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1. Angular Frequency

   Linked via "linear frequency ($f$,"

   $$\omega = \frac{2\pi}{T}$$
   This definition establishes the proportionality constant between linear frequency ($f$, measured in Hertz, where $f = 1/T$) and angular frequency. The factor of $2\pi$ arises because one complete cycle) corresponds to a phase change of $2\pi$ radians [2].
   Relationship to Velocity

2. Angular Frequency

   Linked via "linear frequency $f$"

   $$y(t) = A \cos(\omega t + \phi)$$
   where $A$ is the amplitude and $\phi$ is the phase constant. When decomposing complex signals using techniques such as the Fourier Transform, the variable conjugate to time ($t$) is often expressed in terms of $\omega$ rather than linear frequency $f$, as this simplifies the imaginary exponent's argument [1].
   The $\omega$ Shift in Electrical Circuits

3. Angular Frequency

   Linked via "linear frequency ($f$)"

   Comparative Metrics
   The choice between using angular frequency ($\omega$) and linear frequency ($f$) often depends on the established conventions of the relevant scientific discipline. The relationship is universally defined by the $2\pi$ factor.
   | Metric | Symbol | Unit (SI) | Primary Context | Relationship to $\omega$ |