I agree with Captain Jerry's analysis. The distance between the lathe centers decreases as the offset is increased, the constant is workpiece length, which is the hypotenuse, with center distance being the cosine side and the offset being the sine side.
With this in mind, the algebra looks like:
Let L = workpiece length and t = taper per foot or t/12 inches per inch.
The taper for the half angle is half that (taper per 'side' rather than taper per diameter), so that the half angle of the taper = arctan t/24
If we agree that this is the angle that the workpiece must make with the lathe centerline, then,
Offset = L sin (arctan t/24)
We can make this even easier by dispensing with the trig altogether. For small angles (the half angles of Morse and B&S self-holding tapers are all less than 1.5 degrees - small enough), sin angle ~ tan angle, or, equivalently, arctan number ~ arcsin number. So,
Offset = L sin (arctan t/24) ~ L sin (arcsin t/24) = Lt/24
Example:
t = 0.6 inches per foot
L = 5 inches
Using the trig explicitly, Offset = 0.12496
The simplified way has Offset = 0.12500, an 'error' of 4 one hundred thousandths of an inch.
I'm trained in math and thus can make spectacular errors. What do y'all think?
Mark
Jeez, first post. Hi, Guys!