What would the force of gravity be on an astronaut who is 12, 800 kilometers above earths surface?
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The earth's mass can be considered to be at its centre. The force between two bodies is given by F=GMm/x^2 (by Newton) where M and m are the masses and x is the distance between them. G is a constant of proportionality. If M=mass of the earth and m the mass of the astronaut, F=GMm/R^2 where R=radius of the earth. On the earth's surface we can write F=mg where g is the acceleration of gravity, so GMm/R^2=mg and g=GM/R^2. The gravitational force at a distance 12800 above earth's surface is g'=GM/(R+12800)^2, so g'/g=R^2/(R+12800)^2, and g'=gR^2/(R+12800)^2.

R=6371 km and g=9.81 m/s^2; g'=9.81*6371^2/19171^2=1.083 m/s^2 approximately or about (1/9)g.

On another matter, 1/6 of 12,800 km=2,133km approx.

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