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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/Trägh. \(m\) \(\Theta = \int_V \vec r^2_\perp \diff m\)
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 \(P = \vec F \bdot \vec v\) \(P = \vec M \bdot \vec \omega\)