Confusion about the "now-you-see-me-now-you-don't" radian

[BP Leonard]

If we have a circle with radius r and a sector of the circle with central angle theta, the generally accepted formula for finding the corresponding arc-length of the sector is;

s = r theta

usually with the stipulation that "theta must be in radians." So let's say that r = 0.1 m and theta = 30 degrees. To find s, we first convert the angle to radians. Since 360 degrees = 2 pi radians (= one revolution), we have 30 degrees = pi/6 radians. Then, using the above well-known formula:

s = (0.1 m) (pi/6 rad) =(pi/60) m rad

Since we know that s is a length (to be expressed in metres), the "rad" in the answer seems anomalous. We therefore "get rid of rad"--using a variety of justifications and explanations. [Note that this is equivalent to dividing the post-calculation expression by 1 radian.] The (correct) answer is then s = pi/60 m.

The inverse problem--finding theta, given s and r--gives an angle that appears to be dimensionless. We must then "insert rad" in order to get the correct answer. [Equivalent to multiplying the post-calculation expression by 1 radian.] The angle (now expressed in radians) can then be converted to other angle units using well-known formulas.

This deletion or insertion of rad (however justified and explained) causes widespread confusion. The problem stems from the fundamental equation, "s = r theta," which is dimensionally inconsistent. A dimensionally consistent relationship must be written:

s = const (r theta)

where, if A represents the dimension "angle," then the constant must have the dimension 1/A. We can evaluate this constant by setting s = 2 pi r and theta = 1 rev (one complete revolution), giving:

const = (2 pi)/rev

Since rev/(2 pi) is one radian, rad, we see that const = 1/rad. Therefore, the dimensionally correct fundamental relationship is:

s = r theta/rad

where theta can be expressed in any angle units. In other words, we have (correctly) explicitly divided the right-hand side by rad before the calculation--rather than post calculation. Similarly for the inverse relationship, theta = (s/r) rad, where we have (correctly) multiplied the RHS by rad before the calculation--rather than post calculation.

The correct equation, s = r theta/rad and its differential vector form ds = (dtheta/rad) x r (noting the order of factors), has far-reaching consequences in rotational kinematics and dynamics, trigonometry, vibrations, and many other areas. In other words, if "rad" is correctly expressed in fundamental equations, it does not have to be deleted or inserted post calculation--thereby avoiding the multiple and varied (and very confusing) justifications and explanations that seem to abound in textbooks and on-line tutorials.

 

[Mark44]

If we have a circle with radius r and a sector of the circle with central angle theta, the generally accepted formula for finding the corresponding arc-length of the sector is;

s = r theta

usually with the stipulation that "theta must be in radians." So let's say that r = 0.1 m and theta = 30 degrees. To find s, we first convert the angle to radians. Since 360 degrees = 2 pi radians (= one revolution), we have 30 degrees = pi/6 radians. Then, using the above well-known formula:

s = (0.1 m) (pi/6 rad) =(pi/60) m rad

Radians are generally considered to be dimensionless, unlike degrees. The reason is that a radian is defined to be the ratio of two lengths -- the arc length along a circle divided by the length of the radius.
So s above is in units of meters, not meter-radians.

Since we know that s is a length (to be expressed in metres), the "rad" in the answer seems anomalous. We therefore "get rid of rad"--using a variety of justifications and explanations. [Note that this is equivalent to dividing the post-calculation expression by 1 radian.] The (correct) answer is then s = pi/60 m.

The inverse problem--finding theta, given s and r--gives an angle that appears to be dimensionless. We must then "insert rad" in order to get the correct answer. [Equivalent to multiplying the post-calculation expression by 1 radian.] The angle (now expressed in radians) can then be converted to other angle units using well-known formulas.

This deletion or insertion of rad (however justified and explained) causes widespread confusion. The problem stems from the fundamental equation, "s = r theta," which is dimensionally inconsistent.

Not if you take the usual position that radians are dimensionless.

A dimensionally consistent relationship must be written:

s = const (r theta)

where, if A represents the dimension "angle," then the constant must have the dimension 1/A. We can evaluate this constant by setting s = 2 pi r and theta = 1 rev (one complete revolution), giving:

const = (2 pi)/rev

Since rev/(2 pi) is one radian, rad, we see that const = 1/rad. Therefore, the dimensionally correct fundamental relationship is:

s = r theta/rad

where theta can be expressed in any angle units. In other words, we have (correctly) explicitly divided the right-hand side by rad before the calculation--rather than post calculation. Similarly for the inverse relationship, theta = (s/r) rad, where we have (correctly) multiplied the RHS by rad before the calculation--rather than post calculation.

The correct equation, s = r theta/rad and its differential vector form ds = (dtheta/rad) x r (noting the order of factors), has far-reaching consequences in rotational kinematics and dynamics, trigonometry, vibrations, and many other areas. In other words, if "rad" is correctly expressed in fundamental equations, it does not have to be deleted or inserted post calculation--thereby avoiding the multiple and varied (and very confusing) justifications and explanations that seem to abound in textbooks and on-line tutorials.

 

[BP Leonard]

To Mark44: Angles are dimensional physical quantities and have been recognised as such, either implicitly or explicitly, for millennia. Actually, in the International System of Quantities (ISQ), "angles" themselves are considered to be dimensionless and are "defined" as the ratio of the arc-length of any sector of a circle (in a plane) for which the "angle" is the central "angle" divided by the corresponding radius; thus: "theta" = s/r.

The answer is said to be in "radians"--as defined by the International System of Units (SI). A real (dimensional) radian--as described in elementary textbooks whenever radians are first introduced--is the real (dimensional) angle that is the central angle of a circular sector whose arc-length is equal to its radius. To avoid confusion, let me write angle* for "angle" as defined by the ISQ and rad* for the "radian" defined in the SI. Whereas angle (without the asterisk) is the real dimensional angle and radian (without the asterisk), symbol rad, is a real dimensional radian--both angle and radian have the dimension: angle, symbol A--not recognised by the ISQ or SI.

The ISQ angle* is actually the (dimensionless) number of radians in the angle: angle* = angle/rad. The SI radian is the number of radians in one radian: rad* = rad/rad = 1. Since any quantity can be divided or multiplied by 1 without changing its value, we see the "justification" of, respectively, deleting or inserting the SI radian*--in order to "get the correct answer."

Similarly for solid angle. The ISQ "solid angle" is actually the number of square-radians in the real (dimensional) solid angle, where solid angle has dimension A squared. A real (dimensional) steradian is a square radian. In the SI, the "steradian" is sr* = sr/(rad-squared) = (rad-squared)/(rad-squared) = 1. This can similarly be deleted or inserted to "get the correct answer."

 

[Mark44]

To Mark44: Angles are dimensional physical quantities and have been recognised as such, either implicitly or explicitly, for millennia. Actually, in the International System of Quantities (ISQ), "angles" themselves are considered to be dimensionless and are "defined" as the ratio of the arc-length of any sector of a circle (in a plane) for which the "angle" is the central "angle" divided by the corresponding radius; thus: "theta" = s/r.

Which is what I said.

The answer is said to be in "radians"--as defined by the International System of Units (SI). A real (dimensional) radian--as described in elementary textbooks whenever radians are first introduced

The textbooks I've seen during my 20+ years of teaching college mathematics treat radians as dimensionless.

--is the real (dimensional) angle that is the central angle of a circular sector whose arc-length is equal to its radius. To avoid confusion, let me write angle* for "angle" as defined by the ISQ and rad* for the "radian" defined in the SI. Whereas angle (without the asterisk) is the real dimensional angle and radian (without the asterisk), symbol rad, is a real dimensional radian--both angle and radian have the dimension: angle, symbol A--not recognised by the ISQ or SI. The ISQ angle* is actually the (dimensionless) number of radians in the angle: angle* = angle/rad. The SI radian is the number of radians in one radian

Unless you have this backwards, this statement supports what I said about radians being dimensionless. The number of apples in one apple is one, a pure (and dimensionless) number.
Regarding SI units, this wikipedia page (https://en.wikipedia.org/wiki/International_System_of_Units#Derived_units) defines the units of radians in terms of SI base units as $(m \cdot m^{-1})$; implying that radians are dimensionless.

So both systems, ISQ and SI, are consistent in saying that radians are dimensionless.

: rad* = rad/rad = 1. Since any quantity can be divided or multiplied by 1 without changing its value, we see the "justification" of, respectively, deleting or inserting the SI radian*--in order to "get the correct answer." Similarly for solid angle. The ISQ "solid angle" is actually the number of square-radians in the real (dimensional) solid angle, where solid angle has dimension A squared. A real (dimensional) steradian is a square radian. In the SI, the "steradian" is sr* = sr/(rad-squared) = (rad-squared)/(rad-squared) = 1. This can similarly be deleted or inserted to "get the correct answer."

 

[anorlunda]

@BP Leonard , have a look at this PF Insights article https://www.physicsforums.com/insights/can-angles-assigned-dimension/

The author made a spirited case for dimensional angles. But the follow-up discussion showed that he didn't convince most of the others here.

 

[Stavros Kiri]

Radians are generally considered to be dimensionless, unlike degrees. The reason is that a radian is defined to be the ratio of two lengths -- the arc length along a circle divided by the length of the radius.

I think that sais it mostly all, that's why:
s = rθ holds only in rad.
Just minor phrasing correction:

a radian is defined to be the ratio of two lengths -- the arc length along a circle divided by the length of the radius.

You mean "an angle measured in rad".
Definition of rad is "an angle which corresponds to arc length equal to the radius".

Note: @BP Leonard, welcome to PF!

 

[Mark44]

a radian is defined to be the ratio of two lengths -- the arc length along a circle divided by the length of the radius.

You mean "an angle measured in rad".
Definition of rad is "an angle which corresponds to arc length equal to the radius".

When I said "a radian is defined to be ..." of course I was talking about an angle, but my main concern was the units, or rather the lack of them, since it would be length unit divided by length unit.

 

[BP Leonard]

The "controversy" about whether angles should be treated as dimensional quantities (with an independent dimension called angle, with, for example, symbol A) or as dimensionless numbers has been going on for a long time. In fact, there should be no controversy. When I turn my head from looking straight ahead to looking straight left, I have turned my gaze through one (dimensional) right-angle or a quarter of a revolution, rev/4; I have not turned my gaze through the (dimensionless) number pi/2 (= 1.570 . . .). Similarly, the angle in each corner of an equilateral triangle is a (dimensional) angle equal to one sixth of a revolution, rev/6; it is not the number pi/3 (= 1.047 . . .). And, in a well-known construction, a radian is the (dimensional) central angle of a circular sector for which the arc-length is equal to the length of the radius; this means that rad = rev/(2 pi), a little bit smaller than the angles in an equilateral triangle. This (real physical dimensional) radian is not a (dimensionless) number.

What is called "angle" in the ISQ and SI is not the (real physical dimensional) angle as described above but, rather, the (dimensionless) number of (real physical dimensional) radians in the (real physical dimensional) angle: angle/rad. Once again, let me use the asterisk to represent the ISQ/SI "angle." Then angle* = angle/rad. Similarly, the SI "radian" is rad* = rad/rad = 1. The SI claims that the radian is a derived unit: rad* = m/m (metre per metre), where, by definition, derived units are uniquely represented in SI base units. But an ISQ/SI central "angle" of a circular sector is also equal to twice the sector area divided by the radius squared, in which case, the SI "radian" (using the SI argument about derived units) is equal to metre-squared per metre-squared--i.e. the SI "radian" is not uniquely represented in terms of base units and therefore cannot be a derived uint. [Similarly for the SI "steradian."] The only thing that can be said about the SI "radian" is that it is the number 1--being the number of real physical dimensional radians in one real physical dimensional radian. [Similarly, the SI "steradian" is 1, not metre-squared per metre-squared.]

 

[Mark44]

The SI claims that the radian is a derived unit: rad* = m/m (metre per metre), where, by definition, derived units are uniquely represented in SI base units.

What units remain after the division in $\frac m m$? What you have left is a dimensionless number.

But an ISQ/SI central "angle" of a circular sector is also equal to twice the sector area divided by the radius squared, in which case, the SI "radian" (using the SI argument about derived units) is equal to metre-squared per metre-squared--i.e. the SI "radian" is not uniquely represented in terms of base units and therefore cannot be a derived uint.

Both of these statements above validate what I'm saying about radians being dimensionless. I'm not at all concerned that two different standards bodies can't agree whether a radian can or cannot be represented in SI base units.

Your argument to this point of inserting and deleting a radian "unit" doesn't make sense to me. This seems to be a solution in search of a problem.