Motion Around An Ellipse

  • Thread starter pollytree
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  • #1
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Homework Statement



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]

Homework 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.

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!!
 
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Answers and Replies

  • #2
rl.bhat
Homework Helper
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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
 
  • #3
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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.
 
  • #4
rl.bhat
Homework Helper
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To find the angle, find the slope tanθ = dy/dx at x = 0.6 m, and then proceed.
 

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