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; pgftransformxshiftcm; 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; pgftransformyshift-cm; pgftransformxshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; tikzpicture center
Solution:
Für die drei jeweiligen Koordinaten des Schwerpunktes gilt: itemize item x-Koordinate center tikzpicturelatex draw rectangle ; node at .. ; draw - rectangle ; node at -.. ; draw - rectangle -; node at -.. ; draw - rectangle -; node at -.. ; drawcolorgreen!!black- -.--. nodeabove y; drawcolorgreen!!black- -.--. noderight x; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black ..--.-.nodebelow .; tikzpicture center sscxS fracM _i^ x_i m_i frac -. -. - . + . -. item y-Koordinate center tikzpicturelatex draw rectangle ; node at .. ; draw rectangle ; node at .. ; draw rectangle ; node at .. ; draw - rectangle ; node at -.. ; draw - rectangle -; node at -.. ; drawcolorgreen!!black- .---. nodebelow x; drawcolorgreen!!black- -.--. noderight y; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black ..--.-. nodebelow .; drawcolorgreen!!black ..--.-. nodebelow .; drawcolorgreen!!black ..--.-.nodebelow .; tikzpicture center sscyS fracM _i^ y_i m_i frac -. + -. +. + . + . . item z-Koordinate center tikzpicturelatex draw rectangle ; node at .. ; draw rectangle ; node at .. ; draw - rectangle ; node at -.. ; drawcolorgreen!!black- -.--. nodeabove y; drawcolorgreen!!black- -.--. noderight z; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black ..--.-. nodebelow .; drawcolorgreen!!black ..--.-. nodebelow .; tikzpicture center ssczS fracM _i^ z_i m_i frac -. +. + . . itemize sscxS -. sscyS . ssczS .
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; pgftransformxshiftcm; 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; pgftransformyshift-cm; pgftransformxshiftcm; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; filldrawcolorblack fillyellow!!red --------cycle; drawdashed ----; drawdashed --; tikzpicture center
Solution:
Für die drei jeweiligen Koordinaten des Schwerpunktes gilt: itemize item x-Koordinate center tikzpicturelatex draw rectangle ; node at .. ; draw - rectangle ; node at -.. ; draw - rectangle -; node at -.. ; draw - rectangle -; node at -.. ; drawcolorgreen!!black- -.--. nodeabove y; drawcolorgreen!!black- -.--. noderight x; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black ..--.-.nodebelow .; tikzpicture center sscxS fracM _i^ x_i m_i frac -. -. - . + . -. item y-Koordinate center tikzpicturelatex draw rectangle ; node at .. ; draw rectangle ; node at .. ; draw rectangle ; node at .. ; draw - rectangle ; node at -.. ; draw - rectangle -; node at -.. ; drawcolorgreen!!black- .---. nodebelow x; drawcolorgreen!!black- -.--. noderight y; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black ..--.-. nodebelow .; drawcolorgreen!!black ..--.-. nodebelow .; drawcolorgreen!!black ..--.-.nodebelow .; tikzpicture center sscyS fracM _i^ y_i m_i frac -. + -. +. + . + . . item z-Koordinate center tikzpicturelatex draw rectangle ; node at .. ; draw rectangle ; node at .. ; draw - rectangle ; node at -.. ; drawcolorgreen!!black- -.--. nodeabove y; drawcolorgreen!!black- -.--. noderight z; drawcolorgreen!!black -..---.-. nodebelow -.; drawcolorgreen!!black ..--.-. nodebelow .; drawcolorgreen!!black ..--.-. nodebelow .; tikzpicture center ssczS fracM _i^ z_i m_i frac -. +. + . . itemize sscxS -. sscyS . ssczS .