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I've been dealing with a problem, that is depicted on the picture. I had to calculate the gravitational force exerted on a point mass m at coordinates (0,0,h) by a cylinder with radius R, height R and constant density ro0 as illustrated in the picture. Now I've calculated the force as presented, but what puzzles me now, is how I would calculate the limit of this force, when R->0. L'Hospital rule could only take me one step, and the expected solution is $F_g$ between two point bodies, which then means $F=GmM/{h^2}$. However, the $R^{-3}$ won't go away and is thus popping out the infinity.

All the answers are greatly appreciated.

Cheers!

Attempted solution

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1 Answer 1

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It works!

You need to expand your square roots binomially to third order in $\left(\frac{R}{h}\right)$. The first and second order terms cancel (if you've been careful with your algebra) and your big square bracket boils down to $h \times \frac{1}{2} \frac{R^3}{h^3}$, which is just what you need to give the expected limit.

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  • $\begingroup$ "to third order in $\frac{R}{h}$", right? $\endgroup$
    – secavara
    Jan 27, 2018 at 13:24
  • $\begingroup$ Indeed so. That is the case. $\endgroup$ Jan 27, 2018 at 14:22
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    $\begingroup$ Meant to write "in $\frac{R}{h}$ ". Now corrected. $\endgroup$ Jan 27, 2018 at 14:31

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