Problem: Use proportionality to find a formula of the form g = k/r ^ 2 for the gravitational field strength of the Earth at distance r from its center, assuming that r is at least as great as the 6400 km radius of the Earth. Give the proportionality constant k, then give the field strength at distances 2.000 * 10 ^ 4, 2.000 * 10 ^ 5 and 2.000 * 10 ^ 6 kilometers from the center of the Earth.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Solution: We assume the proportionality g = k/r ^ 2. Substituting the known field strength g = 9.8 m/s ^ 2 at distance 6400 kilometers, we obtain 9.8 m/s ^ 2 = k / (6400 km) ^ 2. Solving for k we obtain k = 4.014 * 10 ^ 6 km m / s ^ 2. This gives us the equation g = [ 4.014 * 10 ^ 6 km m / s ^ 2] / r ^ 2.
Substituting r = 2.000 * 10 ^ 4 kilometers we obtain
g = 4.014 * 10 ^ 6 kg m /s ^ 2 / ( 2.000 * 10 ^ 4 km) ^ 2 = .1003 m/s ^ 2.
We substitute the other two distances and obtain .0001003 m/s ^ 2 and 1.003 * 10 ^ -6 m/s ^ 2.
[Note that we could have found these results by using the fact that the second radius is 10 times that of the first and the third is 100 times that of the first. A little reflection tells us that the spheres over which the gravitational effect is spread will have areas 100 and 10000 times as great as that of the first. The fields will therefore have 1/100 and 1/10000 times the strength of the first. These results coincide with the results of substitution.]