From 25ad1fc0fe3c468c86cf4fc492b82b0bd5d021e1 Mon Sep 17 00:00:00 2001
From: Nao Pross <naopross@thearcway.org>
Date: Sun, 22 Dec 2019 15:10:00 +0100
Subject: First commit

---
 Ph1Mech-zf.tex | 262 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 262 insertions(+)
 create mode 100644 Ph1Mech-zf.tex

(limited to 'Ph1Mech-zf.tex')

diff --git a/Ph1Mech-zf.tex b/Ph1Mech-zf.tex
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+\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{arydshln}
+\usepackage{enumitem}
+\usepackage{booktabs}
+\usepackage[tikz]{mdframed}
+
+\usetikzlibrary{calc}
+\pgfplotsset{compat=newest}
+
+\mdfsetup{
+    linecolor=black,
+    linewidth=2pt,
+%
+    innertopmargin=.5em,
+    innerbottommargin=.5em,
+    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}
+
+
+\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}]
+\end{mdframed}
+
+\begin{mdframed}[frametitle={Translationsbewegung}]
+  \begin{center}
+    \begin{minipage}{.45\textwidth}
+      Physikalische Gr\"o{\ss}en
+      \begin{align*}
+        \v{v} &= \dot{\v{x}}                 & \v{p} &= m\,\v{v} \\
+        \v{a} &= \dot{\v{v}} = \ddot{\v{x}}  & \v{F} &= \dot{\v{p}} = m\,\v{a}
+      \end{align*}
+    \end{minipage}
+    \begin{minipage}{.45\textwidth}
+      Mit konstante Beschleudigung \(\v{a}\)
+      \begin{align*}
+        \v{v} &= \v{v}_0 + \v{a}\,t \\
+        \v{x} &= \v{x}_0 + \v{v}_0\,t + \frac{\v{a}}{2}\,t^2
+      \end{align*}
+    \end{minipage}
+  \end{center}
+
+  \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]
+  \[
+  \sum_k \v{F}_k = m\cdot\v{a} \qquad \sum_k \v{M}_k = J\vs{\alpha}
+  \]
+\end{mdframed}
+
+\begin{mdframed}[frametitle=Reibung]
+\end{mdframed}
+
+\begin{mdframed}[frametitle={St\"o{\ss}e}]
+\end{mdframed}
+
+\end{document}
-- 
cgit v1.2.1

