The Formula for the Force of Gravity
Newton rightly saw this as a confirmation of the "inverse square law". He proposed that a "universal" force of gravitation F existed between any two masses
m and M, directed from each to the other, proportional to each of them and inversely proportional to the square of their separation distance
r. In a formula (ignoring for now the vector character of the force):
F = G mM/r2
Suppose M is the mass of the Earth, R its radius and m is the mass of some falling object near the Earth's surface. Then one may write
F = m GM/R2 = m g
g = GM/R2
The capital G is known as the constant of universal gravitation. That is the number we need to know in order to calculate the gravitational attraction between, say, two spheres of 1 kilogram each. Unlike the attraction of the Earth, which has a huge mass M, such a force is quite small, and the number G is likewise very, very small. Measuring that small force in the lab is a delicate and difficult feat.
It took more than a century before it was first achieved. Only in 1796 did Newton's countryman Henry Cavendish actually measure such weak gravitational attraction, by noting the slight twist of a dumbbell suspended by a long thread, when on of its weights was attracted by the gravity of heavy objects. His instrument ("torsion balance") is actually very similar to the one devised in France by by Charles Augustin Coulomb to measure the distance dependence of magnetic and electric forces. The gravitational force is much weaker, however, making its direct observation much more challenging.
A century later (as already noted) the Hungarian physicist Roland Eötvös greatly improved the accuracy of such measurements.
Gravity in our Galaxy (Optional)
Gravity obviously extends much further than the Moon. Newton himself showed the inverse-square law also explained Kepler's laws--for instance, the 3rd law, by which the motion of planets slows down, the further they are from the Sun.
What about still larger distances? The solar system belongs to the Milky Way galaxy, a huge wheel-like swirl of stars with a radius around 100,000 light years. Being located in the wheel itself, we view it edge-on, so that the glow of its distant stars appears to us as a glowing ring circling the heavens, known since ancient times as the Milky Way. Many more distant galaxies are seen by telescopes, as far as one can see in any direction. Their light shows (by the "Doppler effect") that they are slowly rotating.
Gravity apparently holds galaxies together. At least our galaxy seems to have a huge black hole in its middle, a mass several million times that of our Sun, with gravity so intense that even light cannot escape it. Stars are much denser near the center of our galaxy, and their rotation near their center suggests Kepler's third law holds there, slower motion with increasing distance.
The rotation of galaxies away from their centers does not follow Kepler's 3rd law--indeed, outer fringes of galaxies seem to rotate almost uniformly. This observed fact has been attributed to invisible "dark matter" whose main attribute is mass and therefore, gravitational attraction (see link above). It does not seem to react to electromagnetic or nuclear forces, and scientists are still seeking more information about it.
A site about the story that Newton's inspiration about the force of gravity came from observing an apple drop from a tree.
A detailed article: Keesing, R.G., The History of Newton's apple tree, Contemporary Physics, 39, 377-91, 1998
Richard Feynman's calculations can be found in the book "Feynman's Lost Lecture: The Motion of Planets Around the Sun" by D. L Goodstein and J. R. Goodstein (Norton, 1996; reviewed by Paul Murdin in Nature, vol. 380, p. 680, 25 April 1996). The calculation is also described and expanded in "On Feynman's analysis of the geometry of Keplerian orbits" by M. Kowen and H. Mathur, Amer. J. of Physics, 71, 397-401, April 2003.
An article in an educational journal about the subjects discussed above: The great law by V. Kuznetsov. Quantum, Sept-Oct. 1999, p. 38-41.
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