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Basic electrostatics problem; analytical solution?

  1. Feb 3, 2017 #1
    1. The problem statement, all variables and given/known data
    Capture.JPG
    For the system given, both objects have the same charge and same mass (both given). I'm also given string length, L. I need to solve for θ.

    2. Relevant equations
    Coulomb's Law, W=mg

    3. The attempt at a solution
    Using simple equilibrium force analysis (with weight, tension, and electrostatics forces), I get:
    Capture1.JPG
    Is there a way to solve for θ analytically, or do I have to find a graphical solution?
    Thank you!
     
  2. jcsd
  3. Feb 3, 2017 #2

    gneill

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    Staff: Mentor

    It might be manipulated into the form of a cubic equation. Cubics have a closed form solution for their roots. It'll involve a couple of changes of variables I think. Try expressing cos(θ) in terms of sin(θ), and then call x = sin(θ) so you're working with x rather than trig.
     
  4. Feb 3, 2017 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    The equation ##\sin^3(\theta)/\cos(\theta) = a## is not difficult to solve analytically. For ##a > 0## we have ##0 < \theta < \pi/2##, so ##\cos(\theta) = \sqrt{1 - \sin^2(\theta)} > 0##. Therefore, the new variable ##x = \sin^2(\theta)## obeys the cubic equation ##x^3/(1-x) = a^2##. The exact solution of this cubic is not too complicated or difficult to work with. From ##x## we can recover ##\theta = \arcsin(\sqrt{x})##.
     
    Last edited: Feb 3, 2017
  5. Feb 6, 2017 #4

    andrevdh

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    Homework Helper

    I get that
    sin(θ) cos(θ) = kq2/4L2mg
    ?
    which, if correct, then comes to

    ½ sin(2θ) = kq2/4L2mg
     
  6. Feb 7, 2017 #5
    I believe this is incorrect. I'm pretty sure of the solution I posted, since it does agree with computational results.
     
  7. Feb 7, 2017 #6
    Thank you so much!
    I went down this path and saw the enormously complexity of solving cubics, which is - I guess - why we're taught to use graphical solutions methods instead.

     
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