0

Why so the oscillations which are not circular also have angular frequency which is a quantity related to the circular motion? I have referred many articles related to simple harmonic motion where they directly include the term angular frequency. Is every periodic oscillation correlated to circular motion such that it has the angular frequency term?

Qmechanic
  • 220,844
Karthikeyan s
  • 1

2 Answers2

1

I think the root of this is Euler's relationship in complex numbers.

A complex number can be represented as $R\exp(i\omega t)$, where $R$ is an amplitude and $t$ is time and $i^2=-1$. This complex number can be drawn as a coordinate in the complex plane (real numbers vs imaginary numbers) and will trace out a circular path with angular frequency $\omega$.

A linear oscillation (e.g. SHM) can then easily be represented as the projection of this circular motion onto the real or imaginary axis through Euler's relation.

$$R\exp(i\omega t) = R\cos \omega t + iR\sin \omega t$$

I.e. the real part is a (co)sinusoidal oscillation, whilst the imaginary part is an oscillation with the same amplitude and angular frequency, but 90 degrees out of phase.

Possibly this explanation doesn't help if you've never come across complex numbers, but even if you haven't, you should be able to see that a linear sinusoidal oscillation can be viewed as the projection of a 2-D circular motion onto one axis.

ProfRob
  • 141,325
0

More context might help, but I can think of two possible interpretations:

  1. The angular frequency as defined by $\omega = 2\pi f = 2\pi/T$ can be applied to any periodic motion ($T$ is the period and $f=1/T$ the “regular” frequency).

  2. The definition as the time derivative of the angle (as a coordinate) $\omega = \dot\phi = d\phi/dt$ could be applied to any motion. (Though it might more properly be called angular velocity.)

xebtl
  • 367