Andrew's git - notes.git/blob

e22575b2f61b1945861644d505f4a40a0feb3a43
   1\documentclass[a4paper, tikz]{article}
   2\usepackage[a4paper,margin=2cm]{geometry}
   3\usepackage{array}
   4\usepackage{amsmath}
   5\usepackage{amssymb}
   6\usepackage{tcolorbox}
   7\usepackage{fancyhdr}
   8\usepackage{pgfplots}
   9\usepackage{tikz}
  10\usetikzlibrary{arrows,
  11    calc,
  12    decorations,
  13    scopes,
  14}
  15\usepackage{tabularx}
  16\usepackage{keystroke}
  17\usepackage{listings}
  18\usepackage{xcolor} % used only to show the phantomed stuff
  19\definecolor{cas}{HTML}{e6f0fe}
  20
  21\pagestyle{fancy}
  22\fancyhead[LO,LE]{Year 12 Specialist - Dynamics}
  23\fancyhead[CO,CE]{Andrew Lorimer}
  24
  25\setlength\parindent{0pt}
  26
  27\begin{document}
  28
  29  \title{Dynamics}
  30  \author{}
  31  \date{}
  32  \maketitle
  33
  34  \section{Resolution of forces}
  35
  36  \textbf{Resultant force} is sum of force vectors
  37
  38  \subsection{In angle-magnitude form}
  39
  40  \makebox[3cm]{Cosine rule:} \(c^2=a^2+b^2-2ab\cos\theta\)
  41  \makebox[3cm]{Sine rule:} \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
  42  
  43  \subsection{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form}
  44
  45  Vector of \(a\) N at \(\theta\) to \(x\) axis is equal to \(a \cos \theta \boldsymbol{i} + a \sin \theta \boldsymbol{j}\). Convert all force vectors then add.
  46
  47  To find angle of an \(a\boldsymbol{i} + b\boldsymbol{j}\) vector, use \(\theta = \tan^{-1} \frac{b}{a}\)
  48
  49  \subsection{Resolving in a given direction}
  50
  51  The resolved part of a force \(P\) at angle \(\theta\) is has magnitude \(P \cos \theta\)
  52
  53  \section{Newton's laws}
  54  
  55  \begin{enumerate}
  56    \item Velocity is constant without a net external velocity
  57    \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\)
  58    \item Equal and opposite forces
  59  \end{enumerate}
  60
  61  \subsection{Weight}
  62  A mass of \(m\) kg has force of \(mg\) acting on it
  63
  64  \subsection{Momentum \(\rho\)}
  65  \[ \rho = mv \tag{units kg m/s or Ns} \]
  66
  67  \subsection{Reaction force \(R\)}
  68
  69  \begin{itemize}
  70    \item With no vertical velocity, \(R=mg\)
  71    \item With upwards acceleration, \(R-mg=ma\)
  72    \item With force \(F\) at angle \(\theta\), then \(R=mg-F\sin\theta\)
  73  \end{itemize}
  74
  75  \subsection{Friction}
  76
  77  \[ F_R = \mu R \tag{friction coefficient} \]
  78
  79  \section{Inclined planes}
  80
  81  \[ \boldsymbol{F} = |\boldsymbol{F}| \cos \theta \boldsymbol{i} + |\boldsymbol{F}| \sin \theta \boldsymbol{j} \]
  82 \def\iangle{30} % Angle of the inclined plane
  83
  84    \def\down{-90}
  85    \def\arcr{0.5cm} % Radius of the arc used to indicate angles
  86
  87\begin{tikzpicture}[
  88        >=latex',
  89        scale=1,
  90        force/.style={->,draw=blue,fill=blue},
  91        axis/.style={densely dashed,gray,font=\small},
  92        M/.style={rectangle,draw,fill=lightgray,minimum size=0.5cm,thin},
  93        m/.style={rectangle,draw=black,fill=lightgray,minimum size=0.3cm,thin},
  94        plane/.style={draw=black,fill=blue!10},
  95        string/.style={draw=red, thick},
  96        pulley/.style={thick},
  97        ]
  98        \pgfmathsetmacro{\Fnorme}{2}
  99        \pgfmathsetmacro{\Fangle}{30}
 100        \begin{scope}[rotate=\iangle]
 101            \node[M,transform shape] (M) {};
 102            \coordinate (xmin) at ($(M.south west)-({abs(1.1*\Fnorme*sin(-\Fangle))},0)$);
 103            \coordinate (xmax) at ($(M.south east)+({abs(1.1*\Fnorme*sin(-\Fangle))},0)$);
 104            \coordinate (ymax) at ($(M.north)+(0, {abs(1.1*\Fnorme*cos(-\Fangle))})$);
 105            \coordinate (ymin) at ($(M.south)-(0, 1cm)$);
 106            \coordinate (axiscentre) at ($(M.south)+(0.5cm, 0.5cm)$);
 107            \draw[postaction={decorate, decoration={border, segment length=2pt, angle=-45},draw,red}] (xmin) -- (xmax);
 108            \coordinate (N) at ($(M.center)+(0,{\Fnorme*cos(-\Fangle)})$);
 109            \coordinate (fr) at ($(M.center)+({\Fnorme*sin(-\Fangle)}, 0)$);
 110            % Draw axes and help lines
 111
 112            {[axis,->]
 113                \draw (ymin) -- (ymax) node[right] {\(\boldsymbol{j}\)};
 114                \draw (M) --(M-|xmax) node[right] {\(\boldsymbol{i}\)};    % mental note for me: change "right" to "above"
 115            }
 116
 117            % Forces
 118            {[force,->]
 119                % Assuming that Mg = 1. The normal force will therefore be cos(alpha)
 120                \draw (M.center) -- (N) node [right] {\(R\)};
 121                \draw (M.center) -- (fr) node [left] {\(\mu R\)};
 122            }
 123%            \draw [densely dotted, gray] (fr) |- (N) node [pos=.25, left] {\tiny$\lVert \vec F\rVert\cos\theta$} node [pos=.75, above] {\tiny$\lVert \vec F\rVert\sin\theta$};
 124        \end{scope}
 125        % Draw gravity force. The code is put outside the rotated
 126        % scope for simplicity. No need to do any angle calculations. 
 127        \draw[force,->] (M.center) -- ++(0,-1) node[below] {$mg$};
 128        \draw (M.center)+(-90:\arcr) arc [start angle=-90,end angle=\iangle-90,radius=\arcr] node [below, pos=.5] {\tiny\(\theta\)};
 129    \end{tikzpicture}
 130
 131  \subsection{Connected particles}
 132
 133  \begin{itemize}
 134    \item \textbf{Suspended pulley:} tension in both sections of rope are equal
 135    \item \textbf{Linear connection:} find acceleration of system first
 136    \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane
 137  \end{itemize}
 138
 139\end{document}