Simple Harmonic Motion: conceptual idea of angular frequency

[nehamalcom]

One of the conditions to distinguish Simple Harmonic Motion from other harmonic motions is by the relation that

a∝x

where x is the displacement from the point that acceleration is directed towards​

But what confuses me is the constant of proportionality introduced to this relation: ω²

ω is angular frequency which seems relevant when there is actual angular displacement as in the case of a SHM observing pendulum; but what about a block connected to a spring observing SHM on a frictionless surface? How are we associating ω² here?

I need to know the actual concept of having angular frequency in SHM.

[Mentors' note: this post has been edited to correct some formatting problems]​

 

[BvU]

Hello nehamal, ,

In the phase plane ($x$ horizontal, $\dot x$ -- i.e. $dx\over dt$ -- vertical ) the trajectory is an ellipse and one cycle takes $2\pi\over \omega$ units of time. Divide the abscissa by $\omega$ and you get a circular trajectory.

Does that answer your question ?

 

[sophiecentaur]

when there is actual angular displacement

That's displacement angle not the angle that's considered in SHM. In fact the nature of the oscillation is independent of that angle (for small displacements). The 'Angle' in 'Angular Frequency" concerns the Phase of the oscillation or the fraction of the cycles of oscillation And ω is the number of radians of phase per second.
If the time period of the oscillation is T (in seconds), then the frequency f is the number of times it oscillates in 1s
So f = i/T
and, as there are 2π radians in one cycle, we use ω where ω = f/2π
EDIT: I was writing with my head up my backside last night : New version: For annoying Mathematical reasons we don't tend to use f when doing the Maths of oscillations because of what happens when you differentiate Sin(x) when x is in degrees. It gives you 180Cos(x)/π. But differentiation Sin(y), when y is in radians, gives Cos(y)There's a lot of Calculus associated with SHM and if you do a lot of that operation, you can end up with a lot of odd 2π's littered about the page. If you use ω instead (where ω = 2πf because there are 2π radians of phase in one cycle) then differentiating Sin(ωt) just gives ωCos (ωt) with no odd extra constants outside the brackets. WHY BOTHER? you may ask but it seriously is worth while when you get beyond the very basics of oscillations.
For starters it it best to try to just go along with this. This link shows you how it's done.