## Uniform Circular Motion – Simple and Complex

In reply to Matt, here’s a very short way to find the acceleration of an object in uniform circular motion.

Let the object move in the complex plane, so its position is given by

$z = R e^{\imath \omega t}$

Two time derivatives give the acceleration.

$\dot{z} = \imath \omega z$

$\ddot{z} = (\imath \omega)^2 z = - \omega^2 z$

That’s it. The acceleration points opposite the position, towards the center of the circle. The velocity $\dot{z}$ has a factor $\imath$, indicating a 90-degree rotation from the position. Hence the velocity is tangent to the circle.

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