Download Jim - IEEE Standards working groups

Clearing up some confusion about angle and the radian
I don't think SI-10 can adequately address the confusion surrounding plane (and solid)
angle and the “disappearing-and-reappearing” radian, but I think that the “SI-10
community” should be aware that there is a very straightforward fix that would
completely eliminate the confusion—outlined below. Given that major changes to the SI
are scheduled to be adopted three years from right about now, perhaps people might start
thinking about how the well-documented problems with the SI (mis)treatment of angle
might be fixed. [There is a CCU working group on angles and dimensionless quantities
in the SI (CCU-WGADQ); but, according to their mission, they're mostly concerned
about avoiding errors of 2π in calculations involving periodic phenomena, and with
introducing new “units” to represent dimensionless “items” and “counts.”]
In the well-known formula for a circular sector, “s = r θ,” and all the other
formulas stemming from this involving rotational-to-translational (and vice versa)
geometry, we are told (in fact, warned!) that the “angle θ” must be expressed in
radians. [Similarly for the arguments of circular functions.] Unfortunately, this
“instruction” is incomplete—in fact, we must express the angle in radians and then use its
numerical value. This is equivalent to using the alternate formula: s = r θ/rad, where  is
the actual angle. [And sin(θ/rad), etc., in trigonometry.] In other words, in the formula
“s = r θ,” we are not using the (real physical) angle, , but rather the dimensionless
number of radians in the angle, θ/rad. [The dimensionless ratio, /rad, is (correctly)
called the “radian measure of .” The “” appearing in “s = r θ” is actually the radian
measure of the real angle, , i.e., θ/rad.]* The SI base unit is then the number of (real
physical) radians in one (real physical) radian, i.e., the radian measure of one radian,
(1 rad)/rad—which, of course, is identically equal to 1. This is very confusing, and leads
to a lot of gibberish—as a glance at any textbook or on-line tutorial will confirm.
The obvious question is: why not use the correct formula, s = r θ/rad, to begin with?
1
The current (mis)treatment of angle by the SI dates from when Anders Thor and
Erik Rudberg and colleagues sketched out the idea on the back of a dinner bill at a pizza
restaurant in Copenhagen in 1978; after considerable debate throughout the alphabet-soup
of subcommittees of the BIPM, it was finally adopted by the SI several years later. In my
view, it is well past the time that the (obvious physical quantity) angle should be treated
as a bona fide (eighth) base physical quantity with its own independent dimension, angle
(symbol A), and base unit radian. It has been conceptualized as a physical quantity (with
a status comparable to that of time and length) for millennia—easily conceptualized by
young infants (who can match cut-outs of similar triangles without knowing anything
about numbers), and more easily conceptualized than “mass” and other physical
quantities by many adults. Everybody still (really) thinks of angle this way today—I
don't know anybody who (really) thinks that an angle of a square corner is the
number 1.570 796 . . . .
********
Let the dimension “angle” be represented by A. From the axioms of Euclidean space, we
know that the arc-length, s, of a plane circular sector is proportional to the radius, r, and
the vertex angle, θ, i.e., proportional (not necessarily equal) to their product:
(1)
s = r θ/k
where k is a reference constant to be determined. Since s and r both have the dimension
L, the reference constant must have the dimension A. We can find this reference angle, k,
in terms of an easily conceptualized defining constant by setting s equal to the
circumference of the circle, 2πr, and θ correspondingly equal to one full revolution, rev,
giving k = rev/(2π) in the denominator of the right-hand side of (1). So we now have:
(2)
s = r θ /[rev/(2π)]
Clearly, rev/(2π) is the Reference Angle Denominator Intrinsically Appearing Naturally
in relating s to r and θ. Since this is a bit of a mouthful, we use the acronym RADIAN—
shortened to RAD for Reference Angle Denominator and written in lower case as rad.
2
So the fundamental relationship is:
s = r θ /rad
(3)
where dimension(s) = L, dimension(r) = L, dimension(θ) = A, and dimension(rad) = A.
So this relationship is dimensionally consistent—and independent of the unit used to
express θ.
To understand what the reference angle denominator, rad, looks like, we can set
s = r in the sector, giving the well-known geometrical construction for a (real physical)
radian. We can use rad as a suitable base unit for plane angle and define other angle
units in terms of rad, such as:
(4)
rev = 2π rad
(5)
deg = (π/180) rad
(revolution, dimension: A)
(degree, dimension: A)
and so on.
[Solid angle is a derived quantity, plane angle squared, with dimension A2. Its
unit, the steradian, is a coherent derived unit: sr = rad2.]
We note that the Thor-Rudberg (SI) interpretation of “plane angle” as s/r is, in
fact:
(6)
θ* = s/r = θ /rad
This is not plane angle at all, but rather the dimensionless “number of radians” in the
angle θ, or the “radian measure of .” And the SI base unit, which I will write as rad*, is
*(s = r), i.e., the “number of radians” in one radian (the “radian measure of one radian”):
(7)
rad* = (1 rad)/rad ≡ 1
The SI interprets this as m/m; but θ* can also be written in terms of the sector area, Asect =
sr/2, and the radius, r, as:
3
θ* = 2Asect/r2
(8)
(dimension: L2/L2)
When s = r, Asect = r2/2, so rad* = 2(r2/2)/r2 = r2/r2, so the SI radian, rad*, could also be
interpreted as m2/m2. This means that the “definition” of the SI radian as a “coherent
derived unit” is not unique! [Similarly for the steradian.]
********
The well-known “problems” associated with plane (and solid) angle and the “now-yousee-me-now-you-don’t” radian can easily be eliminated by:
(A)
Acknowledging that (plane) angle is a bona fide base physical quantity, with
dimension “angle” (symbol A).
(B)
Writing the rotational-to-translational relationship correctly as s = r θ/rad, where
(C)
The base unit rad is the angle corresponding to s = r, i.e., rad = rev/(2π), where
one revolution, rev, is an easily conceptualized “defining constant,” with dimension A.
(D)
Defining solid angle as the derived quantity (plane) angle squared, with
dimension A2, and with the coherent derived unit steradian, sr = rad2.
Then everything else falls naturally into place with dimensional and unit consistency.
Rotational-only quantities
Angular displacement:
θ (right-hand rule for vector direction) dimension: A
Angular velocity:
ω = dθ/dt
dimension: A/T
Angular acceleration:
α = dω/dt
dimension: A/T2
4
Rotational↔translational formulas
Circular sector:
(i)
s = r θ/rad
Differential vector form (note order of factors):
(ii)
ds = d(θ/rad) × r
where r is a position vector relative to an origin anywhere along the rotation axis.
Tangential velocity:
(iii)
v = ds/dt = [d(θ/rad)/dt] × r = (ω/rad) × r
Differential mechanical work
(iv)
dW = F ds = F d(θ/rad) × r = d(θ/rad)  r × F = d(θ/rad)  τ
where τ is the applied torque.
Transferred power:
(v)
P = dW/dt = (ω/rad)  τ
Trigonometric derivatives—geometric-construction proof follows from (i):
(vi)
dsin(θ/rad)/d(θ/rad) = cos(θ/rad)
Polar form of complex number:
(vii)
z = r exp(i θ/rad)
5
********
In the meantime, simply replacing “θ” in the “well-known” (but incorrect) formulas by
θ/rad (and “ω” by ω/rad, etc.) and treating the radian as a real physical angle—thereby
guaranteeing unit and dimensional consistency—would go a long way toward clearing up
the widespread confusion.
Benny.
__________
*The “specified-unit measure” of a quantity is a dimensionless ratio: the quantity divided by the
specified unit. For example, if we have a mass m = 5 kg, the kilogram measure of m is m/kg = 5;
this is the number of kilograms in m. For an angle  = 1.5 rad, the radian measure of  is θ/rad =
1.5; this is the number of radians in .
6