# Mechanical Motion 

Rotation Translation

##### Definition

Motion is a change in position of an object over time.

## Newton's Laws of Motion

1. If the vector sum of all forces acting on an object is zero, then the velocity of the object is constant. $$\sum {\boldsymbol F} = 0 \; \Leftrightarrow \; \frac{{\,\text{d}}{\boldsymbol v} }{ {\,\text{d}}t } = 0$$
2. If an object is accelerating, then there is a force on it. $${\boldsymbol F} = m \cdot {\boldsymbol a}$$
3. If one object $A$ exerts a force $F_A$ on a second object $B$, then $B$ simultaneously exerts a force $F_B = - F_A$ on $A$, and the two forces are equal in magnitude and opposite in direction.

## Classical Mechanics

Motion Translation { Rotation} (Radius $r$)
Strecke/Winkel ${\boldsymbol x}$ ${\boldsymbol \varphi} = \frac{{\boldsymbol x}}{r}$
Geschwindigkeit ${\boldsymbol v} = \dot{{\boldsymbol x}}$ ${\boldsymbol \omega} = \dot{{\boldsymbol \varphi}} = \frac{{\boldsymbol v}}{r}$
Beschleunigung ${\boldsymbol a} = \dot{{\boldsymbol v}} = \ddot{{\boldsymbol x}}$ ${\boldsymbol \alpha} = \dot{{\boldsymbol \omega}} = \ddot{{\boldsymbol \varphi}} = \frac{{\boldsymbol a}}{r}$
Masse/Trägh. $m$ $\Theta = \int_V {\boldsymbol r}^2_\perp {\,\text{d}}m$
Impuls/Drall ${\boldsymbol p} =m {\boldsymbol v}$ ${\boldsymbol L} = {\boldsymbol{\Theta}} {\boldsymbol \omega} = {\boldsymbol r} \times {\boldsymbol p}$
Kraft/Moment ${\boldsymbol F} = \dot{{\boldsymbol p}} = m {\boldsymbol a}$ ${\boldsymbol M} = \dot{{\boldsymbol L}} = {\boldsymbol{\Theta}} {\boldsymbol \alpha} = {\boldsymbol r} \times {\boldsymbol F}$
Energie $E_{{\rm}kin}=\frac12mv^2$ $E_{{\rm}rot}=\frac12 \Theta \omega ^2$
Leistung $P = {\boldsymbol F} {\pmb{\cdot}}{\boldsymbol v}$ $P = {\boldsymbol M} {\pmb{\cdot}}{\boldsymbol \omega}$
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