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I have recently been introduced to the topic of simple harmonic motion for the first time (I'm currently an A-level physics student). I feel that I have understood the fundamental ideas behind SHM very well. However, I have one question which has been bugging me and I can't seem to find a valid answer.

I will consider the example of a simple pendulum which has been set in motion and therefore oscillates about a fixed equilibrium position.

If the displacement of the pendulum bob is considered as a function of time, then the graph of x/t is analogous to a sine curve (assuming that the bob is released from a point of maximum displacement - since I believe that starting from the equilibrium position would produce a cosine curve [please correct me if I'm wrong here])

The graph will have the equation x = Asin(wt). Now I try to break this down in order to understand why this equation is true for SHM.

Firstly, as I said the variation of x with t produces a sine curve, explaining why X is a function of sin(t). I'm fine with that. Next, I understand that A (the amplitude or max. displacement) is a coefficient on the outside because it has the effect of 'stretching' the sine curve (since the bob oscillates between positions of max. displacement either side.) Again, that all makes sense. However, what I do not understand is why sin(wt) [I know omega is the letter used but I only have w on my keyboard] is used instead of sin(t). It is time which is plotted on the horizontal axis, so surely the y axis represents displacement (x) and the x axis represents time (t).

Why is the equation not:

x = Asin(t)?

If somebody could clear this up for me it would be greatly appreciated.

Thank you in advance.