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Curvature problem

  1. Jul 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Given the polar function r = 4cos(3θ) find the curvature.



    2. Relevant equations



    3. The attempt at a solution

    I know there is a formula for curvature of a polar function but I was never given that equation and was told to convert to parametric. and use ||v x a|| / ||v||^3. So this is my work
    Converting to parametric:

    x = 4cos(3θ)cos(θ)
    y= 4cos(3θ)sin(θ)
    z = 0

    to get v i took the derivative

    x' = -12sin(3θ)cos(θ) - 4cos(3θ)sin(θ)
    y' = -12sin(3θ)sin(θ) + 4 cos(4θ)cos(θ)
    z' = 0

    I then took the second derivative to get a

    x'' = -40cos(3θ)cos(θ) + 24sin(3θ)sin(θ)
    y'' = -40cos(3θ)sin(θ) - 24sin(3θ)cos(θ)
    z'' = 0

    However I feel like i did something wrong because to computer ||v x a|| would be really messy. Is there any way to simplify before I do the cross product or is there is easier way and I'm just doing the problem wrong? Thanks any help is appreciated.
     
  2. jcsd
  3. Jul 8, 2012 #2

    tiny-tim

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  4. Jul 8, 2012 #3

    Filip Larsen

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    Gold Member

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