Kartonschachteln
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The following quantities appear in the problem:
Länge \(\ell\) / Kraft \(F\) / Drehmoment \(\vec M\) /
The following formulas must be used to solve the exercise:
\(\vec M = \vec \ell \times \vec F \quad \) \(\sum \stackrel{\curvearrowleft}{M} \stackrel{!}{=} \sum \stackrel{\curvearrowright}{M} \quad \)
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Exercise:
Berechne die xyz-Koordinaten des Schwerpunktes des abgebildeten Stapels von jeweils gleich schweren Kartonschachteln. Jede Schachtel sei ein Würfel mit cm Kantenlänge center tikzpicturescale. draw ----------------------------; draw --------; draw ----; draw --; draw --; draw --; draw --; draw --; draw --; draw ----; draw --; draw ----; draw --; draw --; draw --; draw --; draw --; draw --; tikzpicture center
Solution:
Es gelte folges Koordinatensystem: center tikzpicturescale. draw ----------------------------; draw --------; draw ----; draw --; draw --; draw --; draw --; draw --; draw --; draw ----; draw --; draw ----; draw --; draw --; draw --; draw --; draw --; draw --; draw-latex colorgreen!!black -- nodeabove x; draw-latex colorgreen!!black -- nodeabove y; draw-latex colorgreen!!black -- nodeabove z; tikzpicture center Mit der Schwerpunktformel finden wir dann newqtylcm solqtyxfracell/*lncm solqtyyelllncm solqtyzfracell/*lncm al x^* fraclimits_i^m_ix_ilimits_i^ m_i fracfracm fracell + fracmfracell + fracmfracellm xf fracl xTT y^* fraclimits_i^m_iy_ilimits_i^ m_i fracfracmfracell + fracmfracell+fracmfracellm yf l yTT z^* fraclimits_i^m_iz_ilimits_i^ m_i fracfracmfracell+fracmfracell+fracmfracellm zf fracl zTT.
Berechne die xyz-Koordinaten des Schwerpunktes des abgebildeten Stapels von jeweils gleich schweren Kartonschachteln. Jede Schachtel sei ein Würfel mit cm Kantenlänge center tikzpicturescale. draw ----------------------------; draw --------; draw ----; draw --; draw --; draw --; draw --; draw --; draw --; draw ----; draw --; draw ----; draw --; draw --; draw --; draw --; draw --; draw --; tikzpicture center
Solution:
Es gelte folges Koordinatensystem: center tikzpicturescale. draw ----------------------------; draw --------; draw ----; draw --; draw --; draw --; draw --; draw --; draw --; draw ----; draw --; draw ----; draw --; draw --; draw --; draw --; draw --; draw --; draw-latex colorgreen!!black -- nodeabove x; draw-latex colorgreen!!black -- nodeabove y; draw-latex colorgreen!!black -- nodeabove z; tikzpicture center Mit der Schwerpunktformel finden wir dann newqtylcm solqtyxfracell/*lncm solqtyyelllncm solqtyzfracell/*lncm al x^* fraclimits_i^m_ix_ilimits_i^ m_i fracfracm fracell + fracmfracell + fracmfracellm xf fracl xTT y^* fraclimits_i^m_iy_ilimits_i^ m_i fracfracmfracell + fracmfracell+fracmfracellm yf l yTT z^* fraclimits_i^m_iz_ilimits_i^ m_i fracfracmfracell+fracmfracell+fracmfracellm zf fracl zTT.