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

Abstract

The Comment by B P Leonard [Metrologia, XX (2022)] primarily proposes that if angle is treated as a base quantity, with the radian as its base unit, it would be wrong to change the units for torque (from N m to J rad-1), angular momentum (from J s to J s rad-1) and moment of inertia (from kg m2 to kg m2 rad-2), as was proposed in the Letter being commented on [Quincey, Metrologia 58 053002 (2021)]. This Reply clarifies the situation by looking directly at the consequences of the two proposals. Apart from the comfort of retaining the familiar units for these quantities, the benefit of Leonard’s proposal would be the preservation of a few favoured equations used in specific situations, while the general relationships between many physical quantities would need to change. The revised units proposed in the Letter would leave all the established general relationships unchanged, and are the best option for allowing the longstanding problem of angles being wrongly treated as numbers within the SI to be resolved. This Reply includes some historical context, which describes how Euler implicitly introduced the idea that “the radian is another name for the number one” into the mathematics used for rotating objects, at a time long before anyone had thought about unit systems.

Unit one is intrusive

“…In 2016, Mills argued that the radian and the cycle should be treated as units of dimension angle, thereby clarifying that certain pairs of definitions refer to the same quantity expressed with different units [24]. Discussion of proposals to add dimension angle continued as a separate thread from the treatment of dimensionless quantities and has remained active [25][26][27][28][29][30][31].…”

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

“…Equations throughout science have treated angles as dimensionless quantities for a very long time [1]. 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.…”

“…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 .…”

“…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 . Fortunately, this topic is relatively simple.…”