What is the expression of gravitational acceleration at an altitude h when h < R and g is gravitational acceleration on the surface of the planet R radius of the planet?

Answer the following question in detail.

Show that acceleration due to gravity at height h above the Earth’s surface is `"g"_"h" = "g"("R"/"R + h")^2`

Answer the following question in detail.

Discuss the variation of acceleration due to gravity with altitude.

Let, R = radius of the Earth, M = mass of the Earth.
g = acceleration due to gravity at the surface of the Earth.
Consider a body of mass m on the surface of the Earth. The acceleration due to gravity on the Earth’s surface is given by,`"g" = "GM"/"R"^2` ....(1)
The body is taken at height h above the surface of the Earth as shown in the figure. The acceleration due to gravity now changes to,
`"g"_"h" = "GM"/("R + h")^2` .....(2)
Dividing equation (2) by equation (1), we get,`"g"_"h"/"g" = ("GM"/("R + h")^2)/("GM"/"R"^2)``therefore "g"_"h"/"g" = "R"^2/("R + h")^2``therefore "g"_"h" = "gR"^2/("R + h")^2`We can rewrite,`therefore "g"_"h" = "gR"^2/("R"^2(1 + "h"/"R")^2)`
`therefore "g"_"h" = "g"(1 + "h"/"R")^-2`
For small altitude h, i.e., for `"h"/"R"` << 1, by neglecting higher power terms of `"h"/"R"`, `"g"_"h" = "g"(1 - "2h"/"R")`

This expression can be used to calculate the value of g at height h above the surface of the Earth as long as h << R.

Concept: Variation in the Acceleration Due to Gravity with Altitude, Depth, Latitude and Shape

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In this post, we will discuss and derive the formula for the acceleration due to gravity at height h (where h<<Radius of the earth). As height or altitude from the earth’s surface increases the value of g falls. Now we are to discuss and derive the equation of g that describes this change in g with the increase in height.

The formula for the acceleration due to gravity at height h

The formula for the acceleration due to gravity at height h (where h<<R) is expressed by the formula:
g1 = g (1 – 2h/R).Here g1 is the acceleration due to gravity at height h and R is the radius of the earth.g denotes acceleration due to gravity on the earth’s surface.

For example, considering g = 9.8 m/s^2 on the earth’s surface, g1 at a height of 1000 meters from the surface of the earth becomes 9.7969 m/s^2. [ check with online calculator ]