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# Angular motion equations

From the Motion discipline we have four motion equations1 (for when acceleration is constant) with which we can figure out almost everything about a linear motion.

In rotation, lenghts are just replaced with angles. Other than that, everything is equivalent: Rotate the ice-crusher handle with $2\;\mathrm{s^{-1}}$ (angular start speed $\omega_0$) while speeding up with $1\;\mathrm{s^{-2}}$ (one extra radian-per-second each second, so angular acceleration $\alpha$) for 3 seconds (time $t$), and your new angular speed $\omega$ will of course be $5\;\mathrm{s^{-1}}$.

$$\underbrace{5\,\mathrm{s^{-1}}}_\omega= \underbrace{2\,\mathrm{ s^{-1} }}_{\omega_0}+ \underbrace{1\,\mathrm{ s^{-2} }}_\alpha\cdot \underbrace{3\,\mathrm s}_t$$

This is the exact equivalent of the Motion equations skill. All in all, we get exactly equivalent angular motion equations:2

$$\omega=\omega_0+\alpha t\\ \theta=\theta_0+\omega_0 t+\frac12 \alpha t^2\\ \omega^2=\omega_0^2+2\alpha(\theta- \theta_0)\\ \theta = \theta_0+\frac12 (\omega+\omega_0)t$$