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Rotation Translation


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 $P = {\boldsymbol F} {\pmb{\cdot}}{\boldsymbol v}$ $P = {\boldsymbol M} {\pmb{\cdot}}{\boldsymbol \omega}$