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Natural Turning Radius due to Stagger

The term stagger refers to the use of two different size tires/wheels on the rear axle of a car to improve the ability to turn on an oval track.  The term "Natural Turning Radius" refers to the radius that this combination would make if it were allowed to roll down the track by itself, with no other factors affecting it's direction.   Assuming r1 and r2 are the radius of the two tires, and y is the vehicle track, the natural turning radius may be expressed as:

This formula is derived as follows:

Assuming that the two wheels fit into an imaginary cone, the natural turning radius (x) may be calculated.  The radius of the inner (r1) and outer (r2) wheels are separated by the vehicle track (y).   Working in two dimensions, two similar triangles can formed by the inner and outer wheels, and the vertex of the natural radius.   Therefore the following proportional formula is valid:   

  Solving for x we get:   





Example:   On a vintage sprint car, assuming the inner tire is a 600-16 (27.91”) and the outer is 650-16 (29.38”), and the track is 60”.   Applying the formula we get 1140”, or approximately 95 foot natural turning radius. 

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