The expression e p = mgh applies only close to the surface of the earth. the general expression for the potential energy of a mass m at a distance r from the center of the earth (of mass m e ) is e p = − g m e m / r . write r = r e + h , where r e is the radius of the earth; show that, when h ≪ r e , this general expression reduces to the special case, and find an expression for g. you will need the expansion ( 1 + x ) − 1 = 1 − x + … .

g = G mE /Re^2

Explanation:

Hello!

First of all we need to find an equation where we can apply the series expansion of (1+x)^-1 where the variable x be a small number.

Since we are considering that  h<<Re  this implies that (h/Re)<<1 so this is the small number we are looking for.

Then in the equation for the earth's potential we can take of the earths radius as:

Now we can define x=h/Re and use the series expansion. Therefore:

Since x is a small number, x^2 will be even smaller, tehrefore we will only consider the first power of x:

and:

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