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Homework Help: Motion Around An Ellipse

  1. Aug 20, 2010 #1
    1. The problem statement, all variables and given/known data

    There is an elliptical path and pegs A and B are restricted to move around it. If the link moves with a constant speed of 10m/s, determine the magnitude of velocity when x=0.6m

    [PLAIN]http://users.adam.com.au/shortround/Prob.12-78.jpg [Broken]

    2. Relevant equations

    [tex]\frac{x^2}{4}[/tex]+y2=1

    [tex]\rho[/tex]=[tex]\frac{(1+(\frac{dy}{dx})^2)^\frac{3}{2}}{\left|\frac{dy}{dx}\right|}[/tex]

    a=[tex]\frac{dv}{dt}[/tex] [tex]\vec{e}[/tex]t+[tex]\frac{v^2}{\rho}[/tex] [tex]\vec{e}[/tex]n

    Where [tex]\rho[/tex] is the radius of curvature.

    3. The attempt at a solution

    I rearranged [tex]\frac{x^2}{4}[/tex]+y2=1 to get x2+4y2=4

    I then differentiated this to get: [tex]\frac{dy}{dx}[/tex]=[tex]\frac{-x}{4y}[/tex] and [tex]\frac{d^2y}{dx^2}[/tex]=[tex]\frac{(x/y)-1}{4y}[/tex]

    Using x=0.6m, y=[tex]\sqrt{0.91}[/tex]=0.9539

    By substituting this into the derivative and second derivatives and then putting these into the radius of curvature equation, I found the radius of curvature. However I am not sure if this is the correct way to do it. Also once I have the radius of curvature, how do I find the velocity?

    Thanks!!
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 20, 2010 #2

    rl.bhat

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    Homework Helper

    Υou need not calculate the radius of curvature.
    When x = 0.6m find y using the equation of ellipse. If O is the center of the ellipse, find the angle AOC. Then the velocity of A at that position is v*sinθ
     
  4. Aug 20, 2010 #3
    That doesn't work. I have an example in my book with x=1.0m and hence y=sqrt(0.75). They give v=10.4m/s, however your method gives v=6.55m/s.
     
  5. Aug 20, 2010 #4

    rl.bhat

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    Homework Helper

    To find the angle, find the slope tanθ = dy/dx at x = 0.6 m, and then proceed.
     
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