From ee54163f000ff92135c3d4dd12ab746ce900cc33 Mon Sep 17 00:00:00 2001 From: Darius Peters <38152878+dariusptrs@users.noreply.github.com> Date: Sat, 2 Nov 2024 15:24:57 +0100 Subject: [PATCH] first release (#1) * add main structure * add coriolis, thermo and quanten * add Brechung * begin Linsen * Add schwingungen diagram and Bewegungen * Update Physik.tex * Add Hydromechanik and complete Linsen * update font * add Kraft Energie Impuls * add Schwingung and Wellen * some formatting * formatting * small changes * fix typo in Linsen * fix typo in Wellen * fix typo in Harmonische Schwingung * Change authors * Update Physik.tex I would suggest removing the 'emphbox' and replacing it with Textcolor where we find the formulas necessarily important. In my opinion, we can highlight other formulas as well, without only focusing on those 4 formulas. Otherwise, we can box the formulas that we find important, but then maybe other ones. At the first glance, I can just focus on those 'emphboxes' since they are really visually highlighted. It could act as an advantage as well as the opposite. Let me know what you think. * remove emphbox * Change email - remove textcolor --------- Co-authored-by: stevensalloum --- Physik.tex | 493 ++++++++++++++++++++++++++++++++++++++- img/.texpadtmp/Linsen.ps | Bin 0 -> 423416 bytes img/Linsen.pdf | Bin 0 -> 14280 bytes img/Linsen.ps | Bin 0 -> 423416 bytes img/Linsen.tex | 57 +++++ img/Linsen_crop.pdf | Bin 0 -> 13417 bytes img/Schwebung.pdf | Bin 0 -> 26960 bytes img/Schwebung.tex | 40 ++++ img/Schwebung_crop.pdf | Bin 0 -> 27521 bytes img/Schwingungen.pdf | Bin 0 -> 19631 bytes img/Schwingungen.tex | 37 +++ img/coriolis.pdf | Bin 0 -> 11534 bytes img/coriolis.tex | 36 +++ 13 files changed, 654 insertions(+), 9 deletions(-) create mode 100644 img/.texpadtmp/Linsen.ps create mode 100644 img/Linsen.pdf create mode 100644 img/Linsen.ps create mode 100644 img/Linsen.tex create mode 100644 img/Linsen_crop.pdf create mode 100644 img/Schwebung.pdf create mode 100644 img/Schwebung.tex create mode 100644 img/Schwebung_crop.pdf create mode 100644 img/Schwingungen.pdf create mode 100644 img/Schwingungen.tex create mode 100644 img/coriolis.pdf create mode 100644 img/coriolis.tex diff --git a/Physik.tex b/Physik.tex index 4c1b337..d333984 100644 --- a/Physik.tex +++ b/Physik.tex @@ -13,14 +13,23 @@ % possible options: color/nocolor, english/german, threecolumn % defaults: color, english -\documentclass[english]{latex4ei/latex4ei_sheet} +\documentclass[german]{latex4ei/latex4ei_sheet} % set document information -\title{LaTeX4EI \\ Cheat Sheet} -\author{LaTeX4EI} % optional, delete if unchanged +\title{Physik} +\author{Darius Peters, Steven Salloum} % optional, delete if unchanged \myemail{info@latex4ei.de} % optional, delete if unchanged \mywebsite{www.latex4ei.de} % optional, delete if unchanged +%custom packages +\usepackage{xcolor} + +%custom settings +\titlespacing*{\subsection}{.5pt}{.5pt}{2pt} +\titlespacing*{\subsubsection}{0pt}{.5pt}{2pt} + +%custom font +\renewcommand{\ttdefault}{cmtt} % ====================================================================== % Begin @@ -31,28 +40,494 @@ % ---------------------------------------------------------------------- \maketitle % requires ./img/Logo.pdf +% Klassische Mechanik +% ---------------------------------------------------------------------- +\section{Klassische Mechanik} + +\subsection{Bewegungen} +$x={\ir const.} \rightarrow v=0 \qquad v={\ir const.} \rightarrow a=0$ \\ +$v=\dot{x} \qquad a=\ddot{x}$ \\ +$x=\frac{1}{2}at^2+v_0t+x_0$ \\ +$v=a\cdot t + v_0$ +\subsubsection{Freier Fall} +$h=\frac{1}{2}gt^2 \qquad v_E=\sqrt{2gh}$\\ +$\Delta t = \sqrt{2h/g}$ +\subsubsection{Ebene} +$a=g\sin \alpha \qquad \Delta s = \frac{1}{2}g\sin \alpha$\\ +$v_E=\sqrt{2gh}$ +\subsubsection{Senkrechter Wurf} +$h=v_0t-\frac{1}{2}gt^2+h_0$\\ +$v_M=\sqrt{0,5v_0} \qquad v^2 = v_0^2-2g(h_0-h)$\\ +$v=v_0-g\cdot t$\\ +$t_h=\frac{v_0}{g} \qquad x_h=\frac{v_0^2}{2g}$ +\subsubsection{Waagrechter Wurf} +$v_y=gt \qquad v_x = v_0$\\ +$s_y=h_0-\frac{1}{2}gt^2 \qquad s_x=v_0\cdot t_F$\\ +$v=\sqrt{v_0^2+v_y^2}$\\ +$s_y=h_0-\frac{g\cdot s_x^2}{2v_0^2}$ +\subsubsection{Schräger Wurf} +$v_x=v_0\cos \alpha \qquad v_y=v_0\sin \alpha -gt$\\ +$v_h=\sqrt{v_0^2-2gh}$\\ +$s_x=v_0\cos \alpha t \qquad s_y=v_0 \sin \alpha t - \frac{1}{2} gt^2 +h_0$\\ +$s_{x,\ir max}=\frac{v_0^2 \sin{2\alpha}}{g}$\\ +$t_w=\frac{2v_0\sin \alpha}{g}$ \qquad +$h_{\ir max}=\frac{v_0^2 \sin^2 \alpha}{2g}+h_0$ + +\subsection{Newton'sche Axiome} +\begin{sectionbox} +\begin{enumerate} + \item Masse ist träge. Jeder Körper behält seinen Bewegungszustand bei. + \item $F=m\cdot a=\frac{\diff p}{\diff t}=\dot{p}$ + \item $F_{12}=-F_{21}$ +\end{enumerate} +\end{sectionbox} +\textbf{Intertialsystem:} Koordinatensysteme die sich mit $v=\it const.$ zueinander bewegen. +\subsection{Kraft, Energie, Impuls} +\subsubsection{Kraft} +Federkraft: $F_F=-k(x-x_0)$ \\ +Gewichtskraft: $F_G=mg$ \\ +$F_N=mg\cos \alpha$ \\ +$F_H=mg\sin \alpha$ \\ +Zentripetalkraft: $F_Z=m\cdot \frac{v^2}{r}=m\omega^2r=p\omega$ \\ +Reibungskraft: $F_R=\mu F_{(N)}=\mu m g \cos \alpha$ +\subsubsection{Energie} +$E_{\ir kin}=\frac{1}{2}mv^2=\frac{p^2}{2m}$\\ +$E_{\ir pot}=mgh$\\ +$E_{F}=\frac{1}{2}k(x-x_0)^2$\\ +$E_{\ir rot}=\frac{1}{2}I\omega^2$ +\subsubsection{Arbeit} +$W=\Delta E=F\cdot s \cdot \cos \alpha$ +\subsubsection{Leistung} +$P= +\underbrace{\frac{\Delta W}{\Delta t}}_{\text{mittlere Leistung}} += \, \underbrace{\frac{F\cdot \Delta s}{\Delta t}=F\cdot r}_{\text{momentane Leistung}}$\\ +$\eta=\frac{E_{\it nutz}}{E_{\ir zu}}=\frac{P}{P}$ +\subsubsection{Impuls} +$p_{\ir vorher}=p_{\ir nachher}$\\ +$P=mv$\\ +Mittlere Kraft: $\overline{F}=\frac{\Delta p}{\Delta t}$\\ +Elastischer Stoß: $m_1v_1+m_2v_2=m_1u_1+m_2u_2$\\ +Unelastischer Stoß: $m_1v_1+m_2v_2=(m_1+m_2)u$ +\subsubsection{Raketengleichung} +$F=ma=\frac{\diff m}{\diff t}v_a+F_{\ir ext}$\\ +$\vec{v}=\vec{v_0}+\vec{v_a}\ln | \frac{M_G}{m} | $\\ +$m_0=M+m_a$ +\subsection{Planetenbewegungen} +\subsubsection{Gravitation} +$U_{\ir pot}=-G\frac{m_E\cdot m}{r} \qquad r>r_E$\\ +$\vec{F}_G=G\frac{m_1m_2}{r^2}\cdot \vec{e_r}$\\ +$g=G\cdot \frac{m}{r^2}$\\ +$W_G=Gm_1m_2(\frac{1}{r_1}-\frac{1}{r_2})$\\ +$F_G \, || \, r \rightarrow M=r\times F=0$\\ + +Satellit-Erde: $\frac{1}{2}m_Sv^2=G\frac{m_Sm_E}{R_E}$ \\ +Stabile Umlaufbahn: $\omega_S=\sqrt{\frac{G\cdot m_E}{R_E^3}}\qquad v=\omega R_E$\\ +Geostat. Umlaufbahn: $r_g=\sqrt[3]{G\cdot m_E \cdot \frac{T_g^2}{4\pi^2}} \qquad h_g=r_g-R_E$ -% Section +\subsubsection{Keplersche Gesetze} +\begin{sectionbox} +\begin{itemize} +\item[1.] Keplersches Gesetz: Planetenbahnen sind Ellipsen um Stern in einem der beiden Brennpunkte +\item[2.] Keplersches Gesetz: In gleicher Zeit wird die gleiche Fläche an einer Bahn aufgespannt\\$\frac{dA}{dt} = \frac{1}{2} \vec {r} \vec {v} sin \alpha = \frac{1}{2}\vec {r} \times \vec {v} = \frac{1}{2}m|\vec {L}| \Rightarrow$ Der Drehimpuls ist zeitlich konstant +\item[3.] Keplersches Gesetz: $\frac{T_1^2}{T_2^2} = \frac{a_1^3}{a_2^3}$ mit T: Umlaufzeit, a: Große Halbachse +\end{itemize} +\end{sectionbox} +\subsection{Drehungen} +\subsubsection{Drehmoment} +$M=F\times r=F\cdot r\cdot \sin \alpha$\\ +$M=I\cdot \dot{\omega}=\frac{\diff L}{\diff t}$ +\subsubsection{Trägheitsmoment} +Zylindermantel: $J = mr^2$ (Drehachse ist Körperachse)\\ +Hohlzylinder: $J = \frac{1}{2}m(r_a^2+r_i^2)$ (Drehachse ist Körperachse)\\ +Massive Kugel: $J = \frac{2}{5}mr^2$ (Drehachse durch Mittelpunkt)\\ +\subsubsection{Drehimpuls} +$L=r\times p=I\cdot \omega = mrv$ +\subsection{Corioliskraft} +$\vec{F_c}=2m(v\times \omega)= 2m v w \cdot \sin{(v\angle \omega)}$\\ +Rechte-Hand-Regel: Daumen: $\vec{v}$, Zeigefinger: $\vec{\omega}$ (von Süden nach Norden)\\ +$a_c=-2(\omega \times v)=2v\omega$\\ +$\Delta x= \frac{2}{3}\cos \beta \omega h \sqrt{\frac{2h}{g}}$\\ +$\Delta y = -\omega^2R\cos \beta \sin \beta \frac{h}{g}$\\ +%\includegraphics[width=0.25\columnwidth]{coriolis.pdf} +$R_E=\SI{6370}{\kilo \meter} \qquad r=R_E \cos \varphi \qquad \omega_E=\frac{2\pi}{86400\si{\second}}$\\ +$F_C$: Auf der Nordhalbkugel nach rechts. Auf der Südhalbkugel nach links. + +% Harmonische Schwingungen % ---------------------------------------------------------------------- -\section{Math} +\section{Harmonische Schwingungen} +$x(t)=A_0\cos(\omega t + \varphi)$ \\ +$\dot{x}(t)=-A_0\omega\sin(\omega t + \varphi)$\\ +$\ddot{x}(t)=-A_0\omega^2\sin(\omega t + \varphi)$ + +\subsection{Frei ungedämpft} +Bewegungsgleichung: $\ddot{x}+\omega_0^2x=0$\\ +$f_0=\frac{\omega_0}{2\pi} \qquad y_0=\frac{mg}{k} \qquad v_0 = \hat{x}\sqrt{\frac{k}{m}}$ +\begin{sectionbox} +\subsubsection{Federpendel} +$\omega_0^2=\frac{\text{Rücktreibende Kraft}}{\text{Einheitsmasse} \times \text{Einheitsauslenkung}}=\frac{k}{m} \rightarrow \omega = \sqrt{\frac{k}{m}}$\\ +$A_0=v_0\sqrt{\frac{m}{k}}=\frac{v_0}{\omega_0}$\\ +Bewegungsgleichung: $\ddot{x}+\frac{k}{m}x=0$ +\subsubsection{Fadenpendel} +Kleinwinkelnäherung: $\alpha << 1$\\ +$\omega_0=\sqrt{g}{l}$\\ +$l=\frac{T_0^2g}{4\pi^2} \qquad h=l-l\cos \alpha$\\ +$A_0 = v_0 \sqrt{\frac{l}{g}} \qquad E_{\ir ges} = mgl(\frac{3}{2}-\cos \alpha)$\\ +Bewegungsgleichung: $\ddot{x}+\frac{g}{l}x=0$ +\subsubsection{Torisionsschwingungen} +$\omega_0 = \sqrt{\frac{k_t}{I}}=\sqrt{\frac{k_t}{mr^2}}$\\ +$I=\frac{T_0^2}{4\pi^2}k_t \qquad T_0=2\pi \sqrt{\frac{I}{k_t}}$\\ +Bewegungsgleichung: $\ddot{x}+\frac{k_t}{I}x=0$ +\end{sectionbox} +\subsection{Frei gedämpft} +Stokesche Reibungskraft: $F_R=-6 \pi \eta R \vec{v}=-2m\delta\vec{v}$\\ +Bewegungsgleichung: $\ddot{x}+2\delta \dot{x}+\omega_0^2x=0$\\ + Resonanzfrequenz $\omega'=\sqrt{\omega_0^2-\delta^2}$ \\ + $\delta = \frac{b}{2m}=\frac{3\pi \eta R}{m}$\\ + +\begin{sectionbox} +\subsubsection{Schwingfall} +$\omega_0^2>\delta^2$\\ +$\omega'=\sqrt{\omega_0^2-\delta^2}<\omega_0$\\ +$x(t)=Ae^{-\delta t}\cos(\omega't)$ +\subsubsection{Aperiodischer Grenzfall} +$\omega_0^2=\delta^2$\\ +$\omega'=0$\\ +$x(t)=Ae^{-\delta t}\cdot (1+\delta t)$ +\subsubsection{Kriechfall} +$\omega_0^2<\delta^2$\\ +$\omega'=i\delta \pm i\sqrt{\delta^2-\omega_0^2}$\\ +$x(t)=A_1e^{-(\delta+\sqrt{\delta^2-\omega_0^2})t}+A_2e^{-(\delta - \sqrt{\delta^2-\omega_0^2})t}$\\ +\includegraphics[width=.9\columnwidth]{img/Schwingungen.pdf} +\end{sectionbox} +\subsection{Erzwungen gedämpft} +Bewegungsgleichung: $\ddot{x}+2\delta \dot{x}+\omega_0^2x=F_0e^{-i\Omega t}$\\ +%Resonanzfrequenz $\Omega_{\ir res}=\sqrt{\omega_0^2-2\gamma^2}$\\% +$\hat{x}(t)=Ae^{i(\Omega t + \varphi}$\\ +$A(\Omega)=\frac{F_0/m}{\sqrt{(\omega_0^2-\Omega^2)^2+(\frac{b\Omega}{m})^2}}$\\ +\begin{sectionbox} +\textbf{Resonanzfälle:} +\begin{enumerate} + \item $\Omega < \omega_0$\\ + $A\rightarrow \frac{F_0m^{-1}}{\omega_0^2}=\frac{F_0}{k}$\\ + $\Delta \varphi = 0 \qquad \Omega \rightarrow 0$ + \item $\Omega \approx \omega_0$ (Katastrophe)\\ + $A_{\ir max}=\frac{F_0}{b\omega_0}=\frac{F_0/m}{2\delta \sqrt{\omega_0^2-\delta^2}} \rightarrow \infty$\\ + $\Omega_{\ir max}=\sqrt{\omega_0^2-\frac{1}{2}(\frac{b}{m})^2}=\sqrt{\omega_0^2-2(\frac{3\pi \eta R}{m})^2}\rightarrow \omega_0$ + $\Delta \varphi = \frac{\pi}{2} \qquad \delta \rightarrow 0$ + \item $\Omega > \omega_0$\\ + $A\rightarrow 0$\\ + $\Delta \varphi = \pi \qquad \Omega \rightarrow \infty$ +\end{enumerate} +\end{sectionbox} + +% Wellen +% ---------------------------------------------------------------------- +\section{Wellen} +\subsection{Allgemeines} +Allgemeine Wellengleichung: $\frac{1}{c^2}\frac{\partial^2u}{\partial t^2}=\Delta u$\\ +$c=\lambda \cdot f=\frac{\omega}{k}=\frac{x}{\Delta t}$\\ +$\omega = 2\pi f=\frac{2\pi}{T} \qquad k=\frac{2\pi}{\lambda}=\frac{\omega}{c}$ +Funktion: $y(x,t)=\hat{y}\sin(kx - \omega t + \phi)$ +\subsection{Doppler-Effekt} \begin{sectionbox} - \subsection{Sinus} +\begin{enumerate} + \item Beobachter \textbf{bewegt}, Quelle \textbf{ruht}\\ + Annäherung: $f_B=f_Q(1+\frac{v_B}{c})$\\ + Entfernung: $f_B=f_Q(1-\frac{v_B}{c})$ + \item Beobachter \textbf{ruht}, Quelle \textbf{bewegt}\\ + Annäherung: $f_B=\frac{f_Q}{1-v_Q/c}$\\ + Entfernung: $f_B=\frac{f_Q}{1+v_Q/c}$ + \item Beobachter \textbf{bewegt}, Quelle \textbf{bewegt}\\ + $f_B = f_Q \frac{c \, \underset{\textcolor{green}{-}}{\textcolor{red}{+}} \, V_B}{c \, \underset{\textcolor{green}{+}}{\textcolor{red}{-}} \, V_Q} \qquad \text{Q} \, \underset{\textcolor{green}{\leftarrow} \textcolor{green}{\rightarrow}}{\textcolor{red}{\rightarrow} \textcolor{red}{\leftarrow}} \, \text{B}$\\ + $f_B = f_Q \frac{c \, \underset{\textcolor{cyan}{-}}{\textcolor{purple}{+}} \, V_B}{c \, \underset{\textcolor{cyan}{-}}{\textcolor{purple}{+}} \, V_Q} \qquad \text{Q} \, \underset{\textcolor{cyan}{\rightarrow} \textcolor{cyan}{\rightarrow}}{\textcolor{purple}{\leftarrow} \textcolor{purple}{\leftarrow}} \, \text{B}$ + \item Quelle \textbf{bewegt} ($v>1 \, \ir Ma$)\\ + $\sin \alpha=\frac{\diff w}{\diff s}=\frac{c_s t}{v_Qt}=\frac{1}{\ir Ma}$ +\end{enumerate} +\end{sectionbox} - Text goes here ... +\subsection{Schwebung} +Tritt auf bei $f_1\approx f_2$ ($f_1 \neq f_2$)\\ +$f_S=|f_1-f_2| \qquad f_R= \frac{f_1+f_2}{2}$\\ +$y(t)=\underbrace{2\hat{y}\cos(2\pi \cdot \frac{f_1-f_2}{2}\cdot t)}_{\text{Amplitudenfaktor}} \cdot \underbrace{\sin(2\pi \cdot \frac{f_1+f_2}{2}\cdot t)}_{f_R} $\\ +\includegraphics[width=.9\columnwidth]{Schwebung_crop.pdf} +\begin{minipage}{\columnwidth} +\subsection{Gekoppelte Wellen} +\begin{enumerate} + \item \textbf{Gleichphasig} $(x_1 = x_2)$ : $\omega_1 = \omega_0 = \sqrt{\frac{K}{m}}$ + + \item \textbf{Gegenphasig} $(x_1 \neq x_2)$ : $\omega_2 = \sqrt{\omega_0 + \frac{2K_{12}}{m}}$ + + \item \textbf{Allgemein}\\ + $x_1(t) = x_0 \cos\left( \frac{\omega_0 + \omega_2}{2} t \right) \cos\left( \frac{\omega_0 - \omega_2}{2} t \right)$\\ + $x_2(t) = x_0 \sin\left( \frac{\omega_0 + \omega_2}{2} t \right) \sin\left( \frac{\omega_0 - \omega_2}{2} t \right)$ +\end{enumerate} +\end{minipage} +\subsection{Interferenz} +$f_1=f_2 \qquad A_1 = A_2 \qquad \varphi_1 \neq \varphi_2$\\ +Konstruktiv: $\Delta \varphi = 2\pi \cdot n$\\ +Destruktiv: $\Delta \varphi = (2n+1)\pi$\\ +\textbf{Allgemein:} +\begin{itemize} + \item Gangunterschied $\Delta s = a\sin \alpha = \frac{x}{n}\sin \alpha$ + \item Phasendifferenz $\varphi = 2\pi \frac{\Delta}{\lambda}=2\pi a \frac{\sin \alpha}{n\lambda}$ + \item Winkel Max: $\sin \alpha_n = \pm n \cdot \frac{\lambda}{a}$ + \item Winkel Min: $\sin \alpha_n = \pm (n+\frac{1}{2})\cdot \frac{\lambda}{a}$ +\end{itemize} +\subsubsection{Doppelspalt} +Int max: $\Delta s = \lambda n$ (Konstruktiv)\\ +Int min: $\Delta s = (2n+1)\frac{\lambda}{2}$ (Destruktiv)\\ +$\Delta s = a \sin \alpha' \approx a \frac{x}{d}$\\ +Kleinwinkelnäherung: $\frac{x}{d}\approx \tan \alpha \approx \sin \alpha \approx \alpha \approx \alpha'$\\ +$\Rightarrow \frac{\Delta s}{a} = \frac{x}{d}$ +\subsubsection{Gitter} +Hauptmaximum: $\sin \alpha_{\ir max} = m\frac{\lambda}{d}$\\ +Einfachspalt: $\sin \alpha_{\ir min} = \pm m \frac{\lambda}{a}$ +\subsection{Reflexion und Wellen in einem Medium} +Festes Ende: $\varphi = \pi$ (Knoten am Ende)\\ +Loses Ende: $\varphi = 0$ (Bauch am Ende)\\ +Bei jeder zusätzlichen Oberschwingung wird die Welle gestaucht, sodass die obigen Bedingungen weiterhin gelten müssen. (Tipp: Skizze der Welle im Medium zeichnen) +% Optik +% ---------------------------------------------------------------------- +\section{Optik} + +\subsection{Rexlexion und Brechnung} +$f_m \cdot \lambda_m = c_m$ \qquad +$\lambda_m = \frac{\lambda_0}{m}$\\ +Berchnungsindex $n=\frac{c_0}{c_m}$ +\subsubsection{Brechungsmöglichkeiten} +\begin{enumerate} +\item[A)] \textbf{zum Lot} (dünn nach dicht)\\ +$\sin \alpha > \sin \beta$ \qquad $c_1 > c_2$ +\item[B)] \textbf{vom Lot} (dicht nach dünn)\\ +$\sin \alpha < \sin \beta$ \qquad $c_1 < c_2$ +\end{enumerate} +\subsubsection{Gesetz von Snellius} +$\frac{\sin \alpha}{\sin \beta} = \frac{n_2}{n_1} = \frac {c_1}{c_2}$ +\subsubsection{Totalreflexion} +Nur möglich wenn $n_1 > n_2$ und $\theta > \theta_{\ir krit}$ \\ +$\theta_{\ir krit} = \arcsin \frac{n_2}{n_1}$\\ +Einfallswinkel = Ausfallswinkel + + +\subsection{Linsen} +Linsengleichung +$D=\frac{1}{f}=\frac{1}{g}+\frac{1}{b}$\\ +Vergößerung: $V=\frac{B}{G}= - \frac{b}{g}=\frac{\tan \alpha}{\tan \alpha_0}$\\ +\includegraphics[width=\columnwidth]{img/Linsen_crop.pdf} +\begin{sectionbox} +\subsubsection{Hohlspiegel} +$f=\frac{r}{2}$ \qquad $V=\frac{B}{G}= - \frac{n_1b}{n_2g}$ \\ +$D=\frac{n_1}{g}+\frac{n_2}{b}=\frac{n_2-n_1}{r}$\\ +$g\rightarrow \infty: b=f_B=\frac{n_2r}{n_2-n_1}$\\ +$b\rightarrow \infty: g=f_G=\frac{n_1r}{n_2-n_1}$\\ +%Bild +\end{sectionbox} +\begin{sectionbox} +\subsubsection{Dünne Linsen} +$\frac{1}{f}=\frac{1}{g}+\frac{1}{b}=(\frac{n_2}{n_1}-1)(\frac{1}{r_1}-\frac{1}{r_2})$\\ +\textbf{Konvexe Linse (Sammellinse)} \\ +Korrektur von Weitsichtigkeit \\ +$V_{\ir sammel}= \frac{b}{g}=\frac{b-f}{f}$\\ +%Bild +\textbf{Abbildung} +\begin{itemize} + \item $\infty > g > 2f \Rightarrow 2f>b>f \quad G>B$ (real und invertiert) + \item $g=2f \Rightarrow 2f=b \quad G=B$ (real und invertiert) + \item $2f>g>f \Rightarrow \infty >b>2f \quad Gg \quad GB$) +\item[$\rightarrow$] Bild immer virtuell und aufrecht +\end{itemize} +\end{sectionbox} +\begin{sectionbox} +\subsubsection{Dicke Linsen} +$\frac{1}{f}=(\frac{n}{n_0}-1)(\frac{1}{r_1}-\frac{1}{r_2}+\frac{(n-n_0)d}{n\cdot r_1 \cdot r_2})$\\ +$h_1=-\frac{f(n-1)d}{n\cdot r_2}$ \qquad $h_2=-\frac{f(n-1)d}{n\cdot r_1}$ \end{sectionbox} +\begin{sectionbox} +\subsubsection{Linsensysteme} +\begin{itemize} + \item \textbf{Lochkamera} \\ + $S=\frac{D(b+g)}{g}$ + \item \textbf{Lupe} \\ + $V=\frac{\tan \alpha_L}{\alpha_0}=\frac{G/f}{G/s_0}=\frac{s_0}{f}$ mit $s_0=\SI{25}{\centi \meter}$ + \item \textbf{Fernrohr} \\ + $V=\frac{f_{ob}}{f_{ok}}$, $g\approx \infty$, $b\approx f$ \\ + $V_T=\frac{f}{g-f}=\frac{b-f}{f}\approx 0$ + \item \textbf{Mikroskop} \\ + $V=V_{ok}\cdot V_{ob}=\frac{(d-f_{ob})s_0}{f_ob \cdot f_ok}=-\frac{l}{f_{ob}}\cdot \frac{s_0}{f_{ok}}$ + \item \textbf{Auflösung Mikroskop} \\ + mit Spalt b: $\delta_{\ir{min}} = \alpha = \arcsin\frac{\lambda}{b} \approx \frac{\lambda}{b}$ (Abbé Limit)\\ + für runde Linse mit Durchmesser D: $\delta_{\ir{min}} = D\cdot\sin\alpha = 1,22\frac{\lambda}{D}$ (Rayleigh-Kriterium)\\ +\end{itemize} +\end{sectionbox} +\subsection{Absorption und Polarisation} + +\subsubsection{Absorption} +Berchungsindex mit Absorption: $n\rightarrow \tilde{n}=n-i\kappa$\\ +Re($\tilde{n}$)$=n$: Brechung \\ +Im($\tilde{n}$)$=\kappa$: Absorption \\ +Absorptionsgesetz von Lambert-Beer: $I=I_0e^{-\alpha \cdot d}=I_0e^{-\frac{2\omega}{c}\kappa d}$ + +\subsubsection{Polarisation} +Gesetz von Malus (für polarisiertes Licht): $I = I_0 \cos^2 \alpha$ + +Für unpolarisiertes Licht gilt nach einem Polarisationsfilter: $I = \frac{I_0}{2}$ + +%Malus, Lambert-Beer, etc. + +% Hydromechanik +% ---------------------------------------------------------------------- +\section{Hydromechanik} +Dichte: $\rho=\frac{m}{V}$\\ +Homogener Druck: $P=\frac{F_N}{A}$\\ +Kompressibilität: $\kappa = -\frac{1}{\Delta P}\cdot \frac{\Delta V}{V}$\\ +Kompressionsmodul: $K=\frac{1}{\kappa}$\\ +$R_E= \frac{V\cdot \rho \cdot L}{\eta}$ +\subsection{Hydraulische Presse} +$\frac{F_1}{A_1}=\frac{F_2}{A_2}$ \\ +$\frac{\Delta x_1}{\Delta x_2}=\frac{F_2}{F_1}=\frac{A_2}{A_1}$ +\subsection{Kontinuitätsgleichung} +$\dot{m}=\rho_1 A_1 \vec{v_1}=\rho_2 A_2 \vec{v_2}$\\ +$\dot{V}=\frac{\dot{m}}{\rho}=A\vec{v}$ + +\subsection{Bernoulli Gleichung} +$p_B=\underbrace{p}_{\text{statischer Druck}} ++ \underbrace{\frac{1}{2} \rho v^2}_{\text{dynamischer Druck}} ++ \underbrace{\rho gh}_{\text{hydrostatischer Druck}} = \ir{const.}$\\ +$p_{\ir B,1}\stackrel{!}{=}p_{\ir B,2}$ +(Energieerhaltung der Hydromechanik) + +\subsection{Torricelli} +$\vec{v_2}=\sqrt{2(gh+\frac{P_1-P_2}{\rho})}$\\ +$P_1=P_2=P_\infty = \SI{10e5}{\pascal}$ + +\subsection{Gesetz von Hagen-Poiseulle} +Strömung durch ein Rohr\\ +$\dot{V}=\frac{\diff V}{\diff t}=\frac{\pi (P_1-P_2)}{8\eta \cdot l}R^4$\\ +$v(r)=\frac{P_1-P_2}{4\eta \cdot l}(R^2-r^2)$ + +\subsection{Strömungen} +\begin{enumerate} + \item \textbf{Turbulente Strömung:} \\ + Größe Geschwindigkeiten, geringe innere Reibung, hohe Reibung an den Wänden.\\ + Newtonsches Reibungsgesetz:\\ + $F_R=\eta A \frac{\diff v}{\diff x}$\\ + $\eta=\eta_0e^{T/T_0}$ \\ + $R_e >> 1$: $F_w=F_R+F_D=c_w \cdot A \cdot \frac{\rho v^2}{2}$ + \item \textbf{Laminare Strömung:} \\ + $R_e<1$: Stokesche Reibung\\ + $F=b\cdot v = 6\pi \eta r v$ +\end{enumerate} +\textbf{Hydrostatisches Paradoxon:} Druck am Boden hängt nur von Füllhöhe, nicht Form/Menge ab.\\ +\textbf{Hydrodynamisches Paradoxon:} Gleichgewicht bei $mg=\frac{1}{2}\rho v^2 A$. Wenn $v$ steigt, dann fällt $P$. + + +% Thermodynamik +% ---------------------------------------------------------------------- +\section{Thermodynamik} +%aus alter LaTeX FS + Entropie und ein paar kleine Änderungen + +\textbf{Ideale Gasgleichung:} +$p \cdot V = \nu \cdot R \cdot T$\\ +Stoffmenge $\nu = \frac{m}{M}$ \qquad $M$: mol. Masse\\ +\textbf{Wärmemenge Q} \\ +$Q=C_p(T_2-T_1)=c \cdot m \cdot \Delta T$ \\ $c$: spez. Wärmekapazität \\ + +\textbf{Gesetz von Boyle + Mariotte} \\ +für $N=\ir const$ und $T=\ir const.$ gilt: $p_1V_1=p_2V_2$ \\ + +\textbf{2. Gesetz von Gay Lussac} \\ +feste Gasmenge; $V=\ir const.$\\ +$\frac{p_1}{T_1}=\frac{p_2}{T_2}$ \\ +\begin{sectionbox} +\textbf{Thermodynamische Systeme:} +\begin{itemize} + \item \textbf{Offen} (Verbrennungsmotor) \\ + Austausch von Materie, Arbeit, Wärme + \item \textbf{Geschlossen} (Stirlingmotor) \\ + Austausch von Arbeit, Wärme + \item \textbf{Abgeschlossen} (Thermoskanne) \\ + kein Austausch mit der Umgebung + +\end{itemize} +\end{sectionbox} +\textbf{Mittlere kin. Energie eines Gases} \\ +$\overline{E}_{\ir kin}=\frac{3}{2}k_B \cdot T$ \\ + +\textbf{Gesamte Translationsenergie} \\ +$U=\frac{3RT}{2M}$ \, $U$: innere Energie + +\textbf{Energien} \begin{sectionbox} - \subsection{Cosinus} +$\Delta U = \int_{T_1}^{T_2} c \, \diff T$\\ +$W=-\int_{V_1}^{V_2} p \diff V$ \qquad $\partial W= -p \cdot \partial V$ +\end{sectionbox} - Text goes here ... +\subsection{Hauptsätze der Thermodynamik} +\begin{itemize} +\item[0.] Zwei Körper im thermischen Gleichgewicht zu einem dritten\\ $\rightarrow$ Alle stehen untereinander im Gleichgewicht\\ +\item[1.] $\Delta U = \Delta Q + \Delta W \rightarrow $ Es gibt kein Perpetuum mobile erster Art - Maschine mit $>$100\% Wirkungsgrad\\ \\ +$\textbf{Verschiedene Möglichkeiten für Zustandsänderung:}$ +%TODO BESSERE LISTE %TODO +\begin{sectionbox} +$\textbf{a)}$ Isobarer Prozess, $p =$ const.\\ $\rightarrow$ im idealen Gas ist $C_p$ konstant $\Rightarrow Q_{12} = C_p \Delta T$\\ +$\textbf{b)}$ Isochorer Prozess: $V =$ const.\\ $\rightarrow$ im idealen Gas ist $C_v$ konstant $\Rightarrow Q_{12} = \Delta U$\\ +$\textbf{c)}$ Isothermer Prozess: $T =$ const.\\ $\Rightarrow$ $W_{12} = -Q_{12}$ %Umsetzung der Wärmezufuhr in Arbeit $\Rightarrow W_{12} = \\$ +$ =\nu RT \ln\frac{V_1}{V_2}$ +\\Freiwerdende Wärme: $Q_{12} = -W_{12}$\\ +$\textbf{d)}$ Adiabatischer Prozess: $\Delta Q = 0$\\ +%TODO Isochorer Prozess, Isothermer Prozess, adiabatischer Prozess verbessern\\ +In differentieller Schreibweise: $\partial U = \partial W + \partial Q $\\ +$\frac{T_2}{T_1}=({\nu_1}{\nu_2})^{\gamma -1}$\\ +$\Delta W = \Delta U = \eta \cdot C_u(T_2-T_1)$ \end{sectionbox} +\item[2.] Thermische Energie ist nicht in beliebigem Maße in andere Energiearten umwandelbar. $\eta < 1$ +\item[3.] Nernst'sches Theorem: $\lim_{T \rightarrow 0} S(T) = 0$ (Entropie bei 0 K ist 0)\\ +\end{itemize} + +\subsection{Adiabatengleichung} +$p\cdot V^\kappa = \ir const.$ \qquad $T\cdot V^{\kappa -1} = \ir const.$ \\ +Carnotscher Kreisprozess: Idee der Wärmekraftmaschine\\ +Wirkungsgrad $\eta = \frac{\vert W\vert}{Q_{12}}$, $\eta_{\text{Carnot}} = \frac{T_2-T_1}{T_2} < 1$\\ +Entropieänderung ist null + +\subsection{Entropie} +Maß der Unordnung in einem System +\begin{itemize} + \item Es ist wahrscheinlicher, dass die Unordnung zunimmt + \item Kann spontan nur in eine Richtung ablaufen +\end{itemize} +$\Rightarrow$ Entropie kann nur steigen oder konstant bleiben\\ +$S=-k_B\sum p_i \ln p_i$ \\ +$p_i$: Wahrscheinlichkeiten der Mikrozustände\\ +$\partial S = \frac{\partial Q_{rev}}{T}$\\ +Ideales Gas: $\Delta S= \int \partial S = n \cdot c_v \ln \frac{T_2}{T_1}+n\cdot R \ln \frac{V_2}{V_1}$ + + +% Quantenmechanik +% ---------------------------------------------------------------------- +\section{Quantenmechanik} +\subsection{Stefan Boltzmann} +$P=\epsilon \sigma A T^4 = 4 \pi r^2 E_0$\\ +Für schwarze Körper gilt: $\epsilon = 1$\\ +Wienscher Verschiebungssatz: $\lambda_{\ir max}=\frac{\SI{2,898e-3}{\kelvin}}{T} \si{\meter}$ 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+\geometry{paperwidth=11in, paperheight=3in} +\pagestyle{empty} +\begin{document} +\noindent +\centering +\begin{tabular}{|c|c|c|c|c|c|c|} +\hline +\textbf{Linsenform} & +\begin{pspicture}(2,-1) + \lens[lensradiusleft=1.5, lensradiusright=1.5, lensheight=1.5](0,0)(2,0) +\end{pspicture} & +\begin{pspicture}(2,-1) + \lens[lensradiusleft=0, lensradiusright=1.5, lensheight=1.5](0,0)(2,0) +\end{pspicture} & +\begin{pspicture}(2,-1) + \psarc(-0.3,0){1.4}{-40}{40} + \psarc(0.4,0){1}{-70}{70} +\end{pspicture} & +\begin{pspicture}(2,-1) + \lens[lensradiusleft=-1.5, lensradiusright=-1.5, lensheight=1.5](0,0)(2,0) +\end{pspicture} & +\begin{pspicture}(2,-1) + \lens[lensradiusleft=0, lensradiusright=-1.5, lensheight=1.5](0,0)(2,0) +\end{pspicture} & +\begin{pspicture}(2,-1) + \lens[lensradiusleft=-1.0, lensradiusright=2.5, lensheight=1.5](0,0)(2,0) +\end{pspicture} +\rule{0pt}{2.5cm} +\\ +\hline +\textbf{Bezeichnung} & bikonvex & plankonvex & konkav-konvex & bikonkav & plankonkav & konvex-konkav \\ +\hline +\textbf{Radien} & +$r_1 > 0$ \, $r_2 < 0$ & +$r_1 = \infty$ \, $r_2 < 0$ & +$r_1 < r_2 < 0$ & +$r_1 < 0$ \, $r_2 > 0$ & +$r_1 = \infty$ \, $r_2 > 0$ & +$r_2 < r_1 < 0$ \\ +\hline +\textbf{Brennweite} & +$f' > 0$ & +$f' > 0$ & +$f' > 0$ & +$f' < 0$ & +$f' < 0$ & +$f' < 0$ \\ +\hline +\end{tabular} + +\end{document} diff --git a/img/Linsen_crop.pdf b/img/Linsen_crop.pdf new file mode 100644 index 0000000000000000000000000000000000000000..33cb4ebae99b0c60f6305401a66850b8cf89e20d GIT binary patch literal 13417 zcmb_@2|Uza_djLHQg&r+BKtD?jJ@ppI+V2x1`}qO8AO)IR-{DAR-qz72qC0WArum_ zmQW;FvZnuMXwmY0zW?WW{brcg-21u9Iq!S!S?)a}W~QYh2bD)Ki#_XX=w?QMARwaC z5oQ$?X0Q>C;6`x=!4*K?Ud&(}ye9=m0)cfru@syZ&Y9?fV^&jRCR0c_tQYgqUYQY? zC4@#pliop7luYkHCP$9g%LljFYtml`D%>wmn)(bV3yj}@&MyF z6RZ~wBqCLKu{_?KqVT$=UXKJ)jEhkOB4u7T>%RKDB$n=vO|%I zivPOc=`U;yiEmKsQ#N%Py6+sLMp&cwKKp#u@A-B2Fal~nn>O34$OeJPz6_0PzEbA3H>rQ?{#fr?8U3z1PQT;8lHSJf!MM~Zk;-j zX9+5GxvlacdHe3JEZ$aHxrTbtv+!vTl|a`~bnom1xUQ_dFJCTOD#kTDM4`a1&P#gO 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