Mechanical Motion#

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 \vec F = 0 \; \Leftrightarrow \; \frac{\diff \vec{v} }{ \diff t } = 0$$\)
2. If an object is accelerating, then there is a force on it. $$\(\vec F = m \cdot \vec 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 $$\vec x$$ $$\vec \varphi = \frac{\vec x}{r}$$
Geschwindigkeit $$\vec v = \dot{\vec x}$$ $$\vec \omega = \dot{\vec \varphi} = \frac{\vec v}{r}$$
Beschleunigung $$\vec a = \dot{\vec v} = \ddot{\vec x}$$ $$\vec \alpha = \dot{\vec \omega} = \ddot{\vec \varphi} = \frac{\vec a}{r}$$
Masse/TrĂ¤gh. $$m$$ $$\Theta = \int_V \vec r^2_\perp \diff m$$
Impuls/Drall $$\vec p =m \vec v$$ $$\vec L = \ma \Theta \vec \omega = \vec r \times \vec p$$
Kraft/Moment $$\vec F = \dot{\vec p} = m \vec a$$ $$\vec M = \dot{\vec L} = \ma \Theta \vec \alpha = \vec r \times \vec F$$
Energie $$E_{\ir kin}=\frac12mv^2$$ $$E_{\ir rot}=\frac12 \Theta \omega ^2$$
Leistung $$P = \vec F \bdot \vec v$$ $$P = \vec M \bdot \vec \omega$$