Given,
Distance from the surface to the center of the earth, d=4000 miles
Distance from the center to you at a height of 8000 miles, a= 8000+4000=12000 miles
The gravitational force acting on a person at the surface is equal to his weight.
From Newton's Universal Law of Gravitation, the gravitational force is
[tex]F=\frac{G\times M\times m}{r^2}[/tex]Where G is the gravitational constant, M is the mass of the earth, m is the mass of the object/person, r is the distance between the center of the earth and the object/person
At the surface, this force is equal to the weight of the person, W=mg
i.e.
[tex]F_s=\frac{G\times M\times m}{d^2}=W[/tex]On substituting the of d,
[tex]W=\frac{\text{GMm}}{4000^2}[/tex]At a height of 8000 miles from the surface, the gravitational force is equal to,
[tex]F_a=\frac{GMm}{12000^2}[/tex]On dividing the above two equations,
[tex]\frac{F_a}{W}=\frac{4000^2^{}}{12000^2}=\frac{1}{9}[/tex]Therefore,
[tex]F_a=\frac{1}{9}W[/tex]Therefore at a height of 8000 miles above the surface of the earth, the force of gravity becomes 1/9 time your weight.
