Question

During a balloon ascension, wearing an oxygen mask, you measure the weight of a calibrated 2.00-kg mass and find that the value of the gravitational field strength at your location is 9.792 n/kg. how high above sea level, where the gravitational field strength was measured to be 9.803 n/kg, are you located?

Answer 1

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