Periodic motion problem, how to tackle this beast?

You can see the periodic motion as the projection over the diameter of a point moving with a circular motion.

The Amplitude will be the radius of the circumference and ω is the angular frequency (or speed) for both motions.

In the periodic motion, you will have maximum velocity at the center and it will be zero at the extremities, where the projection changes direction, while the acceleration will be maximum at the extremities and zero at the center.

The displacement will then be:
x(t) = A · cos(ωt)
And from this (using a little bit of calculus):
v(t) = A · ω · sin(ωt)
a(t) = A · ω² · cos(ωt)

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