Angular frequency in the International System of Units. Physicist’s view

Abstract

Currently, in the International System of Units (SI), reciprocal second (s −1 ) and radian per second (rad s −1 ) are the units of frequency ν and angular frequency ω, respectively. At the same time, these quantities are related in the SI by the simple mathematical relationship ω = 2πν. And here an evident conflict between this relation and the dimensions of the quantities included in it arises. When using the units in question, the two sides of this relation have different dimensions, which is not admissible in either physics or metrology. However, this situation has existed in metrology for many years. In 2019, the angular frequency with unit rad s −1 was included in the SI brochure. This article analyzes the reasons that have led to this conflict and establishes the relationship between frequency, angular frequency and angular velocity. An example of a simple oscillatory system shows that ω and ν have the same dimension, inverse to the dimension of the time. Therefore, they are measured by the same unit s −1 . This does not coincide with the current definition of the unit of angular frequency in the SI, but fully corresponds to the laws of physics and removes the contradiction in the relation ω = 2πν. It is also shown that the angular frequency and angular velocity are independent quantities of different dimensions.

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

“…The appropriate unit for h (= E/ ν) is therefore J/Hz [28, table 1, note a]. By contrast, ω is the time rate of change of the (dimensionless) phase: the phase rate [16], with dimension 1/T and unit s −1 (but not Hz) [22]. Then from equation (3), h (= E/ω) has the unit J/(s −1 ) = J s [28], not J/(rad/s).…”

“…from equation (3), where h = E/ω, that the unit for h should be J/(rad/s). The widespread confusion about the dimension and unit of ω in similar sinusoidal relationships has recently been discussed and clarified in the paper by Mikhail Kalinin [22].…”