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Question
Explain the variation of g with altitude.
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Solution
Consider an object of mass m at a height h from the surface of the Earth. Acceleration experienced by the object due to Earth is
g’ = `("GM")/(("R"_"e" + "h")^2)`
Mass at a height h from the center of the Earth
g’ = `("GM")/("R"_"e"^2 (1 + "h"/"R"_"e")^2)`
g’ = `("GM")/("R"_"e"^2) (1 + "h"/"R"_"e")^-2`
If h << Re
We can use Binomial expansion. Taking the terms up to first order
g’ = `("GM")/("R"_"e"^2) (1 - 2"h"/"R"_"e")`
g’ = `"g" (1 - 2"h"/"R"_"e")`
We find that g’ < g. This means that as altitude h increases the acceleration due to gravity g decreases.
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