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Electric force question

  1. Aug 23, 2012 #1
    First of all, hey everyone.
    I was wondering if any of you could help me solve this. I've tried but i keep getting insane equations so I might be doing something wrong.

    Two equal spheres with a mass of m have equal charges q. They're suspended by two ropes with length L in points distanced d apart. Calculate the distance between the (centers) of the two spheres when this setup is in balance.
    Picture: https://dl.dropbox.com/u/29642931/phys.png [Broken]



    What I've tried so far is:
    The forces acting on the ball are the electric forces from the charges, the gravity and the tension frmo the rope. I figured the sum of these forces should be 0 for this setup to be balanced, which led me to: Fg + Fel+Frope with |Frope| = |Fg|/cos(θ) + |Fel|/sin(θ), but this is insanely hard to solve..
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Aug 23, 2012 #2

    Doc Al

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    Staff: Mentor

    Good.
    I don't understand that last equation. What's the direction of the electric force?
     
  4. Aug 23, 2012 #3
    The charges are equal, so the electrical forces act 'outwards', right?
    I got that last equation by fiddling a bit, I'm not too sure how to find the rope tension force..
     
  5. Aug 23, 2012 #4
    use the diagram to find the forces at equillibrium
     

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  6. Aug 23, 2012 #5

    Doc Al

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    Right, which is horizontal.
    Redo it by considering horizontal and vertical components separately.
     
  7. Aug 23, 2012 #6
    Using this diagram, I get the conditions:
    |Fel|= q²/(4π*ε*r²) = |T|sin(θ)
    |Fg| = mg = |T|cos(θ)

    And I'm stuck here: how do I find r (or θ?) from this?

    Edit : Dividing top by bottom I get : tanθ = q²/(4π*ε*r²*m*g)
    I can write r = d + 2Lsin(θ) -> r² = d²+4dLsinθ + 4L²sin²(θ)
    If I denote q²/(4π*ε*m*g) = C (for my ease of writing mainly), I get:
    C*cotθ = d²+4dLsinθ + 4L²sin²(θ), which I can then solve for θ (somehow)?
     
    Last edited: Aug 23, 2012
  8. Aug 23, 2012 #7
    you can try to eliminate T , so that you are left with only one variable (θ)
     
  9. Aug 23, 2012 #8

    Doc Al

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    I assume you are given θ and L, and must solve for d in terms of θ and L.
     
  10. Aug 23, 2012 #9
    What I assume is L,d,q and m are given. (d being the distance between the fixpoints of the two ropes). I then have to solve for the distance r between the centers of the spheres (which can be found by finding θ.)
     
  11. Aug 23, 2012 #10

    Doc Al

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    Ah, OK. Do you need to solve it analytically, or are you given values for those givens? (Which you can then plug into a smart calculator, which can solve for θ.)
     
  12. Aug 23, 2012 #11
    No values are given, so analytically I suppose. But I get the feeling this spits out a very ugly equation if I do it exactly..
    So maybe the answer to this question just is "For a θ which satisfies this equation, the distance is given by r = d + 2Lsinθ"?

    Or, maybe, if I assume θ is small, I can get a neat approximate solution?
    Maybe something like: C*cotθ = (d+2Lsinθ)² ~ d² for small θ, so θ ~ arctan(C/d²) = arctan(q²/(4π*ε*m*g*d²))?
     
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