Crazy Cross Products! Radial × Normal?

Question

Okay so I've been working for fun on the air resistance on a sphere with both translational and rotational motion.

Note: I'm using ∂ to denote "d" as in a differential quantity to try to make it a little more readable. I don't mean partial differential (I don't think), I mean normal...

I've taken the equation from this:

F=½*ρ*C*v^2*A

where F and v are vectors, and A is the projectional area in the direction of the velocity...

to this:

∂F=(½*ρ*C*|v+ω×r|) ∂A•(v+ω×r)

where F, v, ω, r, and A are all vectors. (ω is a pseudo-vector)

I'm trying to solve for F. I've got 2 problems:

1. The equation seems wrong to me because when the differential area and velocity are off at an angle of greater than 90°, the differential force should be 0, as it is the inside of the surface that would be experiencing drag, but it cannot, because it is the inside of the surface and therefore doesn't exhibit air resistance.

2. How can I do a cross product between the radius, a radial vector, and another normal vector?

Thanks! If you need more info, just tell me.

Answer 1

An ingenious idea but it won't work.

The equation you started with is the equation for "form drag" which does not include rotation:

http://en.wikipedia.org/wiki/Drag_(physi...

With rotation comes lift (hence curve balls, etc.) which generates lift drag:

http://en.wikipedia.org/wiki/Induced_dra...

Since the form drag equation uses only the cross-sectional area, I don't think it includes skin friction either (a ball and a round nosed bullet with the same cross-sectional area will have quite different skin friction drags).

http://en.wikipedia.org/wiki/Skin_fricti...

Basically, the Rayleigh equation is an empirical simplification, and you are trying to take it beyond its limits of applicability.