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Taylor expanding equation

  • #1

Homework Statement


Consider the position vector of a mass m at height h above the Earth's surface to be [itex] \underline{r}=(R+h)\underline{e}_z [/itex] where R is the radius of the Earth.
Make a Tylor expansion in h/R <<1 of the gravitational field
[tex] \underline{G}(\underline{r})=\frac{-GM\underline{r}}{r^3}[/tex]
to show that the gravitational force on a mass m can be written as [tex] \underline{F} \approx mg\underline{e}_z[/tex] giving an expression for g in terms of R and M, the mass of the Earth.
Find the first correction (h-dependent) to the gravitational force.


Homework Equations




The Attempt at a Solution


I am having trouble understanding how to expand the function in h/R<<1.

We can rewrite [tex] \underline{G}(\underline{r})=\frac{-GM(R+h)}{(R+h)^3}\underline{e}_z[/tex] which then can then be expressed as [tex]\underline{G}(\underline{r})=\frac{-GM}{R^2}(\frac{h}{R}+1)^{-2}\underline{e}_z[/tex]
I am not sure how to proceed with the expansion.
Any explanation will be appreciated.
 

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
12,393
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Let ##x = h/R##. You need to make a Taylor series expansion of the function ##f(x) = (x+1)^{-2}##.
 

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