Newton's Law of Gravitation
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Key Questions

The force of gravity (which is not the same as little g  the acceleration due to gravity on earth) is calculated using:
# F = ("Gm"_1*m_2)/(R^2)# F is the force, and it is always an attractive force (objects attract all of the other objects around them).
G is a constant that is the same anywhere in the universe (at least we think so).
#m_1 and m_2# are the masses of the two objects.R is the radius (distance between the centers of the two objects).

Newtons Law of Gravitation states that the Gravitational force between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Therefore,
#Fpropm_1*m_2#
and,
#Fprop1/r^2# From the rules of joint proportionality we have
#Fprop(m_1*m_2)/r^2# Thus
#F=G(m_1*m_2)/r^2# , where#G# is the Universal Gravitational constant. Its value is#6.67*10^11 m^3 kg^1 sec^2# 
As planets move around the sun, the gravitational force pulls the planets toward the sun (and also the sun toward the planets), therefore the gravitational force of attraction is providing the centripetal, or centerseeking, force that is required for an object to move in a circular path.
The following videos provide a bit more detail and examples:
Kepler's Laws: http://socratic.org/physics/circularmotionandgravitation/keplerslaws/apphysicscorbits
Newton's Law of Universal Gravitation states that there is a force of mutual gravitational attraction between any 2objects in the universe. The magnitude of this force is directly proportional to the product of the 2 masses and inversely proportional to the square of the distance between their centres.
#F=G(m_1m_2)/r^2#
This force by the sun on a planet provides the required centripetal force to maintain the planet in a circular motion around the sun.
#G(m_1m_2)/r^2=(mv^2)/r# However, this theory is made more accurate and precise according to Kepler's 1st Law of Planetary motion, which states that all planet moves in elliptical orbits with the sun at one focus.

In a stronger gravitational field a given mass will have a larger weight.
An object's weight is given by this equation:
#w=mg# Gravitational field strength at a distance, r, from a centre of mass is given by this equation:
#g=(GM)/r^2# If an object is nearer a centre of mass then r will be smaller so g will be larger. In that case the weight of the object will be larger. If an object is moved from a distance R away from one centre of mass and placed a distance R away from the centre of a smaller mass then g will be smaller. In that case the weight of the object will be smaller.
A practical example: if an astronaut on earth has a weight of 833 N, on the moon he/she will have a weight of 136 N.
Using
#m=w/g# the mass of the astronaut is#833/9.8=85 kg# .So the weight on the moon (
#g_M=1.6 Nkg^1# ) is:#w_M=mg_M=85Ã—1.6=136 N#