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Universal Gravitation
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 Nm2kg-2
The value of G has been found, by experiment, to be very small: G = 6.7 × 10-11 Nm2kg-2

This reminds us that gravitational forces tend to be pretty small compared with other forces (electro-magnetic, 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 re-write 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...
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