Derive the equation for variation of g due to height from the surface of earth.

Explanation

`implies` Consider a point mass m at a height h above the surface of the earth as shown in figure. This body placed at point P from the distance `h (r gt R_E)` from the surface of earth.
`implies` Radius of earth is `R_E`
`implies` Its distance from the centre of the earth is `r = R_E+h`
`implies` The magnitude of force on the body
`F(h)=(GM_Em)/((R_E+h)^2)`
`:. (F(h))/m=(GM_E)/(R_E+h)^2`
`:.`   Acceleration due to gravity at height h from the surface of earth,
`g(h) = (GM_E)/(R_E+h)^(2) " ".....(1)`
`implies` Acceleration due to gravity on the surface of earth,
`g(R_E) = (GM_E)/R_E^2" "....(2)`
`implies` Taking ratio of equation (1) and (2),
`(g(h))/(g(R_E))=(GM_E)/(R_E+h)^(2)xx(R_E^2)/(GM_E)`
`:. g(h)=g((R_E^2)/((R_E+h)^2))" " [ :. g(R_E) = g ]`
`:. g(h)=g""(1)/((1+h/R_E)^2) " ".....(3)`
`= g (1+h/R_E)^(-2)`
`:. g(h) = g (1 - (2h)/(R_E))" "....(4)`
( `:.` Taking two terms in using binomial expression)
`implies` Equation (4) tells that for small heights h above the value of g decreased by a factor ` (1-(2h)/R_E)`
`implies` Equation (3) can be used for any height, while equation (4) used only for `h lt lt R_E`