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Taylor expanding equation

  1. Apr 27, 2016 #1
    1. The problem statement, all variables and given/known data
    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.


    2. Relevant equations


    3. 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.
     
  2. jcsd
  3. Apr 27, 2016 #2

    TSny

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    Homework Helper
    Gold Member

    Let ##x = h/R##. You need to make a Taylor series expansion of the function ##f(x) = (x+1)^{-2}##.
     
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