bugbear
Established Member
Jacob":2fu6a2r3 said:Oh you have got it!! Well done!!! Though I doubt it was proved by Archimedes.bugbear":2fu6a2r3 said:.....We have:
A) The effort required for a cut depends purely on the length of the cut.
We also have
B) The shape of cut that removes most material for a given effort is a semi-circle.
This follows nicely from (A) and was first proved by Archimedes. Very basic,
well established maths.
I got it first time; it's obvious. Google "isoperimetric" ; the greeks were all over this one.
If you bother to read what I wrote I said "in theory" and "it's not quite that simple" etc etcBut we also have
C) Factors other than the length of the cut effect the effort required.
...
But it does account for why a scrub plane removes more material faster for a given effort.
Maybe you have misunderstood what I meant by length of cut - I meant the length of the cutting edge.
No misunderstanding; that is the only interpretation where (B) makes any sense at all.
So a flat 1" chisel will require a similar effort to remove a thin flat shaving, as the same chisel bent into a gouge ploughing much more material from a semi circular trough. Not quite that simple - as I keep repeating, but approximately true in principle or theory.
Well, no. The distinction is expressed as clearly as I can in (A) and (C). If (A) is true, your theory holds. If (A) is false, your theory is not even true "in theory", let alone practice. In your 1" chisel example common workshop experience tells us that a 1" chisel taking a 1 thou shaving will require far less effort than your 1" gouge, so effort is overwhelmingly effected by factors other than length of cut. You can't just skate over this by saying "in theory".
If (for realistic example) depth of cut were to effect the effort required, it would follow ("in theory") that a shallower curve than a semi circle would be better, and this theory is a much better fit with reality. In this case, the maths to work out the exact optimal curve is rather complex, and even then, we'd still be assuming a homogenous workpiece, not wood with grain.
BugBear