Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By MaxJasper

Math Help - Radius of Curvature

  1. #1
    Junior Member
    Joined
    May 2012
    From
    Bristol
    Posts
    29
    Thanks
    4

    Radius of Curvature

    I'm having some difficulties with this seemingly basic question:

    Find the radius of curvature at the point \left(\sqrt{2},2\right) on the curve with equation y=x^2.

    The steps I'm comfortable with are:

    \frac{\mathrm{d}y}{\mathrm{d}x}=2x

    and obviously the differential relations

    \frac{\mathrm{d}y}{\mathrm{d}x}=\tan\psi \qquad\frac{\mathrm{d}x}{\mathrm{d}s}=\cos\psi \qquad\frac{\mathrm{d}y}{\mathrm{d}s}=\sin\psi

    therefore

    \begin{align*} \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2&=4x  ^2\\ \left( \frac{\mathrm{d}s}{\mathrm{d}x}\right)^2&=1+ \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)^2 \end{align*}

    Given that \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 is 4y, I've considered proceeding using each of:

     s&=\int \left[ 1+ \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)^2\right]^{\frac{1}{2}} \mathrm{d}x

    and

    \begin{align*} s&=\int \left[ 1+ \left( \frac{\mathrm{d}x}{\mathrm{d}y} \right)^2\right]^{\frac{1}{2}} \mathrm{d}y \end{align*}

    but neither method seems to help me, as I'm not sure how to deal with the integral of the square root in either case (typically in these types of questions the square root can be eliminated, or it is in a simple form \sqrt{ax+b}).

    Any help on what I'm missing (I'm guessing its either an identity or substitution, but I can't see it) will be greatly appreciated (especially as a similar question is causing me the same basic problem!). Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member MaxJasper's Avatar
    Joined
    Aug 2012
    From
    Canada
    Posts
    482
    Thanks
    54

    Lightbulb Re: Radius of Curvature

    Use this for radius of curvature:

    \rho =|\frac{\left(y'^2+1\right)^{3/2}}{y''}| = \frac{27}{2}
    Last edited by MaxJasper; September 5th 2012 at 03:00 PM.
    Thanks from britmath
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Radius of Curvature
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 11th 2010, 12:26 PM
  2. What is the instantaneous radius of curvature?
    Posted in the Math Topics Forum
    Replies: 0
    Last Post: November 11th 2009, 03:37 AM
  3. nstantaneous radius of curvature
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 5th 2009, 12:09 AM
  4. Radius of curvature problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 14th 2009, 02:48 AM
  5. radius of curvature
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 8th 2007, 11:07 AM

Search Tags


/mathhelpforum @mathhelpforum