Dreidimensionales Gebilde
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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 folg abgebildeten Körpers. Er besteht aus lauter identischen Würfeln mit der Kantenlänge a. center tikzpicturescale. latex pgftransformyshift-cm; pgftransformxshift-cm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; pgftransformyshiftcm; pgftransformxshiftcm; pgftransformxshift-cm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; pgftransformxshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; pgftransformyshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; pgftransformyshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; pgftransformxshift-cm; pgftransformyshift-cm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; pgftransformxshift-cm; pgftransformyshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; pgftransformyshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; pgftransformyshift-cm; pgftransformxshiftcm; drawthick- colorgreen!!black -.-- noderight x; drawthick- colorgreen!!black -.--. nodeabovept leftpt y; drawthick- colorgreen!!black -.-- nodebelowpt leftpt z; tikzpicture center
Solution:
itemize item Die bf x-Koordinate des Schwerpunktes findet man folgermassen: center tikzpicture draw-latex colorgreen!!black ---; drawcolorgreen!!black -.--.; nodecolorgreen!!black at . x; node at .. ; node at -.. ; node at -.. ; tikzpicture center Das führt auf folge Gleichung: x_S fracM m_i x_i frac left fraca - fraca - fraca right -.a item Die bf y-Koordinate des Schwerpunktes findet man folgermassen: center tikzpicture draw-latex colorgreen!!black ---; drawcolorgreen!!black -.--.; nodecolorgreen!!black at . y; node at .. ; node at .. ; node at .. ; node at -.. ; tikzpicture center Das führt auf folge Gleichung: y_S fracM m_i x_i frac left fraca + fraca + fraca-fraca right .a item Die bf z-Koordinate des Schwerpunktes findet man folgermassen: center tikzpicture draw-latex colorgreen!!black ---; drawcolorgreen!!black --.---.; nodecolorgreen!!black at . z; node at .. ; node at -.. ; node at -.. ; tikzpicture center Das führt auf folge Gleichung: z_S fracM m_i x_i frac left-fraca + fraca + fraca right .a itemize
Berechne die xyz-Koordinaten des Schwerpunktes des folg abgebildeten Körpers. Er besteht aus lauter identischen Würfeln mit der Kantenlänge a. center tikzpicturescale. latex pgftransformyshift-cm; pgftransformxshift-cm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; pgftransformyshiftcm; pgftransformxshiftcm; pgftransformxshift-cm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; pgftransformxshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; pgftransformyshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; pgftransformyshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red ----------cycle; filldrawcolorblack fillyellow!!red ----------cycle; drawdashed ------; drawdashed ----; pgftransformxshift-cm; pgftransformyshift-cm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; pgftransformxshift-cm; pgftransformyshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; pgftransformyshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; pgftransformyshift-cm; pgftransformxshiftcm; drawthick- colorgreen!!black -.-- noderight x; drawthick- colorgreen!!black -.--. nodeabovept leftpt y; drawthick- colorgreen!!black -.-- nodebelowpt leftpt z; tikzpicture center
Solution:
itemize item Die bf x-Koordinate des Schwerpunktes findet man folgermassen: center tikzpicture draw-latex colorgreen!!black ---; drawcolorgreen!!black -.--.; nodecolorgreen!!black at . x; node at .. ; node at -.. ; node at -.. ; tikzpicture center Das führt auf folge Gleichung: x_S fracM m_i x_i frac left fraca - fraca - fraca right -.a item Die bf y-Koordinate des Schwerpunktes findet man folgermassen: center tikzpicture draw-latex colorgreen!!black ---; drawcolorgreen!!black -.--.; nodecolorgreen!!black at . y; node at .. ; node at .. ; node at .. ; node at -.. ; tikzpicture center Das führt auf folge Gleichung: y_S fracM m_i x_i frac left fraca + fraca + fraca-fraca right .a item Die bf z-Koordinate des Schwerpunktes findet man folgermassen: center tikzpicture draw-latex colorgreen!!black ---; drawcolorgreen!!black --.---.; nodecolorgreen!!black at . z; node at .. ; node at -.. ; node at -.. ; tikzpicture center Das führt auf folge Gleichung: z_S fracM m_i x_i frac left-fraca + fraca + fraca right .a itemize