Angular Acceleration

Here’s a quick proof for the formula for the magnitude of angular acceleration using Euler’s Identity, that I haven’t seen anywhere else online before.

The equation for the position of an object rotating at constant speed about the origin in the complex plane is $s = re^{i\omega t}$, where $\omega$ is the angular speed of said object, $r$ is the radius of the path, and $t$ is time.

Since $\omega = \frac{v}{r}$, where $v$ is tangential velocity, we can differentiate with respect to $t$ and find that [\frac{\mathrm{d}s}{\mathrm{d}t} = v = ive^{i\omega t}.]

Differentiating again, we find that [ \frac{\mathrm{d}v}{\mathrm{d}t} = a = -\frac{v^2}{r}e^{i\omega t}.]

Since $\lvert -e^{i\omega t}\rvert = 1$, the magnitude of acceleration is $\lvert a \rvert = \frac{v^2}{r}$ and we are done. $\square$