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Thin Rod Gravitational Potential and Field Vector

  1. Jun 19, 2015 #1
    1. The problem statement, all variables and given/known data
    I am having a hard time understanding where to begin with this problem. Here it is:

    Consider a thin rod of length L and constant density n that lies on the x-axis with endpoints at x=0 and x=L.
    (i) Find a formula for the gravitational potential Φ = Φ(x) at the (variable) point P located on the positive x-axis at coordinate x>L.
    (ii) Find the gravitational field vector E=E(x) = -E(x)i at the point P and show that the gravitational intensity is E(x)=m/x(x-L), where m is the mass of the rod.


    2. Relevant equations

    Total mass due to mass density μ over Volume V m=∫μ dv

    Potential Due to Mass Density η over Curve C Φ=∫η/r dl

    3. The attempt at a solution
    I do not know how to start a solution because first I am confused as to which equation I would use and then second I get confused as to how I find the variables. I have done this problem with a circular disk instead of a thin rod and it was easier because I could find r but now I am lost.
     
  2. jcsd
  3. Jun 19, 2015 #2

    LCKurtz

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    Consider a segment of the rod of length ##ds## located at ##(s,0)## on the rod, where ##s## is between ##0## and ##L##. What is its mass? What force is exerted at the point ##(x,0)## by that segment? Then add up (integrate) that over the length of the rod and see what you get.
     
  4. Jun 19, 2015 #3
    deleted because of formatting
     
  5. Jun 19, 2015 #4
    Okay so if I am looking at s the mass would be ∫(ρ)∫(θ) η/√(x^2+s^2) dΘdρ ?
    And then just split it up and integrate? Also would the bounds be (0,L) for ρ and (0, 2π) for Θ?

    Am I on the right track?
     
  6. Jun 19, 2015 #5

    LCKurtz

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    No. You have introduced a new variable (what is ##\theta##?) and you didn't answer either of my questions. Try again:
    1. What is the mass of that ##ds## segment?
    2. What force is exerted at the point ##(x,0)## by it?
    Then we can talk about the integral.
     
  7. Jun 22, 2015 #6
    The reason why I added θ because I thought I had to put it into spherical coordinates. I am honestly so confused I do not know where to go with this.
     
  8. Jun 22, 2015 #7

    Ray Vickson

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    The question said to find the force at a point on the x-axis. So, you have a simple 1-dimensional problem.

    If you had been asked for the gravitational force (or potential) at a general point (x,y,z), then it might be the case that spherical or polar coordinates would be helpful---it might also not be the case. First set up the required integration, then decide what type of coordinates would make the problem easiest.
     
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