Angles in the SI: a detailed proposal for solving the problem

Abstract

A recent letter [1] proposed changing the dimensionless status of the radian and steradian within the SI, while allowing the continued use of the convention to set the angle 1 radian equal to the number 1 within equations, providing this is done explicitly. This would bring the advantages of a physics-based, consistent, and logically-robust unit system, with unambiguous units for all physical quantities, for the first time, while any upheaval to familiar equations and routine practice would be minimised. More details of this proposal are given here. The only notable changes for typical end-users would be: improved units for the quantities torque, angular momentum and moment of inertia; a statement of the convention accompanying some familiar equations; and the use of different symbols for the action and the angular momentum, a small step forward for quantum physics. Some features of the proposal are already established practice for quantities involving the steradian such as radiant intensity and radiance.

Comment on ‘Angles in the SI: a detailed proposal for solving the problem’

“…But Paul Quincey considers this equation to be unit specific [24]-'so that the angle unit must be the radian, and the units on either side of the equation do not match' [14]. This widely believed apparent 'dimensional inconsistency' is typically 'patched up' by using a variety of seemingly ad hoc postcalculation 'rules'-such as, for example, 'drop radians .…”

“…Then by considering the familiar equations in column 2, it would appear that these can be obtained from the explicitradian equations in column 1 by taking the latter and visually 'setting the radian equal to one' everywhere-i.e. by applying the radian convention [12][13][14]. However, as explained above, the variable represented by θ in the familiar equation for arc length is not the physical angle (dimension A) of the corresponding explicit-radian equation, as it would be if the radian (dimension A) could actually be set equal to (the number) 1, but rather the 'angle in radians': {θ} rad = θ/rad = N rad -i.e.…”

“…In the referenced paper, Paul Quincey claims that the moment of force r × F (with dimension ML 2 /T 2 and unit N m) is not 'torque' but, rather, the result of applying the radian convention to the claimed form of 'torque'-defined in table 2 of his paper [14] as (r × F)/θ C or, equivalently, in terms of a modified moment arm [3,4,7], τ Q = (r/θ C ) × F, as in column 3 of table 1 here-stemming from the widely believed analogy (Quincey uses the term 'correspondence') between translational and rotational work: translational work (energy) = (force) • (length) rotational work (energy) = ('torque') • (angle) from which 'torque' would have the dimension ML 2 /(T 2 A) and complete unit J/rad [12].…”

“…Solid angle is then a derived quantity with dimension A 2 ; the steradian is a coherent derived unit, sr = rad 2 . Some authors [1][2][3][4][5][6][7][8][9][10][11][12][13][14] have claimed that, among other things, this would imply that the units for torque, angular momentum and moment of inertia must be, respectively, J/rad, J/(rad/s) and J/(rad/s) 2 -which would be in conflict with all these quantities' traditional angle-independent units: N m, kg m 2 s −1 and kg m 2 . Paul Quincey's most recent paper [14] contains a careful description of these claims promoting, as he also does, the adoption of the angle-dependent (so-called 'improved') units for these quantities.…”

“…In the referenced paper [14] and earlier publications [12,13], Paul Quincey has attempted to resolve the apparent conflict by introducing the 'radian convention': the familiar form of the equations of rotational mechanics can (apparently) be obtained from the dimensionally consistent (explicitradian) equations by 'setting the radian equal to 1'. For torque, Quincey argues, application of the radian convention (to the 'improved' unit joule per radian) results in the familiar conventional unit: [J/rad] rad=1 ⇒ J = kg m 2 s −2 = N m, where the newton metre is currently preferred in order to distinguish the conventional unit for torque from that ( joule) used for various forms of energy [23, section 2.3.4].…”