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