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Algebraic Manipulation

  1. Mar 1, 2007 #1
    1. The problem statement, all variables and given/known data
    Hey I need to rearrange the following, and find x in terms of G, M, m, r
    [tex]\frac{{GM}}{{x^2 }} = \frac{{Gm}}{{\left( {r - x} \right)^2 }}[/tex]

    2. The attempt at a solution

    I haven't manged to get far with this problem as I am confused about the powers of x and how to manage them. This is where I have manged to get to:
    [tex]
    \begin{array}{c}
    \frac{{GM}}{{x^2 }} = \frac{{Gm}}{{\left( {r - x} \right)^2 }} \\
    x^2 \left( {r - x} \right)^2 = GM\left( {Gm} \right) \\
    x^2 \left( {r^2 - 2rx + x^2 } \right) = G^2 Mm \\
    r^2 x^2 - 2rx^3 + x^4 = G^2 Mm \\
    \end{array}
    [/tex]

    Any help is greatly appreciated, many thanks in advance,
    unique_pavadrin
     
  2. jcsd
  3. Mar 1, 2007 #2

    cristo

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    Science Advisor

    What've you done in your first line? i.e. how does the original equation become x2(r-x)2=GM(Gm) ?
     
  4. Mar 1, 2007 #3
    I have an idea.. multiply both sides by (1/G) or divide by G..
    Then you'd get mx^2=M(r-x)^2 and work from there.
     
  5. Mar 2, 2007 #4

    danago

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    Gold Member

    Could use the quadratic formula
     
  6. Mar 2, 2007 #5
    Thanks cristo for having pointed out that stupid mistake.Pugfug90, your method doesn't seem to work, but thanks anyhow. Danago, thanks for your suggestion, as i have used it. Here is what i have managed to come up with:

    [tex]
    \begin{array}{l}
    \frac{{GM}}{{x^2 }} = \frac{{Gm}}{{\left( {r - x} \right)^2 }} \\
    Gmx^2 = GM\left( {r - x} \right)^2 \\
    Gmx^2 = GM\left( {r - x} \right)\left( {r - x} \right) \\
    Gmx^2 = GM\left( {r^2 - 2rx + x^2 } \right) \\
    Gmx^2 = GMr^2 - 2GMrx + GMx^2 \\
    - GMr^2 = - 2GMrx + GMx^2 - Gmx^2 \\
    GMr^2 = 2GMrx - GMx^2 + Gmx^2 \\
    0 = \left( {Gm - GM} \right)x^2 + 2GMrx - GMr^2 \\
    x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}} \\
    x = \frac{{ - 2GMr \pm \sqrt {\left( {2GMr} \right)^2 - 4\left( {Gm - GM} \right)\left( {GMr^2 } \right)} }}{{2\left( {Gm - GM} \right)}} \\
    \end{array}
    [/tex]

    thanks once again for the help from those who replied
     
  7. Mar 2, 2007 #6

    cristo

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    Why not cancel the G on both sides in the first line? There's no need to carry it through the calculation then.
     
  8. Mar 2, 2007 #7
    oh true, thanks
    other than that are my steps right?
    thanks
     
  9. Mar 2, 2007 #8

    cristo

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    Not quite:
    You missed a minus sign in the last line: it should read (without the G's)[tex]x=\frac{-2Mr\pm\sqrt{4r^2M^2+4(m-M)Mr^2}}{2(m-M)}[/tex]
     
  10. Mar 2, 2007 #9
    kill the G!
     
  11. Mar 3, 2007 #10
    okay thanks cristo, that was great help thanks
    unique_pavadrin
     
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