In Angular Velocity and Angular Momentum Miles claims the current definitions of angular velocity and angular momentum are wrong.
To start with, look again at the basic equations
p = mv
L = rmv
Where L is the angular momentum. This equation tells us we can multiply a linear momentum by a radius and achieve an angular momentum. Is that sensible? No. It implies a big problem of scaling, for example. If r is greater than 1, the effective angular velocity is greater than the effective linear velocity. If r is less than 1, the effective angular velocity is less than the effective linear velocity. How is that logical?
First of all momentum and angular momentum are vectors and the correct definitions should read and . Miles took the absolute value of the angular momentum while tacitly assuming that (as in circular motion). It escapes us how he constructs his “problem of scaling”, but it seems he didn’t notice that and are different quantities. We now go straight to the part of the paper where a serious error occurs.
The correction for all this is fairly simple, although it required me to study the Principia very closely. We need a new equation to go from tangential or linear velocity to ω. Newton does not give us that equation, and no one else has supplied it since then. We can find it by following Newton to his ultimate interval, which is the same as going to the limit. We use the Pythagorean Theorem. As t→0,
ω2 → v2 – Δv2
and, v2 + r2 = (Δv + r)2
So, by substitution, ω2 + Δv2 + r2 = Δv2 + 2Δvr + r2
Δv = √ v2 + r2) – r = ω2/2r
ω = √[2r√v2 + r2) – 2r2]
r = √[ω4/(4v2 – 4ω2)]
This is Miles’ replacement for the current defintion of angular velocity. Apparently he got and from Newton’s principia. It suffices to say that the units in the last equation do not match and it is therefore wrong.
Some days ago Miles added a section where he is supposedly clarifying the problems with units. He argues that he simply redefined to be an acceleration. Redefine he did (according to the highlighted text), but is still not an acceleration. It is a quantity which has no physical meaning, because the squares of two different quantities are added.