Plexiglasplastik
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The following quantities appear in the problem:
Länge \(\ell\) / Masse \(m\) / Kraft \(F\) / Strecke \(s\) / Drehmoment \(\vec M\) /
The following formulas must be used to solve the exercise:
\(\vec M = \vec \ell \times \vec F \quad \) \(\sum \stackrel{\curvearrowleft}{M} \stackrel{!}{=} \sum \stackrel{\curvearrowright}{M} \quad \) \(\vec r_{\rm S} = \frac{1}{M} \sum_i m_i \vec r_i \quad \)
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Exercise:
Nebenan ist eine Plexiglasplastik aus quadratischen Prismen gleichen Querschnitts dargestellt. Sie weist überall nur rechte Winkel auf. Die Masse sind aus der Abbildung ersichtlich. Welche Werte haben die Schwerpunktskoordinaten im eingezeichneten System? center tikzpicturescale. latex draw- --; draw- .--.-; draw- .--.; draw- -.---.; draw- ----; draw- .--.; node at -. pqcm; node at - pqcm; node at . pqcm; node at .- pqcm; node at .-. pqcm; node at .-. pqcm; draw --.; drawdashed .--; draw ----; drawdashed --; drawdashed --; draw ----; drawdashed --; draw ------------------------; draw --------; draw --; draw ----; draw --; draw --; draw --; draw --; draw --; drawdashed --; drawdashed --; drawdashed --------; drawdashed ----; draw --; draw --; drawdashed --; draw --; drawdashed --; tikzpicture center
Solution:
tikzpicturescale. latex drawdarkgreen- -- noderightx; drawdarkgreen- -- nodeabovey; drawdarkgreen- -- nodebelowz; draw- --; draw- .--.-; draw- .--.; draw- -.---.; draw- ----; draw- .--.; node at -. pqcm; node at - pqcm; node at . pqcm; node at .- pqcm; node at .-. pqcm; node at .-. pqcm; draw --.; drawdashed .--; draw ----; drawdashed --; drawdashed --; draw ----; drawdashed --; draw ------------------------; draw --------; draw --; draw ----; draw --; draw --; draw --; draw --; draw --; drawdashed --; drawdashed --; drawdashed --------; drawdashed ----; draw --; draw --; drawdashed --; draw --; drawdashed --; tikzpicture Wir teilen die Figuren der Grösse nach in vier Teilfiguren ein und bestimmen damit den Schwerpunkt: NewQtyVecubiccentimeter NewQtyVzcubiccentimeter NewQtyVdcubiccentimeter NewQtyVvcubiccentimeter NewQtyxecm NewQtyxzcm NewQtyxdcm NewQtyxvcm NewQtyyecm NewQtyyzcm NewQtyydcm NewQtyyvcm NewQtyzecm NewQtyzzcm NewQtyzdcm NewQtyzv-cm SolQtyxfracV_x_ + V_x_ + V_x_ + V_x_V_+V_+V_+V_VeX*xeX+VzX*xzX+VdX*xdX+VvX*xvX/VeX+VzX+VdX+VvXcm SolQtyyfracV_y_ + V_y_ + V_y_ + V_y_V_+V_+V_+V_VeX*yeX+VzX*yzX+VdX*ydX+VvX*yvX/VeX+VzX+VdX+VvXcm SolQtyzfracV_z_ + V_z_ + V_z_ + V_z_V_+V_+V_+V_VeX*zeX+VzX*zzX+VdX*zdX+VvX*zvX/VeX+VzX+VdX+VvXcm al x^* fracm_x_+m_x_+m_x_+m_x_m_+m_+m_+m_ xF fracVexe + Vzxz + Vd xd + Vv xvVe+Vz+Vd+Vv xP y^* fracm_y_+m_y_+m_y_+m_y_m_+m_+m_+m_ yF fracVeye + Vzyz + Vd yd + Vv yvVe+Vz+Vd+Vv yP z^* fracm_z_+m_z_+m_z_+m_z_m_+m_+m_+m_ zF fracVeze + Vzzz + Vd zd + Vv qtyzvVe+Vz+Vd+Vv zP.
Nebenan ist eine Plexiglasplastik aus quadratischen Prismen gleichen Querschnitts dargestellt. Sie weist überall nur rechte Winkel auf. Die Masse sind aus der Abbildung ersichtlich. Welche Werte haben die Schwerpunktskoordinaten im eingezeichneten System? center tikzpicturescale. latex draw- --; draw- .--.-; draw- .--.; draw- -.---.; draw- ----; draw- .--.; node at -. pqcm; node at - pqcm; node at . pqcm; node at .- pqcm; node at .-. pqcm; node at .-. pqcm; draw --.; drawdashed .--; draw ----; drawdashed --; drawdashed --; draw ----; drawdashed --; draw ------------------------; draw --------; draw --; draw ----; draw --; draw --; draw --; draw --; draw --; drawdashed --; drawdashed --; drawdashed --------; drawdashed ----; draw --; draw --; drawdashed --; draw --; drawdashed --; tikzpicture center
Solution:
tikzpicturescale. latex drawdarkgreen- -- noderightx; drawdarkgreen- -- nodeabovey; drawdarkgreen- -- nodebelowz; draw- --; draw- .--.-; draw- .--.; draw- -.---.; draw- ----; draw- .--.; node at -. pqcm; node at - pqcm; node at . pqcm; node at .- pqcm; node at .-. pqcm; node at .-. pqcm; draw --.; drawdashed .--; draw ----; drawdashed --; drawdashed --; draw ----; drawdashed --; draw ------------------------; draw --------; draw --; draw ----; draw --; draw --; draw --; draw --; draw --; drawdashed --; drawdashed --; drawdashed --------; drawdashed ----; draw --; draw --; drawdashed --; draw --; drawdashed --; tikzpicture Wir teilen die Figuren der Grösse nach in vier Teilfiguren ein und bestimmen damit den Schwerpunkt: NewQtyVecubiccentimeter NewQtyVzcubiccentimeter NewQtyVdcubiccentimeter NewQtyVvcubiccentimeter NewQtyxecm NewQtyxzcm NewQtyxdcm NewQtyxvcm NewQtyyecm NewQtyyzcm NewQtyydcm NewQtyyvcm NewQtyzecm NewQtyzzcm NewQtyzdcm NewQtyzv-cm SolQtyxfracV_x_ + V_x_ + V_x_ + V_x_V_+V_+V_+V_VeX*xeX+VzX*xzX+VdX*xdX+VvX*xvX/VeX+VzX+VdX+VvXcm SolQtyyfracV_y_ + V_y_ + V_y_ + V_y_V_+V_+V_+V_VeX*yeX+VzX*yzX+VdX*ydX+VvX*yvX/VeX+VzX+VdX+VvXcm SolQtyzfracV_z_ + V_z_ + V_z_ + V_z_V_+V_+V_+V_VeX*zeX+VzX*zzX+VdX*zdX+VvX*zvX/VeX+VzX+VdX+VvXcm al x^* fracm_x_+m_x_+m_x_+m_x_m_+m_+m_+m_ xF fracVexe + Vzxz + Vd xd + Vv xvVe+Vz+Vd+Vv xP y^* fracm_y_+m_y_+m_y_+m_y_m_+m_+m_+m_ yF fracVeye + Vzyz + Vd yd + Vv yvVe+Vz+Vd+Vv yP z^* fracm_z_+m_z_+m_z_+m_z_m_+m_+m_+m_ zF fracVeze + Vzzz + Vd zd + Vv qtyzvVe+Vz+Vd+Vv zP.