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Math Help - Plate dynamics (quadratic equation)

  1. #1
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    Plate dynamics (quadratic equation)

    Hi, I've got a small problem with a quadratric equation I can't seem to be able to re-arrange to give me the correct answer. It's probably quite simple but I can't make it work!

    So the following equation relates horizontal force (P) on a plate (as in tectonic plate) with it's elastic rigidity D, and the densities of mantle (rho_m) and water (rho_w) giving displacement w.

    D(d^4w/dx^4)+P(d^2w/dx^2)+(rho_m-rho_w)gw = 0

    by finding the 2nd and forth differentials of

    w(x) = w_o * sin(2pi/lambda) *x

    you can turn the top equation into:

    D(2pi/lambda)^4 -P(2pi/lambda)^2 + (rho_m - rho_w)g = 0

    Now, this is a quadratic in (2pi/lambda)^2. You're supposed to be able to rearrange this quadratic to give

    P^2 >= 4D(rho_m-rho_w)g

    but I just can't get there!

    Any ideas? It's probably really simple but I've tried a hundred times and it just isn't clicking.

    Thanks!
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  2. #2
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    Quote Originally Posted by chambochae View Post
    Hi, I've got a small problem with a quadratric equation I can't seem to be able to re-arrange to give me the correct answer. It's probably quite simple but I can't make it work!

    So the following equation relates horizontal force (P) on a plate (as in tectonic plate) with it's elastic rigidity D, and the densities of mantle (rho_m) and water (rho_w) giving displacement w.

    D(d^4w/dx^4)+P(d^2w/dx^2)+(rho_m-rho_w)gw = 0

    by finding the 2nd and forth differentials of

    w(x) = w_o * sin(2pi/lambda) *x

    you can turn the top equation into:

    D(2pi/lambda)^4 -P(2pi/lambda)^2 + (rho_m - rho_w)g = 0

    Now, this is a quadratic in (2pi/lambda)^2. You're supposed to be able to rearrange this quadratic to give

    P^2 >= 4D(rho_m-rho_w)g

    but I just can't get there!

    Any ideas? It's probably really simple but I've tried a hundred times and it just isn't clicking.

    Thanks!
    You're missing the point of what they want you to do here. You are to find a condition on the variables, not represent the equation:
    D \left ( \frac{2 \pi}{\lambda} \right ) ^4 -P \left ( \frac{2 \pi}{\lambda} \right ) ^2 + ( \rho_m - \rho_w)g = 0

    This is, indeed, a quadratic in \left ( \frac{2 \pi}{\lambda} \right ) ^2. So define z = \left ( \frac{2 \pi}{\lambda} \right ) ^2. Thus
    Dz^2 - Pz + ( \rho_m - \rho_w)g = 0

    In order for this equation to have real solutions for z (which is required since I would presume that \lambda is a real number), we require the discriminant to be non-negative:
    P^2 - 4D(\rho_m - \rho_w)g \geq 0

    Your condition follows from this.

    -Dan
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  3. #3
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    Ah I see! Thanks ever so much!
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