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b/sose2020/theo/uebungen/theo3.tex @@ -0,0 +1,120 @@ +\documentclass[uebung]{../../../lecture} + +\begin{document} + +\punkte[2] +\author{Christian Merten} +\title{Theo II: Übungsblatt 3} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Falls $l_0 \le 0$: In beiden Fällen findet keine Bewegung statt. + Sei also $l_0 > 0$. Die verallgemeinerte Koordinate sei + $l > 0$, der Abstand des untersten Massepunkts von der Tischkante. Damit ist + $l(0) = l_0$. + \begin{enumerate}[(i)] + \item Zunächst gilt, wegen $m_{\text{ges}} = 2 m$: + \[ + T = m \dot{l}^2 + .\] + Die potentielle Energie ist abhängig von $l$: + \[ + V = \begin{cases} + - m g l & l \le L \\ + - m g l - m g (l - L) & l > L + \end{cases} + .\] Damit folgt + \[ + \mathcal{L} = T - V = \begin{cases} + m\dot{l}^2 + mgl & l \le L \\ + m\dot{l}^2 + 2mgl & l > L + \end{cases} + .\] + \item Für die kinetische Energie gilt + \[ + T = \frac{M}{2} \dot{l}^2 + .\] Da die potentielle Energie mit der Länge der überhängenden Kette + steigt, folgt für $l \le L$: + \begin{align*} + V &= - g \int_{0}^{l} \frac{\d h}{L} Mh = - \frac{Mg}{2L}l^2 + \intertext{Für $l > L$ muss die Kette durch die Länge $L$ begrenzt werden, damit folgt} + V &= - g \int_{l-L}^{l} \frac{M}{L} g h \d h = - Mg \left(l - \frac{L}{2}\right) + .\end{align*} + Insgesamt folgt damit: + \[ + \mathcal{L} = T - V = \begin{cases} + \frac{M}{2} \dot{l}^2 + \frac{Mg}{2L}l^2 & l \le L \\ + \frac{M}{2} \dot{l}^2 + Mg \left( l - \frac{L}{2} \right) & l > L + \end{cases} + .\] + \end{enumerate} + \item Mit den Langrange Gleichungen + \[ + \frac{\mathrm{d}}{\d t} \frac{\partial \mathcal{L}}{\partial{\dot{q}_i}} + - \frac{\partial \mathcal{L}}{\partial{q_i}} = 0 + .\] folgt + \begin{enumerate}[(i)] + \item Mit $l(0) = l_0$ und $v_0 = 0$ folgt für $l \le L$: + \[ + 2 m\ddot{l} - mg = 0 \implies l(t) = \frac{1}{4} g t^2 + l_0 + .\] Für $l > L$ folgt mit $t_e := \sqrt{\frac{4}{g} (L - l_0)}$ und + $v_e := \dot{l}(t_e)$: + \[ + 2 m \ddot{l} - 2mg = 0 \implies l(t-t_e) = \frac{1}{2}g t^2 + v_e t + L + .\] + \item Für $l \le L$ folgt + \begin{align*} + M\ddot{l} - \frac{Mg}{L} l &= 0 + \intertext{Mit $w^2 := \frac{g}{L}$ folgt} + l_{1,2}(t) &= e^{\pm \omega t} \\ + l(t) &= A e^{\omega t} + B e^{-\omega t} + \intertext{Aus den Anfangsbedingungen $l(0) = l_0$ und $v_0 = 0$ folgt} + l(t) &= \frac{l_0}{2} \left(e^{\omega t} + e^{-\omega t} \right) + .\end{align*} + Für $l > L$ folgt für $t \ge t_e$ analog zu (i) eine freie Fallbewegung: + \[ + l(t - t_e) = \frac{1}{2} g t^2 + v_e t + L + .\] + \end{enumerate} + \item Mit $E = T + V$ folgt jeweils für $l \le L$: + \begin{enumerate}[(i)] + \item + \begin{align*} + E &= m \dot{l}^2 - mgl + = mg \left(\frac{1}{4} g t^2 - \frac{1}{4} gt^2 - l_0\right) = - m g l_0 \\ + \implies \frac{\d E}{\d t} &= 0 + .\end{align*} + \item + \begin{align*} + E &= \frac{M}{2} \frac{l_0^2}{4} \omega^2 \left( e^{\omega t} - e^{-\omega t} \right)^2 + - \frac{Mg}{2L} \frac{l_0^2}{4} \left( e^{\omega t} + e^{- \omega t} \right)^2 \\ + &= \frac{M}{2} \frac{l_0^2}{4} \left( \omega^2 \left(e^{\omega t} - e^{- \omega t} \right)^2 + - \omega^2\left(e^{\omega t} + e^{-\omega t}\right)^2 \right) \\ + &= - \frac{M}{2} l_0^2 \omega^2 \\ + \implies \frac{\d E}{\d t} &= 0 + .\end{align*} + \end{enumerate} + Für $l > L$ liegt eine freie Fallbewegung vor, hier ist die Energie offensichtlich erhalten. + Analoge Rechnung zu (i). + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe}[Verständnisfragen] + \begin{enumerate}[a)] + \item Verallgemeinerte Kräfte sind das Analogon zum Übergang von karthesischen + Koordinaten zu verallgemeinerten Koordinaten. Sie haben jetzt genau + $f$ Komponenten und haben, genauso wie die verallgemeinerten Koordinaten, nicht mehr die + Einheit einer Kraft. + \[ + F_j = \sum_{k=1}^{n} F_i \frac{\partial x_i}{\partial q_j} + .\] + \item Die Lagrange-Gleichungen 2. Art sind eine Neuformulierung des 2. Newton'schen Axioms, wobei + die karthesischen Koordinaten aufgegeben werden und anstatt dessen verallgemeinerte + Koordinaten eingesetzt werden, die automatisch alle Zwangsbedingungen erfüllen. Das + 2. Newton'sche Axiom wird damit durch die Zwangsbedingungen eingeschränkt, indem der + $3N$-Konfigurationsraum auf die $f$ dimensionale Untermannigfaltigkeit eingeschränkt wird. + \item + \end{enumerate} +\end{aufgabe} + +\end{document}