Rotational Motion

$r$ $\mathrm{r}$ radius
$\varphi$ $\mathrm{rad}$ angle in radians
$\omega$ $\mathrm{s^{-1}}$ angular velocity
$\alpha$ $\mathrm{s^{-2}}$ angular acceleration
$t$ $\mathrm{s}$ time
$T$ $\mathrm{s}$ period, duration of a rotation
$f$ $\mathrm{s^{-1}, Hz}$ revolution frequency
$n$ $\mathrm{s^{-1}}$ rotational speed

Uniform Rotation

Condition: $\omega = \mathrm{const.}$

$$ \varphi = \omega \, t + \varphi_0 $$

$$ \omega = \frac{2 \,\pi}{T} = 2 \,\pi \, f \qquad
n = \frac{1}{T} $$

Uniformly accelerated Rotation

Condition: $\alpha = \mathrm{const.}$

$$ \varphi = \frac{\alpha}{2} \, t^2 + \omega_0 \, t + \varphi_0 $$

Rotation in general

$$ \varphi = \int\limits_{t_1}^{t_2}\! \omega(t)\;\mathrm{d}t $$

$$ \omega = \int\limits_{t_1}^{t_2}\! \alpha(t)\;\mathrm{d}t \qquad
\omega = \frac{\mathrm{d} \varphi}{\mathrm{d}t} = \varphi'(t) = \dot{\varphi} $$

$$ \alpha = \frac{\mathrm{d} \omega}{\mathrm{d}t} = \omega'(t) \qquad
\alpha = \frac{\mathrm{d^2} \varphi}{\mathrm{d}t^2} = \ddot{\varphi} $$