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I just calculated the gravitational field on the $x$-axis created by a thin rod of mass $M$ and length $L$ by applying the formula:

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I got this:

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However there are some issues that I don't get. If I plug $ x = \pm L/2$ on the equation, gravitational field goes to infinity. Why? What's its physical meaning?

Another thing that bothers me is the fact that when I plug $x=0$ in the equation the result is $\vec{g}(x=0)= \frac{4GM}{L^2}\vec{i}$, but shouldn't it be 0 since $x=0$ is in the middle of the rod and by simmetry it looks like the net field right there is 0?

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  • $\begingroup$ I believe your problems are occurring since you are considering an infinitely thin rod, aka 1d, of you did an area or volume integral for the center it would be 0 $\endgroup$
    – jensen paull
    Commented Jan 24, 2022 at 23:38
  • $\begingroup$ Double check the expression you have for $\vec g$. $\endgroup$
    – joseph h
    Commented Jan 25, 2022 at 1:27

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careful -- you wrote down your $d\vec g$ as though it only points in the $+\hat i$ direction when that's not the case for $x<0$. you need to split up your integral. one interval goes from $-L/2$ to $0$ and the other from $0$ to $L/2$.

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    $\begingroup$ That worked! I first tried to compute the integral your way but I got that it diverged to infinity... Later I realized that I had to take Cauchy principal value in order to get the correct result and I got 0, so everything went right in the end :). Thanks a lot!! $\endgroup$
    – Javieer Picazo
    Commented Jan 25, 2022 at 15:34

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