Answer:
[tex]mg=200.4 N[/tex].
Explanation:
This problem can be solved using Newton's law of universal gravitation: [tex]F=G\frac{m_{1}m_{2}}{r^{2}}[/tex],
where F is the gravitational force between two masses [tex]m_{1}[/tex] and [tex]m_{2}[/tex], [tex]r[/tex] is the distance between the masses (their center of mass), and [tex]G=6.674*10^{-11}(m^{3}kg^{-1}s^{-2})[/tex] is the gravitational constant.
We know the weight of the astronout on the surface, with this we can find his mass. Letting [tex]w_{s}[/tex] be the weight on the surface:
[tex]w_{s}=mg[/tex],
[tex]mg=8*10^{2}[/tex],
[tex]m=(8*10^{2})/g[/tex],
since we now that [tex]g=9.8m/s^{2}[/tex] we get that the mass is
[tex]m=81.6kg[/tex].
Now we can use Newton's law of universal gravitation
[tex]F=G\frac{Mm}{r^{2}}[/tex],
where [tex]m[/tex] is the mass of the astronaut and [tex]M[/tex] is the mass of the earth. From Newton's second law we know that
[tex]F=ma[/tex],
in this case the acceleration is the gravity so
[tex]F=mg[/tex], (becarefull, gravity at this point is no longer [tex]9.8m/s^{2}[/tex] because we are not in the surface anymore)
and this get us to
[tex]mg=G\frac{Mm}{r^{2}}[/tex], where [tex]mg[/tex] is his new weight.
We need to remember that the mass of the earth is [tex]M=5.972*10^{24}kg[/tex] and its radius is [tex]6.37*10^{6}m[/tex].
The total distance between the astronaut and the earth is
[tex]r=(6.37*10^{6}+6.37*10^{6})=2(6.37*10^{6})=12.74*10^{6}[/tex] meters.
Now we can compute his weigh:
[tex]mg=G\frac{Mm}{r^{2}}[/tex],
[tex]mg=(6.674*10^{-11})\frac{(5.972*10^{24})(81.6)}{(12.74*10^{6})^{2}}[/tex],
[tex]mg=200.4 N[/tex].
