The intuition of the universal gravitation law, that is the law describing the
attractive force that acts between two any bodies,was enunciated by Newton.
In accordance with a well known anecdote, such a genial intuition was originated from the abrupt awakening of Newton, who was struck by an apple, while blissfully was sleeping under a tree.
Apart from the anecdote, Newton understood the existence of a universal attractive force,which is responsible for the free fall of bodies on the Earth, and even for the orbital motion of the Moon around the Earth and of the planets around the Sun.
He reached so to the unification of the laws of both the terrestrial and celestial mechanics.
The intensity of the universal attractive force F, the so-called gravity force, between two any masses, M1 and M2 , kept at a distance R each other, is directly proportional to the product M1 M2 and inversely proportional to the square of their distance: F = GM1M2/R2 .
The proportionality constant G is the so-called universal gravitation constant, that may be determined by measuring the extremely weak gravitational force acting between two masses , kept at a certain distance each other.
The measure of the extremely little value of G
( G ~= 6,67 *10-11 newton*metro2/kg2 ) was made for the first time in 1798 by Lord Cavendish, by means of a special torsion balance he invented to perform a so difficult measure.
We consider that the attractive force between two bodies is very little,even if their mass values are equal to hundreds of kilograms.
To explain the great intensity of the gravity force that attracts the bodies toward the Earth center, we must consider the extremely great mass of the Earth.
For the first time Lord Cavendish was able to calculate the mass MT of the Earth, considering our planet as a solid sphere with a radius
RT ~= 6400 km and a uniform density, and assuming that the gravity acceleration g of any body with mass m ,freely falling on the ground from a not too much altitude (less than 1 Km), is equal to the one that would be produced by a pointlike mass MT placed in the Earth center, then at a distance R from the body:
g = F/m = P/m = (weight)/(mass) = (G m MT /RT2)/ m = G MT/RT2 ~= 9,81 meters/seconds2.
Therefore MT = gRT2/G ~= 6 * 1024 kg ( 6 millions of billions of billions of kilograms).
Then, the bodies for which the effects of all the other external forces can be considered very small, they fall freely in close proximity of the Earth ground, with a gravity acceleration g, which is independent on their mass m, because the gravity force (the weight) that produces their free fall, is directly proportional to their mass.
This property is typical only of the gravity force, whose value is directly
proportional to a particular "fundamental charge of Nature" which is common to whole matter and is known as the gravitational mass.
This is the mass that generates the gravitational phenomena and it is numerically identical to the inertial mass, that instead is the mass (inertia) offering a resistance to the acceleration produced by a force.
The famous experiences performed by Galileo by making the bodies fall freely from the
summit of the Pisa tower and moreover by using an inclined plane, brought it to deduce the
independence of the gravity acceleration value ( g ~= 9,8
metri/secondo2 ) on the mass m of a freely-falling
body, provided it is possible to think is negligible the deceleration produced by the
aerodynamic forces (the so-called air resistance).
If the experiments were effected in vacuum,for example by making two bodies with different masses fall freely in a glass pipe (the so called Newton' pipe ), from which the air has been extracted by a vacuum pump, it would be possible to verify the equality of the free-fall times in the field of gravity.
We remember, in this respect, the experiment performed on the lunar ground by the American astronauts of one of the Apollo missions in the early 1970s.
In that occasion, after they had verified the equality of the free-fall times of two bodies with different masses in the vacuum of the lunar environment, they exclaimed: "You were right, Mister Galileo".
We can verify that by means of an experiment, that consists in making two bodies, for example,two small spheres with equal radii and different masses,one of iron and another of plastic material, fall freely from a falling height h, to measure their free-fall times.
In this case the aerodynamic resistance is the same for both spheres, because they have the same surface.
Then they experience the same aerodynamic effects,so that their fall times are equally changed.
Because the motion of the spheres is naturally accelerated, the relation between their free-falling height h and their free-fall time Tc is:
h = ( 1/2 ) a Tc2, from which we can get Tc by extracting the square root of the expression 2h/a.
From having verified the identity of their free-fall times Tc, it is deduced the gravity accelerations a of both bodies are equal.
If instead, to underline the effect of the air resistance,we repeated the experiment using two spheres with the same mass, but having different radii,for example with a 1/10 ratio, we could verify that the fall time would be greater in the case of the sphere with a greater radius.
This sphere indeed, for having a surface 100 times greater than the one of the other, is subjected to an aerodynamic resistance 100 times greater than the other, and then would fall with a smaller resultant acceleration.
In this case the Galilean independence principle of the acceleration a of a body on its mass m isn't valid, because the body is subjected both to the gravity force and to the aerodynamic resistance:
a = [F(gravity) - F(aerodynamic)]/m = g - F(aerodynamic)/m , which is the smaller with respect to g, the greater is the aerodynamic resistance.