If the acceleration due to gravity at a height h above the Earth's surface is given by g ′ = g ( 1 + R h ) − 2 , where R is the radius of Earth and g is the acceleration due to gravity on the s

Question

If the acceleration due to gravity at a height h above the Earth's surface is given by g ′ = g ( 1 + R h ​ ) − 2 , where R is the radius of Earth and g is the acceleration due to gravity on the surface, what is the approximate value of g ′ for h ≪ R ?

TestHub · Accepted Answer

The given expression is g ′ = g ( 1 + R h ​ ) − 2 . For h ≪ R , we can use the binomial approximation ( 1 + x ) n ≈ 1 + n x for ∣ x ∣ ≪ 1 . Here, x = R h ​ and n = − 2 . Therefore, ( 1 + R h ​ ) − 2 ≈ 1 + ( − 2 ) R h ​ = 1 − R 2 h ​ . So, g ′ ≈ g ( 1 − R 2 h ​ ) .

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