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Thread: radius of curvature

  1. #1
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    Question radius of curvature

    Find the radius of curvature of the parabola y^2 = 4px at (0,0).

    Please teach me how to solve this question. Thank you very much.
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  2. #2
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    Please check my work. The answer of this question is 2 lpl.
    I got the answer is 2p. Could you please help me find out the reason? Thank you very much.
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  3. #3
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    Quote Originally Posted by Jenny20 View Post
    Find the radius of curvature of the parabola y^2 = 4px at (0,0).

    Please teach me how to solve this question. Thank you very much.
    In this case take the curvature as:

    $\displaystyle
    \kappa=x''/(1+x')^{3/2}\,
    $

    where we consider $\displaystyle x\,$ as a function of $\displaystyle y\,$ and differentiation is with respect to $\displaystyle y\,$

    This has $\displaystyle x\,$ and $\displaystyle y\,$ interchanged compared to the usual expression but the absolute value of the curvature is the same either way, and this way is easier to handle.

    Here:

    $\displaystyle x=y^2/(4p)\,$

    so:

    $\displaystyle \frac{dx}{dy}=x'=y/(2p)\,$,

    and:

    $\displaystyle
    \frac{d^2x}{dy^2}=x''=1/(2p)\,
    $

    so at $\displaystyle (0, 0)\,$, $\displaystyle x'=0\,$, so:

    $\displaystyle
    \kappa=\frac{[1/(2p)]}{1}=1/(2p)\,
    $

    so the radius of curvature $\displaystyle r=|1/\kappa|=2|p|\,$.

    RonL
    Last edited by ThePerfectHacker; Feb 8th 2007 at 12:10 PM.
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