# Non-Uniform Circular Motion

If an object travels an elliptical path about a point, say a roller coaster with an elliptical bottom, how would one computer the radius of that path when applying the equation: $F = mv^2/r?$ Would it just be the average radius along that elliptic curve?

The radius of curvature of the ellipse varies from point to point. It is not the distance to the center of the ellipse. You need to know the parameters of the ellipse (the two semi-axes) and where on the elliptical path is the point you are interested in. The easiest way to find the formula for the radius of curvature would (in my opinion) to use the parametric equations of the ellipse $$x= a \sin \theta$$ $$y= b \cos \theta$$ and use the formula for radius of curvature $$\rho= \frac{(y'^2 +x'^2)^{3/2}}{x'y''-x''y'}$$ where x',y' are first derivative in respect to $\theta$ and x",y" are the second derivatives. If you get a negative value for the radius you just take the absolute value. You will get a formula for the radius as a function of the polar angle $\theta$. At the two extreme points on the long axis $\theta= 0 \ or \ \pi$ and at the two extreme points on the short axis $\theta= \pi/2 \ or \ 3\pi/2$.