Problem: An asteroid of mass 8 x 1020 kg is traveling toward the Earth-Moon system along a line which will intersect an imaginary line joining the Earth and Moon. When the asteroid is 9 x 108 m from the Earth and the Moon, what is the gravitational field at this point?
Solution: (see the diagram) This is the same problem (mostly) as Example 1. The angle and the magnitudes of the forces are the same:
Now I use Newton's Gravitational Law:
where the subscript "am" represents "asteroid-moon". Likewise, for the asteroid and the earth,
The sum of these forces can be expressed as a magnitude and a direction, or (which is the same thing) in unit vector notation:
All I need to do to get the field is to divide this by the "test" mass; i.e., the mass of the asteroid:
This result is independent of the mass of the asteroid. It is the amount of newtons of force that would be exerted on a kilogram of mass placed where the asteroid is. You'll often use this concept of field in the Physics II course.