Uniform circular motion

In the animation below, the red ball is moving in a circle with constant speed. The blue and magenta balls show how the red ball moves along the x axis and y axis separately.

Along each axis, the projection of the red ball moves as a simple oscillator! Specifically, if x and y are the coordinates of the red ball,

x = r cos (wt)
y = r sin (wt)

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By taking derivatives with respect to time, we find the components of the velocity,

vx = - wr sin (wt)
vy = wr cos (wt)

and the acceleration,

ax = - w2r cos (wt)
ay = - w2r sin (wt)

Looking back at x and y as functions of time, we see that the acceleration can be written more simply as

ax = - w2 x
ay = - w2 y

From this we draw two conclusions:

  1. The vector a is exactly opposite to the position vector r, so it points towards the centre of the circle; and
  2. The magnitude of the acceleration is a = w2r, which can also be written as a = v2 / r.