Relationship

Translational – 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
$s$ $\mathrm{m}$ length of a circular arc
$v$ $\mathrm{m/s}$ path velocity
$v_{E}$ $\mathrm{m/s}$ own speed at the rotation body
$a$ $\mathrm{m/s^2}$ acceleration
$a_{ZP}$ $\mathrm{m/s^2}$ centripetal acceleration
$a_{ZF}$ $\mathrm{m/s^2}$ centrifugal acceleration
$a_{R}$ $\mathrm{m/s^2}$ radial acceleration
$a_{T}$ $\mathrm{m/s^2}$ tangential acceleration
$a_{C}$ $\mathrm{m/s^2}$ coriolis acceleration
$e$   unit vector

Overview

$$ s = \varphi \, r \qquad
\vec{s} = \varphi \; \vec{e}_\omega \times \vec{r} $$

$$ v = \omega \, r \qquad
\vec{v} = \vec{\omega} \times \vec{r} $$

$$ s = \alpha \, r \qquad
\vec{a} = \vec{\alpha} \times \vec{r} $$

Centripetal Acceleration, Radial Acceleration

Acceleration is directed towards center

$$ a_{ZP} = a_{R} = – \omega^2 \, r $$

$$ \vec{a}_{ZP} = \vec{\omega}^2 \times \vec{r} \qquad
\vec{a}_{ZP} = \frac{-v^2}{r} $$

Uniformly accelerated rotation

$$ a_{ZP} = r \, \left[ \,\omega_0 + \alpha_0 \, \left( t – t_0 \right)\, \right]^2 $$

Centrifugal Acceleration

$$ a_{ZF} = – a_{ZP} $$

Tangential Acceleration

Uniformly accelerated rotation

$$ a_T = \alpha_0 \, r $$

Uniform rotation

$$ a_T = 0 $$

Coriolis Acceleration

$$ a_C = -2 \, \left( \, \vec{\omega} \times \vec{v}_E \, \right) \qquad
a_C = -2 \, \omega \, v_E $$