Answer:
There is a 7 % difference between the approximate gravitational force and the actual gravitational force at a height of 240 kilometers.
Explanation:
We can calculate gravity as a function of mass of the Earth ([tex]M[/tex]), measured in kilograms, radial distance from center of the planet ([tex]r[/tex]), measured in meters, and gravitation constant ([tex]G[/tex]), measured in newton-square meters per square kilogram:
[tex]g' = G\cdot \frac{M}{r^{2}}[/tex] (Eq. 1)
And we solve the equation for [tex]r[/tex]:
[tex]r =\sqrt{\frac{G\cdot M}{g'} }[/tex] (Eq. 1b)
If we know that [tex]G = 6.674\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}}[/tex], [tex]M = 5.972\times 10^{24}\,kg[/tex] and [tex]g' = 9.121\,\frac{m}{s^{2}}[/tex], the radial distance from the center of the Earth is:
[tex]r = \sqrt{\frac{(6.674\times 10^{-11}\,\frac{N\cdot m^{2}}{kg^{2}} )\cdot (5.972\times 10^{24}\,kg)}{9.121\,\frac{m}{s^{2}} } }[/tex]
[tex]r \approx 6.611\times 10^{6}\,m[/tex]
The height above the surface of the Earth is the difference between the value above and the radius of the planet. That is:
[tex]h = 6.611\times 10^{6}\,m - 6.371\times 10^{6}\,m[/tex]
[tex]h = 2.40\times 10^{5}\,m[/tex] ([tex]240\,km[/tex])
There is a 7 % difference between the approximate gravitational force and the actual gravitational force at a height of 240 kilometers.
