PS-5.10  Explain how the gravitational force between two objects is affected by the mass of each object and the distance between them.
Key Concepts
Newton ’s Law of Universal Gravitation
  Newton's Law of Universal Gravitation states: " All masses attract each other with a force that is proportional to their masses and inversely proportional to the square of their distances. "
This sounds really complicated, but it is actually quite simple, and it includes two precepts. The first precept states that the greater the mass the more gravitational attraction and object has for other objects. The second precept states that gravitational attraction between masses decreases as the distance between them increases.
One common misconception is that gravity attracts objects to the Earth. In actuality, while the gravity of the Earth does attract other objects, other objects also attract the Earth.  Newton's law of gravitation states that all masses attract all other masses.
The First Precept:
In Newtonian physics, gravity is proportional to an objects mass. This means that the more mass an object contains, the greater its gravitational attraction for other objects. On a graph of gravitational attraction vs mass it would look like this:
diagram A
This is also illustrated by diagram "B".
Diagram B
In diagram "B" above, the more massive object, "B" has a greater gravitational attraction for other objects than "A" does. Since gravitational attraction is proportional to mass, object "B" has twice the gravitational attraction for object "A" than object "A" does for "B". This is clearly illustrated by the Earth and Moon. The Earth with a large mass has much more gravitational attraction for objects than the Moon, with its smaller mass, does. This is why you would only weigh 1/6 as much on the Moon as you do on Earth. Here are some other illustrations that show the proportional relationship between mass and gravitational attraction.
         Diagram C
In diagram "C" mass 1 and mass 2 attract with a force of 1 force unit.
      Diagram D
In diagram "D" the two masses are attracted by a force equal to 2 force units because the mass of M2 has been increased by a factor of 2.
   
   Diagram E
In diagram "E" the masses are attracted by a force equal to 3 force units because the mass of M2 has been increased by a factor of 3.
Diagram F
In diagram "F" the masses are attracted by a force equal to 4 force units because the mass of M1 and M2 have been increased by a factor of 2.
The second Precept:
The second precept states that the force of gravitational attraction between masses is inversely proportional to the square of their distances. In simple terms this means that the force of attraction decreases the further away masses are from each other. The fact that the force of attraction decreases as the square of the distances is called the inverse square law. The following illustrations show this relationship.
Diagram G
In diagram "G", masses M1 and M2 are separated by a distance, "d". The force between the masses is 1Force unit.
Diagram H
In diagram "H" above, the distance between the masses has been increased by a factor of 2. This has the effect of decreasing the attractive force between them to 1/4 the original force of attraction. For example, if the original force of attraction was 4N in diagram "G" then doubling the distance reduces the force of attraction to 1/22 the original value (or 1/4th the original value),  or 1N.
Diagram I
In diagram "I" above, the distance between the masses has been increased by a factor of 3. This has the effect of decreasing the attractive force between them to 1/9 the original force of attraction. For example, if the original force of attraction was 4N in diagram "G", then doubling the distance reduces the force of attraction to 1/32 the original value (or 1/9th the original value),  or .44N.