How Do You Calculate Velocity on an Elliptical Path at a Specific Point?

[pollytree]

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

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!

 

[rl.bhat]

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

 

[pollytree]

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.

 

[rl.bhat]

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

 

[rwisz]

Thank you for sharing your attempt at a solution. It is important to first understand the equations and concepts involved in order to approach a problem like this. It seems like you have correctly identified the equation for an ellipse and have used differentiation to find the derivative and second derivative of the ellipse equation. However, it is not clear how you have used these values to find the radius of curvature.

To find the magnitude of velocity at a specific point on the ellipse, we need to use the equation for acceleration, a=\frac{dv}{dt} \vec{e}t+\frac{v^2}{\rho} \vec{e}n, where \rho is the radius of curvature. In this case, we know the constant speed of 10m/s and the value of x at the point of interest (x=0.6m). We can use these values to solve for the magnitude of velocity, v, at that point.

It is also important to note that the given ellipse equation assumes that the motion is in the x-y plane, so the velocity vector will also be in this plane. This means that the acceleration vector will be tangent to the ellipse at the point of interest.

In summary, to find the magnitude of velocity at x=0.6m, we can use the given constant speed of 10m/s and the radius of curvature at that point, which can be found using the equations you have provided. I hope this helps and good luck with your calculations.