appears.