## 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

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

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

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

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

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