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diff --git a/Ph1Mech-zf.tex b/Ph1Mech-zf.tex new file mode 100644 index 0000000..e726367 --- /dev/null +++ b/Ph1Mech-zf.tex @@ -0,0 +1,262 @@ +\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} |