5891f4e7195bc967116ba2e5f1aaf8e22b49f49b
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   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,
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  16\usetikzlibrary{calc}
  17\usetikzlibrary{angles}
  18\usetikzlibrary{datavisualization.formats.functions}
  19\usetikzlibrary{decorations.markings}
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  21\usetikzlibrary{decorations.markings}
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  30        \uput[-90]{*0}(0,-2){$\vec{w}$}}
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  32
  33\usepackage{tabularx}
  34\usetikzlibrary{angles}
  35\usepackage{keystroke}
  36\usepackage{listings}
  37\usepackage{xcolor} % used only to show the phantomed stuff
  38\definecolor{cas}{HTML}{e6f0fe}
  39
  40\pagestyle{fancy}
  41\fancyhead[LO,LE]{Year 12 Specialist - Dynamics}
  42\fancyhead[CO,CE]{Andrew Lorimer}
  43
  44\setlength\parindent{0pt}
  45
  46\begin{document}
  47
  48\title{Dynamics}
  49\author{}
  50\date{}
  51\maketitle
  52
  53\section{Resolution of forces}
  54
  55\textbf{Resultant force} is sum of force vectors
  56
  57\subsection*{In angle-magnitude form}
  58
  59\makebox[3cm]{Cosine rule:} \(c^2=a^2+b^2-2ab\cos\theta\)
  60\makebox[3cm]{Sine rule:} \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
  61
  62\subsection*{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form}
  63
  64Vector 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.
  65
  66To find angle of an \(a\boldsymbol{i} + b\boldsymbol{j}\) vector, use \(\theta = \tan^{-1} \frac{b}{a}\)
  67
  68\subsection*{Resolving in a given direction}
  69
  70The resolved part of a force \(P\) at angle \(\theta\) is has magnitude \(P \cos \theta\)
  71
  72To convert force \(||\vec{OA}\) to angle-magnitude form, find component \(\perp\vec{OA}\) then \(|\boldsymbol{r}|=\sqrt{\left(||\vec{OA}\right)^2 + \left(\perp\vec{OA}\right)^2},\quad \theta = \tan^{-1}\dfrac{\perp\vec{OA}}{||\vec{OA}}\)
  73
  74\section{Newton's laws}
  75
  76\begin{tcolorbox}
  77  \begin{enumerate}
  78    \item Velocity is constant without a net external velocity
  79    \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\)
  80    \item Equal and opposite forces
  81  \end{enumerate}
  82\end{tcolorbox}
  83
  84\subsection*{Weight}
  85A mass of \(m\) kg has force of \(mg\) acting on it
  86
  87\subsection*{Momentum \(\rho\)}
  88\[ \rho = mv \tag{units kg m/s or Ns} \]
  89
  90\subsection*{Reaction force \(R\)}
  91
  92\begin{itemize}
  93  \item With no vertical velocity, \(R=mg\)
  94  \item With upwards acceleration, \(R-mg=ma\)
  95  \item With force \(F\) at angle \(\theta\), then \(R=mg-F\sin\theta\)
  96\end{itemize}
  97
  98\subsection*{Friction}
  99
 100\[ F_R = \mu R \tag{friction coefficient} \]
 101
 102\section{Inclined planes}
 103
 104\[ \boldsymbol{F} = |\boldsymbol{F}| \cos \theta \boldsymbol{i} + |\boldsymbol{F}| \sin \theta \boldsymbol{j} \]
 105\begin{itemize}
 106  \item Normal force \(R\) is at right angles to plane
 107  \item Let direction up the plane be \(\boldsymbol{i}\) and perpendicular to plane \(\boldsymbol{j}\)
 108\end{itemize}
 109
 110\def\iangle{30} % Angle of the inclined plane
 111
 112\def\down{-90}
 113\def\arcr{0.5cm} % Radius of the arc used to indicate angles
 114
 115\tikzset{
 116  force/.style={->,draw=blue,fill=blue},
 117  axis/.style={densely dashed,gray,font=\small},
 118  M/.style={rectangle,draw,fill=lightgray,minimum size=0.5cm,thin},
 119  m/.style={rectangle,draw=black,fill=lightgray,minimum size=0.3cm,thin},
 120  plane/.style={draw=black,fill=blue!10},
 121  string/.style={draw=red, thick},
 122  pulley/.style={thick}
 123}
 124
 125\begin{figure}[!htb]
 126  \centering
 127  \begin{tikzpicture}
 128
 129    \pgfmathsetmacro{\Fnorme}{2}
 130    \pgfmathsetmacro{\Fangle}{30}
 131
 132    \begin{scope}[rotate=\iangle]
 133      \node[M,transform shape] (M) {};
 134      \coordinate (xmin) at ($(M.south west)-({abs(1.1*\Fnorme*sin(-\Fangle))},0)$);
 135      \coordinate (xmax) at ($(M.south east)+({abs(1.1*\Fnorme*sin(-\Fangle))},0)$);
 136      \coordinate (ymax) at ($(M.north)+(0, {abs(1.1*\Fnorme*cos(-\Fangle))})$);
 137      \coordinate (ymin) at ($(M.south)-(0, 1cm)$);
 138      \coordinate (axiscentre) at ($(M.south)+(0.5cm, 0.5cm)$);
 139      \draw[postaction={decorate, decoration={border, segment length=2pt, angle=-45},draw,red}] (xmin) -- (xmax);
 140      \coordinate (N) at ($(M.center)+(0,{\Fnorme*cos(-\Fangle)})$);
 141      \coordinate (fr) at ($(M.center)+({\Fnorme*sin(-\Fangle)}, 0)$);
 142      {[axis,-]
 143      \draw (ymin) -- (M.center);
 144      }
 145      {[axis,->]
 146      \draw ($(M)+(1,0)$) -- ($(M)+(2,0)$) node[above right] {\(\boldsymbol{i}\)};
 147      \draw ($(M)+(1,0)$) -- ($(M)+(1,1)$) node[above right] {\(\boldsymbol{j}\)};
 148      }
 149      {[force,->]
 150        \draw (M.center) -- (N) node [right] {\(R\)};
 151        \draw (M.center) -- (fr) node [left] {\(\mu R\)};
 152      }
 153    \end{scope}
 154    \draw[force,->] (M.center) -- ++(0,-1) node[below] {$mg$};
 155    \draw (M.center)+(-90:\arcr) arc [start angle=-90,end angle=\iangle-90,radius=\arcr] node [below, pos=.5] {\footnotesize\(\theta\)};
 156  \end{tikzpicture}
 157\end{figure}
 158
 159\section{Connected particles}
 160
 161\begin{itemize}
 162  \item \textbf{Suspended pulley:} tension in both sections of rope are equal
 163  \item \textbf{Linear connection:} find acceleration of system first
 164  \item \textbf{Pulley on right angle:} \(a = \frac{m_2g}{m_1+m_2}\) where \(m_2\) is suspended (frictionless on both surfaces)
 165  \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane
 166\end{itemize}
 167
 168\def\boxwidth{0.5}
 169\tikzset{
 170  box/.style={rectangle,draw,fill=lightgray,minimum width=\boxwidth,thin},
 171  m/.style={rectangle,draw=black,fill=lightgray,minimum size=\boxwidth, font=\footnotesize, thin}
 172}
 173
 174
 175\begin{figure}[!htb]
 176  \centering
 177  \begin{tikzpicture}
 178
 179    \matrix[column sep=1cm] {
 180      \begin{scope}
 181
 182        \coordinate (O) at (0,0);
 183        \coordinate (A) at ($({3*cos(\iangle)},{3*sin(\iangle)})$);
 184        \coordinate (B) at ($({3*cos(\iangle)},0)$);
 185        \coordinate (C) at ($({(1.5-0.5*\boxwidth)*cos(\iangle)},{(1.5-0.5*\boxwidth)*sin(\iangle)})$); % centre of box
 186        \coordinate (D) at ($(C)+(\iangle:\boxwidth)$);
 187        \coordinate (E) at ($(D)+(90+\iangle:0.5*\boxwidth)$);
 188        \coordinate (F) at ($(B)+(0,{1.5*sin(\iangle)})$);
 189        \coordinate (X) at ($(A)+(\iangle:0.5*\boxwidth)$); % centre of pulley
 190        \coordinate (Y) at ($(X)+(90+\iangle:0.5*\boxwidth)$); % chord of pulley
 191
 192        \draw[plane] (O) -- (A) -- (B) -- (O);
 193        \draw (O)+(\arcr,0) arc [start angle=0,end angle=\iangle,radius=\arcr] node [right, pos=.75] {\footnotesize\(\theta\)};
 194
 195        \draw [rotate=\iangle, m] (C) rectangle ++(\boxwidth,\boxwidth) node (z) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
 196        \draw [pulley] (A) -- (X) ++(0.5*\boxwidth, 0) arc[rotate=\iangle, start angle=0, delta angle=360, x radius=0.25, y radius=0.25] node(r) [midway, rotate=\iangle] {};
 197        \draw [string] (E) -- (Y) arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)})$) node[m] {\(m_2\)};
 198
 199      \end{scope}
 200
 201      &
 202
 203      \begin{scope}[rotate=\iangle]
 204
 205        \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node (m1) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
 206
 207        {[axis,-]
 208          \draw (0,-1) -- (0,0);
 209          \draw[solid,shorten >=0.5pt] (\down-\iangle:\arcr) arc(\down-\iangle:\down:\arcr);
 210          \node at (\down-0.5*\iangle:1.3*\arcr) {\(\theta\)};
 211        }
 212
 213        {[force,->]
 214          \draw (M.center) -- ++(0,{cos(\iangle)}) node[above right] {\(R_1\)};
 215          \draw (M.west) -- ++(-0.5,0) node[left] {\(\mu R_1\)};
 216          \draw (M.east) -- ++(1,0) node[above] {\(T_1\)};
 217        }
 218
 219        \draw[force,->, rotate=-\iangle] (M.center) -- ++(0,-1) node[below] {\(m_1 g\)};
 220
 221      \end{scope}
 222
 223      &
 224
 225      \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node [midway, font=\footnotesize] {\(m_2\)};
 226
 227      {[force,->]
 228        \draw (0,0.5*\boxwidth) -- ++(0,1) node[above] {\(T_2\)};
 229        \draw (0,-0.5*\boxwidth) -- ++(0,-1) node[right] {\(m_2 g\)};
 230      }
 231      \\
 232    };
 233  \end{tikzpicture}
 234\end{figure}
 235
 236\section{Equilibrium}
 237
 238\[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\]
 239
 240Three methods:
 241\begin{enumerate}
 242  \item Lami's theorem (sine rule)
 243  \item Triangle of forces or CAS (use to verify)
 244  \item Resolution of forces (\(\Sigma F = 0\) - simultaneous)
 245\end{enumerate}
 246
 247\colorbox{cas}{On CAS:} use Geometry, lock known constants.
 248
 249\section{Variable forces (DEs)}
 250
 251\[ a = \dfrac{d^2x}{dt^2} = \dfrac{dv}{dt} = v\dfrac{dv}{dx} = \dfrac{d}{dx} \left( \frac{1}{2} v^2 \right) \]
 252
 253\end{document}