path: root/Ph1Mech-zf.tex

\documentclass[a4paper,twoside]{article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{mathtools}

\usepackage{float}
\usepackage{calc}
\usepackage[margin=4cm,top=3cm,bottom=3cm]{geometry}
\usepackage{fancyhdr}

\usepackage[german]{babel}

\usepackage[table]{xcolor}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{pgfplots}

\usepackage{multirow}
\usepackage{multicol}
\usepackage{arydshln}
\usepackage{enumitem}
\usepackage{booktabs}
\usepackage[tikz]{mdframed}

\usetikzlibrary{calc}
\pgfplotsset{compat=newest}

\mdfsetup{
    linecolor=black,
    linewidth=2pt,
%
    innertopmargin=.5em,
    innerbottommargin=.75em,
    frametitlefont=\large\bfseries\ttfamily,
    frametitlerule=true,
    frametitlerulewidth=1pt,
    frametitlebackgroundcolor=gray!20,
%
    subtitlefont=\ttfamily,
    subtitleaboveline=true,
    subtitlebackgroundcolor=gray!10,
    subtitlebelowskip=.5em,
    subtitleaboveskip=.5em,
}

\pagestyle{fancy}
\fancyhf{}
\fancyfoot[C]{\thepage}
\fancyhead[C]{Physik 1 Mechanik}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}

\title{Ph1Mech Zusammenfassung}
\author{Naoki Pross}

\setlength{\parindent}{0cm}
% \setlength{\parskip}{0cm}
% \setlength{\columnsep}{1em}


\renewcommand{\v}[1]{\mathbf{#1}}
\newcommand{\vs}[1]{\boldsymbol{#1}}
\newcommand{\dd}[1]{\mathrm{d}#1}

\begin{document}

\begin{mdframed}[frametitle={Physikalischen Gr\"o{\ss}en und Konstanten}]
  \small
  \begin{center}
    \begin{minipage}{.40\textwidth}
      \begin{tabular}{l | >{\(}l<{\)} l}
        Weg             & \v{x}       & m \\
        Geschwindigkeit & \v{v}       & m/s \\
        Beschleunigung  & \v{a}       & m/s\(^2\) \\
        Masse           & m           & kg \\
        Impuls          & \v{p}       & kg \(\cdot\) m/s \\
        Kraft           & \v{F}       & kg \(\cdot\) m/s\(^2\) \\
      \end{tabular}\par
    \end{minipage}
    \hfill
    \begin{minipage}{.55\textwidth}
      \begin{tabular}{l | >{\(}l<{\)} l}
        Winkel                & \vs{\varphi}              & rad \\
        Winkelgeschwindigkeit & \vs{\omega}               & rad/2 \\
        Winkelbeschleunidung  & \vs{\alpha}               & rad/2\(^2\) \\
        Tr\"agheitsmoment     & \underline{\mathbf{J}}, J & kg \(\cdot\) m\(^2\) \\
        Drehimpuls            & \v{L}                     & kg \(\cdot\) m\(^2\)/s \\
        Drehmoment            & \v{M}, \vs{\tau}          & Nm \\
      \end{tabular}\par
    \end{minipage}
  \end{center}
  % \begin{tabular}{l | >{\(}l<{\)} l}
  %   Energie         & E           & J = Ws \\
  %   Arbeit          & \Delta E, W & J \\
  %   Leisung         & P           & W \\
  % \end{tabular}
\end{mdframed}

\begin{mdframed}[frametitle={Postulate f\"ur Newtonsche Mechanik}]
  \begin{multicols}{2}
    \textsc{Absoluter Zeit und Raum} \\
    {\small
      Zeit und Raum sind sowohl vom Beobachter als auch von der darin enthaltenen Objecten und darin stattfindenden physikalischen Vorg\"angen unabh\"angig.
    }\par

    \vspace{.5em}
    \textsc{I. Newtonsche Gesetze} \\
    {\small
      Ein kräftefreier Körper bleibt in Ruhe oder bewegt sich geradlinig mit konstanter Geschwindigkeit
    }\par

    
    \vspace{.5em}
    \textsc{II. Newtonsche Gesetze}
    \[
    \sum\v{F} = m\,\v{a} \qquad \sum\v{M} = J\vs{\alpha} \\
    \]
    \par

    \vspace{.5em}
    \textsc{III. Newtonsche Gesetze} \\
    {\small
      In einem geschlossenen System sind die gesamte Energie und Impuls \emph{immer} erhalten.
    }
    \par

    \vspace{.5em}
    \textsc{Gallilei Invarianz (Boost)} \\
    Beschleunigungen sind vom (nicht drehende) Bezugsystem unabh\"angig.
    \[
    \v{F}' = \v{F} = m\,\ddot{\v{x}}' = m\,\ddot{\v{x}}
    \]
    \par
  \end{multicols}
  \vspace{.5em}
\end{mdframed}

\begin{mdframed}[frametitle={Translationsbewegung}]
  \mdfsubtitle{Spezifische Translationsbewegungen}
  \begin{center}
    \begin{minipage}{.4\textwidth}
      Zweidimensionaler Wurf {\footnotesize (\(\v{a} = \v{g}\))}
      \begin{align*}
        x &= v_0\cdot\cos(\vartheta)\cdot t \\
        y &= v_0\cdot\sin(\vartheta)\cdot t - \frac{g\cdot t^2}{2} \\
        y &= \tan(\vartheta)\cdot x - \frac{g\cdot x^2}{2v_0^2\cos^2(\vartheta)} \\
        d &= \frac{v_0^2}{g}\cdot\sin(2 \vartheta) \quad (y = 0) \\
        h &= \frac{v_0^2}{2g}\cdot\sin^2(\vartheta) \quad (\dot{y} = 0)
      \end{align*}
    \end{minipage}
    \begin{minipage}{.55\textwidth}
      \resizebox{\linewidth}{!}{
        \begin{tikzpicture}
          \pgfmathsetmacro{\g}{9.81}

          \pgfmathsetmacro{\ang}{60.0}
          \pgfmathsetmacro{\vn}{5.0}

          \pgfmathsetmacro{\yn}{1.0}
          \pgfmathsetmacro{\ymax}{\yn + (\vn * \vn) / (2 * \g) * sin(\ang) * sin(\ang)}

          \pgfmathsetmacro{\d}{(\vn * \vn) / (\g) * sin(2.0 * \ang)}
          \pgfmathsetmacro{\dm}{\d/2}
          
          \begin{axis}[
              samples = 80,
              domain=0:2.5,
              xmin=-.2,
              axis equal,
              axis y line = left,
              axis x line = middle,
              axis line shift = 5pt,
              xtick = {0, \dm, \d}, ytick = {\yn, \ymax},
              xticklabels = {0, \(d/2\),\(d\)}, yticklabels = {\(y_0 = 0\), \(h\)},
            ]

            \addplot[dashed, gray] {\yn};
            \addplot[dashed, gray] {\ymax};
            \addplot[ultra thick, gray]{\yn + tan(\ang) * x - (\g * x^2)/(2 * \vn^2 * cos(\ang)^2)};
            \draw[dashed, gray] (axis cs: {\d/2}, 0) -- (axis cs: {\d/2}, {\ymax + .2});
            \draw[dashed, gray] (axis cs: \d, 0) -- (axis cs: \d, {\ymax + .2});

            % angle
            \draw[thick] (axis cs: .5,\yn) arc[
              start angle = 0, end angle = \ang, radius={transformdirectionx(.5)}
            ] (axis cs: .5, \yn) node[above right] {\(\vartheta\)};

            % vectors
            \draw[blue, thick, ->]
            (axis cs: 0,\yn) to node[pos=.8, above left] {\(\v{v}_0\)}
            (axis cs: {cos(\ang)}, {\yn + sin(\ang)});
            
            \draw[thick, ->]
            (axis cs: 0, \yn) to node[pos=.6, right] {\(\v{g}\)}
            (axis cs: 0, {\yn-.5});

            % mass
            \draw[fill=white, black]
            (axis cs: -.1, {\yn -.1}) rectangle (axis cs: .1, {\yn+.1})
            node[pos=.5] {\(m\)};
          \end{axis}
        \end{tikzpicture}
      }
    \end{minipage}
  \end{center}

\end{mdframed}

\begin{mdframed}[frametitle={Rotationsbewegung und Kreisbewegung}]
  \begin{center}
    \begin{minipage}{.5\textwidth}
      \centering
      \resizebox{\linewidth}{!}{
        \tdplotsetmaincoords{70}{110}
        \begin{tikzpicture}[tdplot_main_coords]
          \clip[tdplot_screen_coords] (-1.5,-1.5) rectangle (4.5,4.1);

          \pgfmathsetmacro{\mx}{2}
          \pgfmathsetmacro{\my}{3}
          \pgfmathsetmacro{\mz}{2}
          \pgfmathsetmacro{\mr}{{sqrt(\mx * \mx + \my * \my}}

          \coordinate (O) at (0,0,0);
          \coordinate (M) at (\mx,\my,\mz);

          % axis
          \draw[->] (0,0,0) -- (3,0,0) node[anchor=north east]{\(x\)};
          \draw[->] (0,0,0) -- (0,3,0) node[anchor=west]{\(y\)};
          \draw[->] (0,0,0) -- (0,0,4) node[anchor=south]{\(z\)};

          % arcs
          \tdplotsetrotatedcoords{0}{0}{{atan(\my / \mx)}}
          \tdplotdrawarc[tdplot_rotated_coords, ->, gray]{(0,0,\mz)}{\mr}{10}{350}{}{}
          \tdplotresetrotatedcoordsorigin{}
          
          % axial vectors
          \draw[->, ultra thick, black] (0,0,0) -- (0,0,2.5) node[above left] {\(\vs{\omega}\)};
          \draw[->, thick, black] (0,0,0) -- node[pos=.5, below] {\(\v{r}\)} (M);

          % projections
          \draw[dashed] (O) -- (\mx,\my,0);
          \draw[dashed] (\mx,\my,0) -- (M);
          % angle
          \tdplotdrawarc[->]{(O)}{{\mr/3}}{0}{atan(\my / \mx)}{below}{\(\vs{\varphi}\)}

          \tdplotsetrotatedcoordsorigin{(M)}
          \tdplotsetrotatedcoords{0}{0}{{- 90 + atan(\my / \mx)}}

          % mass vectors
          \draw[tdplot_rotated_coords, ->, thick, blue!70!black]%
          (0,0,0) -- (-2,0,0) node[above] {\(\v{v}_t\)};

          \draw[tdplot_rotated_coords, ->, thick, red!70!black]%
          (0,0,0) -- (0,-1.5,0) node[left] {\(\v{a}_c\)};

          % box
          \draw[black, tdplot_rotated_coords]
          % bottom
          (-.2,-.2,-.2) -- ( .2,-.2,-.2) --
          ( .2, .2,-.2) -- (-.2, .2,-.2) --
          (-.2, .2,-.2) -- (-.2,-.2,-.2)
          % top
          (-.2,-.2, .2) -- ( .2,-.2, .2) --
          ( .2, .2, .2) -- (-.2, .2, .2) --
          (-.2, .2, .2) -- (-.2,-.2, .2)
          % sides
          (-.2,-.2,-.2) -- (-.2,-.2, .2)
          ( .2, .2,-.2) -- ( .2, .2, .2)
          (-.2, .2,-.2) -- (-.2, .2, .2)
          ( .2,-.2,-.2) -- ( .2,-.2, .2);
          %
      \end{tikzpicture}}
    \end{minipage}
    \begin{minipage}{.45\textwidth}
      Physikalische Gr\"o{\ss}en
      \begin{align*}
        \vs{\omega} &= \dot{\vs{\varphi}}                      & \v{L} &= J\vs{\omega} \\
        \vs{\alpha} &= \dot{\vs{\omega}} = \ddot{\vs{\varphi}} & \v{M} &= \dot{\v{L}} = J\vs{\alpha}
      \end{align*}
      Beziehungen mit der Translationsbewegung
      \begin{align*}
        \v{v}_t &= \vs{\omega}\times\v{r} \qquad \v{a}_t = \dot{\v{v}}_t = \vs{\alpha}\times\v{r} \\
        \v{a}_c &= \vs{\omega}\times\v{v}_t = \vs{\omega}\times(\vs{\omega}\times\v{r}) \\
        &= (\vs{\omega}\cdot\v{r})\vs{\omega} - \vs{\omega}^2\v{r}
        \xRightarrow{\vs{\omega}\bot\v{r}\phantom{a}} -\vs{\omega}^2\v{r}
      \end{align*}
    \end{minipage}
  \end{center}
  \mdfsubtitle{Tr\"agheitsmoment}
  \mdfsubtitle{Umlaufbahn}
\end{mdframed}

\begin{mdframed}[frametitle={Energie und Arbeit}]
\end{mdframed}

\begin{mdframed}[frametitle=Statik]
  \[
  \sum_k \v{F}_k = \v{0} \qquad \sum_k \v{M}_k = \v{0}
  \]
\end{mdframed}

\begin{mdframed}[frametitle=Dynamik]
  \[
  \sum_k \v{F}_k = m\cdot\v{a} \qquad \sum_k \v{M}_k = J\vs{\alpha}
  \]

  \mdfsubtitle{Reibung}
  \mdfsubtitle{St\"o{\ss}e}
\end{mdframed}

\begin{mdframed}[frametitle={Deformierb\"are K\"orper}]
\end{mdframed}

\end{document}