Phase in S.H.M.

So far we have described an S.H.M. particle's condition at any instant using its displacement x, or its velocity v, separately. But neither, alone, is enough to fully pin down the "state" of the oscillation. Consider displacement alone: at any given position x (other than the extremes), there are actually TWO possible directions of velocity the particle could have -- it could be passing through that point moving towards positive x, or moving towards negative x -- so knowing x alone leaves this direction ambiguous. Now consider velocity alone: for any given speed |v| (other than at the mean position), there are likewise TWO different positions (symmetric about the mean position) at which the particle could have that speed. And on top of both these ambiguities, both x and v repeat identically in every successive oscillation, so neither one, by itself, even tells you WHICH cycle you are currently in.

What we need is the "state of oscillation" -- known as the PHASE of the motion -- captured by a single quantity that changes CONTINUOUSLY and MONOTONICALLY with time, never repeating a value (unlike x or v, which cycle back). Section 5.7's reference-circle construction supplies exactly this: the reference angle $(\omega t+\phi)$ of the equivalent uniform circular motion increases steadily and without limit as t increases, and (as section 5.7 showed) it fully determines x, v AND a at every instant via the projection formulas. This quantity, $(\omega t+\phi)$, is therefore what we call the phase, or (since it is literally an angle) the phase angle, of the S.H.M.

It is worth working through what a few special phase-angle values physically mean, since these come up constantly in problems (see Examples 5.7 and 5.8):

1. Phase = 0 (or any integer multiple of $2\pi$, i.e. $0, 2\pi, 4\pi, \ldots$): the particle is at the mean position and moving towards the positive side. Phase = 0 marks the very start of the first oscillation; phase = $2\pi$ marks the start of the second oscillation (an identical physical state, one full cycle later); and so on.
2. Phase = $\pi$ (or $\pi + 2n\pi$ for integer n): the particle is at the mean position, but this time moving towards the negative side -- during its first oscillation at phase $\pi$, during its second at phase $\pi + 2\pi = 3\pi$, and so on.
3. Phase = $\pi/2$ (or $\pi/2 + 2n\pi$): the particle is at the positive extreme position, during its first oscillation at phase $\pi/2$, during its second at phase $\pi/2 + 2\pi = 5\pi/2$, and so on. …