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How do I Taylor expand the gravitational field in terms of h/R <<1?
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[QUOTE="spacetimedude, post: 5456521, member: 418958"] [h2]Homework Statement [/h2] 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. [h2]Homework Equations[/h2][h2]The Attempt at a Solution[/h2] 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. [/QUOTE]
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Introductory Physics Homework Help
How do I Taylor expand the gravitational field in terms of h/R <<1?
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