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

Abstract

Paul Quincey makes a compelling argument for recognizing angle as a base quantity with the radian as the base unit. Solid angle is then a derived quantity with the steradian a derived unit equal to one square radian. The author demonstrates how the familiar equations of the SI appear to result from ‘setting the radian equal to one’—the so-called radian convention. He claims, but without any physical foundation (other than by analogy with translational motion), that, for rotation, the ‘improved’ units for torque, angular momentum and moment of inertia must be J/rad, J/(rad/s) and J/(rad/s)2, respectively, and that the conventional units (N m, kg m2 s–1 and kg m2) result from application of the radian convention to these quantities. However, based on sound physical principles, I demonstrate here that the radian convention is simply a (confusing) change of notation, applicable (only) to angle and its time derivatives. It does not apply to torque, angular momentum or moment of inertia. Analogies can be helpful in identifying relationships between quantities, but they do not dictate the physics. The appropriate units for torque, angular momentum and moment of inertia are, respectively, the well-established angle-independent units: newton metre, kilogram metre-squared per second and kilogram metre-squared.

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

“…To end on a lighter note, I enjoyed the little aside in the comment [1] that Archimedes would not have given torque the units J rad −1 because the radian had not been invented. He would not have used the units J rad −1 for a quantity with dimensions energy/angle, because the joule was only invented by electrical engineers in the late 19th century.…”

“…This set follows on from much previous work referenced in that paper. I will call the set preferred by the author of the comment [1] option B. The essentials of these options as they relate to key angular quantities are given in table 1.…”

“…If science were simply a process of agreeing with Great Minds like Paul Dirac, as advocated by the author of the comment [1], we would all follow Leonhard Euler in treating angle as dimensionless.…”

“…The author of the comment [1] and I agree that plane angle should no longer be considered dimensionless within the SI, that equating 1 rad to 1 m/m is wrong, and that 1 sr should be the derived unit 1 rad 2 . However, while he rightly wants to introduce a dimension for angle, he is unwilling to see that this must affect the dimensions of all angular quantities.…”

Comment on ‘On the dimension of angles and their units’

“…Changing their status therefore requires careful consideration of the consequences in all affected areas, some of which may seem superficially detached from a formal change in status for angle units. There has been a recent exchange in this journal focussing on the consequences for the units and equations used in rotational mechanics, involving quantities such as torque, angular momentum and moment of inertia [1,2]. The other key area requiring discussion concerns frequency 1 .…”