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.

This post is 80 words long.

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