formelsamling
Författare
Jonas Vasur
Last Updated
för 10 år sedan
Licens
Creative Commons CC BY 4.0
Sammanfattning
Formelsamling för fysik 1a av Jonas Vasur. Påbörjades på Mediagymnasiet (Nacka Strand), våren 2014.
\title {formelsamling}
\documentclass{article}
\pagenumbering{gobble} %inga bladnummer
%\renewcommand{\familydefault}{\sfdefault} %will switch to using sans-serif for everything except mathematics. The sans-serif will be computer modern sans unless you also put
%\usepackage{helvet} %san serif Helvetica-kopia (typsnitt)
%\documentclass{article}
%\usepackage{mathtools} % \abs{F} absoluta värden? clashes with what?
\usepackage{cancel} %att kunna stryka ut variabler
\usepackage{amssymb}
\usepackage[fleqn]{amsmath} % flush left equations
\mathindent=0pt % ingen indentering
\setlength{\columnsep}{30pt} % så att det skalll finnas "luft" mellan spalterna
\linespread{1.5} %ökar avstånd mellan rader till 1,8
\usepackage{textcomp} %för att kunna skriva \textmu{}
\usepackage [swedish]{babel}
\usepackage[utf8x]{inputenc} %krävs för att å ä ö skall skrivas ut rakt av
\usepackage{multicol} % så att jag kan ha flera spalter
\usepackage[a4paper,landscape]{geometry}
\usepackage{ziffer} %för korrekt avstånd mellan siffror och kommatecken
\usepackage{siunitx} %för att enhetsbeteckningar inte skrivs kursivt & för att kunna skriva Å för ångström i ekvationer
\sisetup{per-mode=fraction} %SI set-up ger horisontellt bråkstreck på alla enheter
%siunitx.pdf page 38
\sisetup{fraction-function=\dfrac} %dfrac skall göra att bråken blir större, luftigare
\sisetup{locale = FR} % SI set-up som gör allt FRanskt (kommatecken, t.ex.)
\let\times\cdot %gör att kommandot "\times" blir en prick varje gång (multiplikation)
\usepackage{tikz}
\usetikzlibrary{calc, intersections, decorations.markings, arrows, positioning}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usetikzlibrary{scopes} %krävs för exemplet med frikroppsdiagrammet
\setlength{\textheight}{450pt}
\setlength{\textwidth}{760pt}
\setlength{\topmargin}{-0.87in}
\setlength{\oddsidemargin}{-0.4in}
\setlength{\evensidemargin}{-0.4in}
\setlength{\parindent}{0pt}
\begin{document}
\begin{multicols}3
%\noindent A \hfill Z
%\setlength{\mathindent}{0pt}
\section*{TRIGONOMETRI}
\begin{minipage}[t]{0.27\textwidth}
\begin{multicols}2
\begin{equation*}
a^2 + b^2 = c^2
\end{equation*}
\begin{tikzpicture}[thick]
\coordinate (O) at (0,0);
\coordinate (A) at (3,2);
\coordinate (R) at (3,0);
\draw (O)--(A)--(R)--cycle;
\tkzLabelSegment[above left=2pt](O,A){$c$};
\tkzLabelSegment[below=2pt](O,R){$b$};
\tkzLabelSegment[right=2pt](A,R){$a$};
\tkzMarkRightAngle[size=0.5](O,R,A)% square angle here
\tkzMarkAngle[size=1cm](R,O,A)% sequence of points describes angle
\tkzLabelAngle[pos = 0.7](R,O,A){$\theta$}
\end{tikzpicture}
%\vfill
\columnbreak
\begin{eqnarray*}
\\ \sin\theta & = & \frac{a}{c}
\\ \cos\theta & = & \frac{b}{c}
\\ \tan\theta & = & \frac{a}{b}
% \tan\theta & = & \frac {\sin\theta}{\cos\theta}
\end{eqnarray*}
\end{multicols}
\end{minipage}
\section*{VEKTORER}
\begin{multicols}2
\begin{minipage}[t]{0.27\textwidth}
\begin{tikzpicture}
\tkzDefPoint(0,0){O} \tkzDefPoint(3,0){X} \tkzDefPoint(0,2){Y} \tkzDefPoint(3,2){P}
\draw [->][thin] (0,0)--(0,2);
\tkzLabelSegment[above=3pt](O,P){$\mathbf{\vec{F}_{res}}$};
\tkzLabelSegment[left=3pt](O,Y){$\vec{F}_y$};
\tkzLabelSegment[below=3pt](O,X){$\vec{F}_x$};
\draw [->][thin] (0,0)--(3,0);
\draw [->][very thick, black] (0,0)--(3,2);
\draw [thin, loosely dotted] (3,0)--(3,2);
\draw [thin, loosely dotted] (0,2)--(3,2);
\end{tikzpicture}
\end{minipage}
Summan av krafter i jämviktat system lika med noll (se kraftdiagram till höger)\\
$\vec{F}_N + \vec{F}_f + \vec{F}_g = 0$
\columnbreak
%\vfill
\begin{eqnarray*}
\mathbf{\vec{F}_{res}} &= \vec{F}_x + \vec{F}_y
\\ F_{res}^2 &= F_x^2 + F_y^2
\\ \Rightarrow F_{res} &= \sqrt{F_x^2 + F_y^2}
\end{eqnarray*}
\\
\begin{tikzpicture}[scale=1]%[dot/.style={draw,circle,minimum size=10mm,inner sep=0pt,outer sep=0pt,fill=black}
% Har inte fått det ifyllda cirkeln i mitten av kraftdiagrammet OMG !!!
%{coordinate [dot] (mitt) at (0,0)};
\tkzDefPoint(0,0){O} \tkzDefPoint(1,0.577){f} \tkzDefPoint(-1,1.73){N} \tkzDefPoint(0,-2.31){g}
\tkzLabelSegment[right=9pt](O,f){$\vec{F}_{f}$};
\tkzLabelSegment[left=1pt](O,N){$\vec{F}_N$};
\tkzLabelSegment[left=1pt](O,g){$\vec{F}_g$};
\filldraw [black] (0,0) circle (4pt);
\draw [->][thick] (0,0)--(1,0.577);
\draw [->][thick] (0,0)--(-1,1.73);
\draw [->][thick] (0,0)--(0,-2.31);
\end{tikzpicture}
\end{multicols}
\section*{POTENSER}
%\noindent A \hfill Z
%\\
\begin{align*}
a^x \times a^y &= a^{x+y} & 2^3 \times 2^2 &= 2^{(3+2)}
\\ \frac{a^x}{a^y} &= a^{x-y} & \frac{2^3}{2^2} &= 2^{(3-2)}
\\ \left(a^x \right)^y &= a^{xy} & (2^3)^2 &= 2^{(3 \times 2)}
\\ a^{-x} &= \frac{1}{a^x} & 2^{-3} &= \frac{1}{2^3}
\\ a^x \times b^x &= \left(a \times b \right)^x & 2^3 \times 5^3 &= (2 \times 5)^3
\\ \frac {a^x}{b^x} &= \left(\frac{a}{b}\right)^x & \frac {2^2}{5^2} &= \left(\frac{2}{5}\right)^2
\\ a^{\frac{1}{n}} &= \sqrt[n]{a} & 8^{\frac{1}{3}} &= \sqrt[3]{8}
\\ a^0 &= 1 & 2^0 &= 1
\end{align*}
\begin{tabular}{ >{$}l<{$} >{$}r<{$}@{,}>{$}l<{$} r l} % >{$}l<{$} definierar vilka spalter som är i math mode; @ tillåter alignering runt kommatecknet
%\text{tiopotens}& \text{siffra} & &&prefix \\
%\hline
10^{-12}& 0 & 000 000 000 001 & p & pico- \\
10^{-9} &0 & 000 000 001 & n & nano- \\
10^{ -6} &0 & 000 001 & \textmu{} & micro-\\
10^{ -3} &0 &001& m & milli-\\
10^{-2} &0 &01 & c & centi-\\
10^{ -1} &0 & 1 & d & deci-\\
10^{ 0} &1 & & & \\
10^{ 2} &100 & & h & hekto-\\
10^{ 3} &1000 & & k & kilo-\\
10^{ 6} &1000 000 & & M & mega-\\
10^{ 9} &1000 000 000 & & G & giga-\\
10^{ 12} &1000 000 000 000& & T & tera-\\
%\hline
\end{tabular}
%\begin{table*}
%\tabular[l l l r]
%\begin{tabular}{r@{.}l}
%\begin{math}
% 3 & 14159 \\
% 16 & 2 \\
% 123 & 456 \\
%\end{math}
%\end{tabular}
%\end{table*}
%
\section*{AREA, VOLYM \& DENSITET}
\begin{align*}
& A_{kvadrat} = x \times y &&A_{cirkel} = \pi \times r^2
%\\ &= \SI{2}{\m} \times \{SI}{2}{\m} &
%\\ &= \SI{4}{\m\squared}
% kvadrat
\\
&\begin{tikzpicture}
\draw[step=1cm][dashed, color=gray] (0,0) grid (2,2);
\draw [thick] (0,0)--(0,2)--(2,2)--(2,0)--(0,0);
\draw [|-|][thin] (2.5,0)--(2.5,2) node[midway, right] {y};
\draw [|-|][thin] (0,-0.5)--(2,-0.5) node[midway, below] {x};
\end{tikzpicture}
% cirkel
&&\begin{tikzpicture}
\draw[step=1cm] [dashed, color=gray](1,0) grid (3,2);
\draw [thick] (2,1)--(3,1) node[midway, above]{$r$};
\draw[thick](2,1) circle [radius=1cm] ;
\draw (0,-0.8) circle [radius=0pt]; %osynlig cirkel för att tvinga fram positionering
\end{tikzpicture}
\\
% rectangular tower !!!!!!!!!!!!!!!!
&\begin{tikzpicture} [scale=0.5]
%\fill[top color=gray!50!black,bottom color=gray!10,middle color=gray,shading=axis,opacity=0.2] (0,0) circle (2cm and 0.5cm);
%\fill[left color=gray!50!black,right color=gray!50!black,middle color=gray!50,shading=axis,opacity=0.2] (2,0) -- (2,6) arc (360:180:2cm and 0.5cm) -- (-2,0) arc (180:360:2cm and 0.5cm);
%\fill[top color=gray!90!,bottom color=gray!2,middle color=gray!30,shading=axis,opacity=0.2] (0,6) circle (2cm and 0.5cm);
%\draw (-2.5,6) -- (-2.5,0) arc (180:360:2.5cm and 0.5cm) -- (2.5,6) ++ (-2.5,0) circle (2.5cm and 0.5cm);
\draw (-2.5,0) -- (-2.5,6) -- (-0.3,5.6) -- (2.5,6) -- (2.5,0); % sidor & fram top
\draw (-2.5,0) -- (-0.3,-0.4) -- (2.5,0); %fram botten
\draw (-2.5,6) -- (0.3,6.4) -- (2.5,6); %bak top
\draw[densely dashed] (-2.5,0) -- (0.3,0.4) -- (2.5,0); %bak botten
\draw[densely dashed] (0.3,6.4) -- (0.3,0.4); %bak vertikalen
\draw (-0.3,5.6) -- (-0.3,-0.4); %framvertikalen
\fill[left color=gray!50!black,right color=gray!50!black,middle color=gray!50,shading=axis,opacity=0.2] (2.5,0) -- (2.5,6);
\draw[|-|][thin] (3.5,0) -- (3.5,6) node[midway, right] {$h$};
%\draw[|-|][densely dashed] (0,0) -- (2.5,0) node[midway, above] {$r$};
\end{tikzpicture}
%shaded cylinder **********
&&\begin{tikzpicture} [scale=0.5]
\fill[top color=gray!50!black,bottom color=gray!10,middle color=gray,shading=axis,opacity=0.2] (0,0) circle (2cm and 0.5cm);
\fill[left color=gray!50!black,right color=gray!50!black,middle color=gray!50,shading=axis,opacity=0.2] (2,0) -- (2,6) arc (360:180:2cm and 0.5cm) -- (-2,0) arc (180:360:2cm and 0.5cm);
\fill[top color=gray!90!,bottom color=gray!2,middle color=gray!30,shading=axis,opacity=0.2] (0,6) circle (2cm and 0.5cm);
\draw (-2,6) -- (-2,0) arc (180:360:2cm and 0.5cm) -- (2,6) ++ (-2,0) circle (2cm and 0.5cm);
\draw[densely dashed] (-2,0) arc (180:0:2cm and 0.5cm);
\draw[|-|][thin] (-3,0) -- (-3,6) node[midway, left] {$h$};
\draw[|-|][densely dashed] (0,0) -- (2,0) node[midway, above] {$r$};
\end{tikzpicture}
% *******************
\\ V &= A \times h & V &= A \times h
\\ &= \SI{4}{\m\squared} \times \SI{6}{\m}
\\ &= \SI{24}{\m\cubed}
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Densitet}
\begin{align*}
\\ \rho &= \frac {m}{V} &
\\ \rho_{vatten} &=\SI{1000}{\kg\per\m\cubed}= \frac{\SI{1000}{\kg}} {\SI{1}{\m\cubed}} \tag{vattnets densitet}
\\ &= \SI{1,00}{\kg\per\liter}
\end{align*}
\\
\subsection*{Omräkning kubikmeter till liter}
\begin{align*}
\SI{1}{m\cubed} &= (\SI{1}{\m})^3 \times 1
\\&= (\SI{1}{\m})^3 \times \left(\frac{\SI{10}{\dm}}{\SI{1}{\m}}\right)^3
\\&= \SI{1}{\cancel\m\cubed} \times \frac{\SI{e3}{\dm\cubed}}{\SI{1}{\cancel\m\cubed}}
%\\&= \SI{1}{\cancelto{\m}{\m\cubed}} \times \frac{\SI{e3}{\dm\cubed}}{\SI{1}{\cancel\m\cubed}}
\\ &= \SI{e3}{\dm\cubed}
\\ &= \SI{e3}{\litre}
\end{align*}
\section*{RÖRELSE}
\begin{align*}
v &= \frac {\Delta s}{\Delta t} & \SI{1}{\m\per\s}&= \frac{\SI{1}{\m}} {\SI{1}{\s}} \tag{hastighet}
\\
\\a &= \frac {\Delta v}{\Delta t} & \SI{1}{\m\per\s\squared}&= \frac{\frac{\SI{1}{\m}} {\SI{1}{\s}}}{\SI{1}{\s}} \tag{acceleration}
\\F &= m \times a & \SI{1}{\N} &= \SI{1}{\kg} \times \SI{1}{\m\per\s\squared} \tag{kraft}
\\W &= F \times \Delta s & \SI{1}{\J} &= \SI{1}{\N} \times \SI{1}{\m} \tag{arbete: Joule}
\\ &= E & \SI{1}{\J} \tag{energi: Joule}
\\P &= \frac{\Delta W}{\Delta t} & \SI{1}{\W} &= \frac{\SI{1}{\J}}{\SI{1}{\s}} \tag{effekt: Watt}
\end{align*}
\begin{align*} %alignera två ggr så att förskjutningen inte blir för stor för hela spalten
\\p=m \times v & &\SI{1}{\kg\m\per\s} &= \SI{1}{\kg} \times \SI{1}{\m\per\s} \tag{rörelsemängd}
\\E_{kinetisk} &= \frac{1}{2} \times m \times v^2 & \SI{0,5}{\J} &= \frac{1}{2} \times \SI{1}{\kg} \times \left(\SI{1}{\m\per\s}\right)^2 \tag{rörelseenergi}
\\E_{potential} &= m \times g \times h & \SI{10}{\J} &= \SI{1}{\kg} \times \SI{10}{\m\per\s\squared} \times \SI{1}{\m} \tag{lägesenergi}
\end{align*}
\section*{TRYCK \& VÄRME}
\begin{align*}
p &= \frac {F}{A} & \SI{1}{\pascal} &= \frac{\SI{1}{\newton}} {\SI{1}{\m\squared}} \tag{tryck: Pascal}
\\ p &= {\rho} \times g \times h & \SI{10000}{\Pa} &= \SI{1000}{\kg\per\m\cubed} \times \SI{10}{\m\per\s\squared} \times \SI{1}{\m} \tag{vätsketryck}
\\ F &= {\rho} \times g \times V & \SI{10000}{\N} &= \SI{1000}{\kg\per\m\cubed} \times \SI{10}{\m\per\s\squared} \times \SI{1}{\m\cubed} \tag{Arkimedes princip: lyftkraft i Newton}
\end{align*}
\begin{align*}
\\p \times V &= n \times R \times T \tag{Allmänna gaslagen}
\\ \SI{2e5}{\Pa} \times \SI{1}{\m\cubed} &= \SI{24,0}{\mol} \times \SI{8,314}{\Pa\m\cubed\per\mol\per\K} \times \SI{1002}{\K}
\end{align*}
\begin{align*}
\\ E &= c \times m \times T & \SI{800}{\kJ} &= \SI{4}{\kJ\per\kg\per\K} \times \SI{4}{\kg} \times \SI{50}{\kelvin} \tag{c = specifik värmekapacitet}
\\ \Delta E &= c \times m \times \Delta T & \SI{1000}{J} &= \SI{1000}{\joule\per\kg\per\K} \times \SI{1}{\kg} \times \SI{1}{\kelvin} \tag{c = specifik värmekapacitet}
\\ \Delta E &= l_s \times m & \SI{1000}{J} &= \SI{1000}{\joule\per\kg} \times \SI{1}{\kg} \tag{smältvärme}
\\ \Delta E &= l_a \times m & \SI{1000}{J} &= \SI{1000}{\joule\per\kg} \times \SI{1}{\kg} \tag{ångbildningsvärme} %OBS: subscript funkar inte att ha med i tags
\end{align*} % no empty line allowed before end
\section*{ELEKTRICITET}
\begin{align*}
F &= k_e \times \frac{Q_1 \times Q_2}{r^2} \tag{Coulombs lag, där $k_e = \SI{8,99e9}{\N\m\squared\per\coulomb\squared}$}
%\\ k_e &= \SI{8,99e9}{\N\m\squared\per\coulomb\squared}
\\ I &= \frac{Q}{t} & \SI{1}{A} &= \frac{\SI{1}{C}}{\SI{1}{s}} \tag{ström: Ampere}
\\ U &= \frac{E}{Q} & \SI{1}{\V} &= \frac{\SI{1}{J}}{\SI{1}{C}} \tag{spänning: Volt}
\\ U &= R \times I & \SI{1}{V} &= \SI{1}{\ohm} \times \SI{1}{A}\tag{Ohms lag, där R=Resistans=motstånd: Ohm}
\\ P &= U \times I & \SI{1}{W} &= \SI{1}{V} \times \SI{1}{A} \tag{Effekt: Watt}
\\ &= R \times I^2 & &= \SI{1}{\ohm} \times \SI{1}{A\squared}
\\ &= R \times I^2 & \SI{1}{W} &= \SI{1}{\ohm} \times (\SI{1}{A})^2)
\\ R_{tot} &= R_1 + R_2 \tag {för två seriekopplade motstånd}
\\ \frac{1}{R_{tot}} &= \frac{1}{R_1} + \frac{1}{R_2} \tag {för två parallelkopplade motstånd}
\end{align*}
\end{multicols}
\end{document}