This is a simple application of Newton's Law of Universal Gravitation.
The force of gravity is inversely proportional to the distance between the two
centers of mass. We don't need to know the mass of earth or the test mass to
solve this, because we'll be setting up a proportionality, which means that all
controlled variables can be expressed as a proportionality constant which will
eventually cancel.
Set up the following proportionality equation from
Newton's Universal Gravitation:
F = k/d²; where k is the constant of proportionality
Plug in values for F and d, making two
equations:
(9.803 N) = k/r²; where r is the radius of earth,
and
(9.792 N) = k/(r+h)²; where h is the height above
sea level.
Divide one by the other, and you get:
9.803 / 9.792 = (r+h)² / r²; the k cancels
Solve for h:
√(9.803 / 9.792) = (r+h) / r
r √(9.803 / 9.792) = r + h
r √(9.803 / 9.792) – r = h
Look up the value for r (radius of earth) and
evaluate:
(6371 km) √(9.803 / 9.792) – (6371 km) = h
h ≈ 3.58 km
