In order to attempt to explain the observed motions of the moon around the
earth, the planets around the sun, as well as the motion of bodies falling
near the earth's surface, Isaac Newton
made a bold suggestion. 

He proposed that every object attracts every other object with a force
which he called gravitation. 



He further suggested that the strength of this force depends on two things: 

1. the masses of the bodies 

2. the distance between their centres of mass. 



His ideas are summarized by what is now called Newton's law of universal
gravitation: 

Every body attracts every other body with a force which is directly
proportional to the product of their masses and inversely
proportional to the square of the distance between their
centres of mass. 




To see why Newton thought the square of the distance is important,
click here. 



Stated mathematically we have 


and 



These two statements can be combined into one proportionality 



and an equation can be obtained by inserting a constant of proportionality,
G 



G is called the universal gravitational constant. 

The units of G (found by rearranging the equation to
have G by itself) are Nm^{2}kg^{2} 

The value of G has been found, by experiment, to be very small: G = 6.7 × 10^{11}
Nm^{2}kg^{2} 



This reminds us that gravitational forces tend to be pretty small compared
with other forces (electromagnetic, nuclear etc) unless at least one of the
masses involved is very big. 



Relation between the Acceleration due to Gravity, g and the Universal Constant of Gravitation, G 

We know that when a body is allowed to fall freely, near the earth's
surface, it falls with acceleration, g (assuming that air resistance is
negligible). 

It should be clear that the value of g must depend on the value of the
universal constant of gravitation, G (because the strength of the force of
gravity in any given situation depends on G). 



From Newton's second law of motion, if a body of mass m has acceleration g
caused by a force F then 


equation 1 


Also, if the body is near the earth's surface, then we can rewrite the
gravitational equation above as 


equation 2 


where M is the mass of the earth and R is the radius of the earth. 



These two equations give us the relation between g and G 



in which we are not surprised to find that m, the mass of the falling body,
does not appear (experiments show that g is independent of m). 



It should be noted that in the above working we have
assumed that the "m" we wrote down in equation 1 and the "m" we wrote in
equation 2 both represent exactly the same value. 



In other words we have assumed that a mass appearing in Newton's second law
(a measure of inertia, so called the inertial mass of the body) has
exactly the same value as the gravitational mass appearing in the
Newton's universal gravitation equation. 



This has been verified by experiment to be true but it is not known
why this is the case... 
