Würfelgebilde
Question
Solution
Short
Video
\(\LaTeX\)
Need help? Yes, please!
The following quantities appear in the problem:
Länge \(\ell\) / Kraft \(F\) / Drehmoment \(\vec M\) /
The following formulas must be used to solve the exercise:
\(\vec M = \vec \ell \times \vec F \quad \) \(\sum \stackrel{\curvearrowleft}{M} \stackrel{!}{=} \sum \stackrel{\curvearrowright}{M} \quad \)
No explanation / solution video to this exercise has yet been created.
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
Berechne die dreidimensionalen Koordinaten des Schwerpunktes des folg abgebildeten Körpers. Er besteht aus lauter identischen Würfeln der Kantenlänge a. center tikzpicturescale. latex drawthick- colorgreen!!black -.-- noderight x; drawthick- colorgreen!!black -.--. nodeabovept leftpt y; drawthick- colorgreen!!black --- nodebelowpt leftpt z; draw --------cycle; draw ------------cycle; draw ---; draw ---; draw ---; draw ---; pgftransformyshiftcm; draw --------cycle; draw ------------cycle; draw ---; draw ---; draw ---; draw ---; pgftransformyshiftcm; draw --------cycle; draw ------------cycle; draw ---; draw ---; draw ---; draw ---; pgftransformxshiftcm; draw --------cycle; draw ------------cycle; draw ---; draw ---; draw ---; draw ---; draw ------------cycle; draw ------------cycle; draw ----; draw ----; draw ----; draw ----; pgftransformxshiftcm; draw ------------cycle; draw ------------cycle; draw ----; draw ----; draw ----; draw ----; pgftransformyshift-cm; draw ------------cycle; draw ------------cycle; draw ----; draw ----; draw ----; draw ----; pgftransformyshift-cm; pgftransformxshift-cm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformxshift-cm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformxshift-cm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformyshiftcm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformyshiftcm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformxshiftcm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; tikzpicture center
Solution:
Das Würfelgebilde besteht aus identischen Würfeln; die Schwerpunktskoordinate in einer beliebigen Richtung berechnet man über folge Formel am Beispiel x-Koordinate: x_S fracM _i^ m_ix_i Konkret hat man für die drei Koordinaten die folgen Situationen der rote Punkt auf den Achsen markiert jeweils den Koordinatenursprung: itemize item x-Koordinate: center tikzpicturelatex filldrawcolorred circle .; drawcolorblack!!green- ---; nodecolorblack!!green at . x; drawcolorblack!!green|- -.--.; node at -.. ; drawcolorblack!!green|- -.--.; node at -.. ; drawcolorblack!!green|- .--.; node at .. ; drawcolorblack!!green|- .--.; node at .. ; drawcolorblack!!green|- .--.; node at .. ; tikzpicture center x_S frac left-.a -.a + .a + .a + .a right frac.a .a item y-Koordinate: center tikzpicturelatex filldrawcolorred circle .; drawcolorblack!!green- --; nodecolorblack!!green at . y; drawcolorblack!!green|- .--.; node at .. ; drawcolorblack!!green|- .--.; node at .. ; drawcolorblack!!green|- .--.; node at .. ; tikzpicture center y_S frac left.a + .a + .a right frac.a .a item z-Koordinate: center tikzpicturelatex filldrawcolorred circle .; drawcolorblack!!green- --; nodecolorblack!!green at . z; drawcolorblack!!green|- -.--.; node at -.. ; drawcolorblack!!green|- -.--.; node at -.. ; drawcolorblack!!green|- .--.; node at .. ; tikzpicture center z_S frac left-.a -.a + .a right frac-.a -.a Der Schwerpunkt dieses Würfelgebildes liegt also beim Punkt S.a.a-.a wobei a die Kantenlänge eines Würfels ist. itemize
Berechne die dreidimensionalen Koordinaten des Schwerpunktes des folg abgebildeten Körpers. Er besteht aus lauter identischen Würfeln der Kantenlänge a. center tikzpicturescale. latex drawthick- colorgreen!!black -.-- noderight x; drawthick- colorgreen!!black -.--. nodeabovept leftpt y; drawthick- colorgreen!!black --- nodebelowpt leftpt z; draw --------cycle; draw ------------cycle; draw ---; draw ---; draw ---; draw ---; pgftransformyshiftcm; draw --------cycle; draw ------------cycle; draw ---; draw ---; draw ---; draw ---; pgftransformyshiftcm; draw --------cycle; draw ------------cycle; draw ---; draw ---; draw ---; draw ---; pgftransformxshiftcm; draw --------cycle; draw ------------cycle; draw ---; draw ---; draw ---; draw ---; draw ------------cycle; draw ------------cycle; draw ----; draw ----; draw ----; draw ----; pgftransformxshiftcm; draw ------------cycle; draw ------------cycle; draw ----; draw ----; draw ----; draw ----; pgftransformyshift-cm; draw ------------cycle; draw ------------cycle; draw ----; draw ----; draw ----; draw ----; pgftransformyshift-cm; pgftransformxshift-cm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformxshift-cm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformxshift-cm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformyshiftcm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformyshiftcm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; pgftransformxshiftcm; draw --------cycle; draw --------cycle; draw --; draw --; draw --; draw --; tikzpicture center
Solution:
Das Würfelgebilde besteht aus identischen Würfeln; die Schwerpunktskoordinate in einer beliebigen Richtung berechnet man über folge Formel am Beispiel x-Koordinate: x_S fracM _i^ m_ix_i Konkret hat man für die drei Koordinaten die folgen Situationen der rote Punkt auf den Achsen markiert jeweils den Koordinatenursprung: itemize item x-Koordinate: center tikzpicturelatex filldrawcolorred circle .; drawcolorblack!!green- ---; nodecolorblack!!green at . x; drawcolorblack!!green|- -.--.; node at -.. ; drawcolorblack!!green|- -.--.; node at -.. ; drawcolorblack!!green|- .--.; node at .. ; drawcolorblack!!green|- .--.; node at .. ; drawcolorblack!!green|- .--.; node at .. ; tikzpicture center x_S frac left-.a -.a + .a + .a + .a right frac.a .a item y-Koordinate: center tikzpicturelatex filldrawcolorred circle .; drawcolorblack!!green- --; nodecolorblack!!green at . y; drawcolorblack!!green|- .--.; node at .. ; drawcolorblack!!green|- .--.; node at .. ; drawcolorblack!!green|- .--.; node at .. ; tikzpicture center y_S frac left.a + .a + .a right frac.a .a item z-Koordinate: center tikzpicturelatex filldrawcolorred circle .; drawcolorblack!!green- --; nodecolorblack!!green at . z; drawcolorblack!!green|- -.--.; node at -.. ; drawcolorblack!!green|- -.--.; node at -.. ; drawcolorblack!!green|- .--.; node at .. ; tikzpicture center z_S frac left-.a -.a + .a right frac-.a -.a Der Schwerpunkt dieses Würfelgebildes liegt also beim Punkt S.a.a-.a wobei a die Kantenlänge eines Würfels ist. itemize