Speed at the top of an elliptical roller coaster loop

[JessicaHelena]

Homework Statement

This isn't really a homework but a project I'm doing that's due soon. In our multivariable calculus class, we're creating a 3D roller coaster, and I need to explain the physics behind the roller coasters.

For a roller coaster loop, if it were perfectly circular, we would have a minimum speed of $v_{min} = \sqrt{gR}$ at the top of the loop where $g=9.8 m/s^2$ and $R$ is the radius of the 'circle'. However, most roller coaster loops are actually not circular but more elliptical. I've been looking for ways to calculate the min. speed at the top for an elliptical loop, but so far I haven't been able to. How could I go about that?

Homework Equations

F_net = ma_c

The Attempt at a Solution

I really didn't know how to do this, so I searched google, but as far as I can see, there aren't any explanations for this...

 

[gneill]

An ellipse is not a circle, but it will have a radius of curvature associated with each point of the curve. You could use the radius of curvature at the top as you would the radius of a circle. Your task will be to find the radius of curvature for the ends of the ellipse.

Start by googling "ellipse radius of curvature".

 

[haruspex]

An ellipse is not a circle, but it will have a radius of curvature associated with each point of the curve. You could use the radius of curvature at the top as you would the radius of a circle. Your task will be to find the radius of curvature for the ends of the ellipse.

Start by googling "ellipse radius of curvature".

Right, but it raises the possibility that the top of the loop is not the only concern.

 

[gneill]

Right, but it raises the possibility that the top of the loop is not the only concern.

Indeed it does. We also don't know how the ellipse is oriented: major axis vertical? Horizontal? Something else?. I've been assuming vertical.

The curvature can be found for any point along the curve, so it might be worth checking. The curvature at the ends of the major axis is a very simple expression as it turns out (also for the ends of the minor axis). I expect it to be much more hairy elsewhere. Finding a general expression for the track normal force around the ellipse might be doable. But I don't think I'm going to do it

 

[CWatters]

Apparently the shape they use is called a clothoid. Its designed to reduce the Max g passengers experience.

Edit: Google found... http://physics.gu.se/LISEBERG/eng/loop_pe.html