7856ca90eab3056143248c1de2479573e0337f72
   1\documentclass[spec-collated.tex]{subfiles}
   2\begin{document}
   3
   4\section{Dynamics}
   5
   6\subsection*{Resolution of forces}
   7
   8\textbf{Resultant force} is sum of force vectors
   9
  10\subsubsection*{In angle-magnitude form}
  11
  12\makebox[3cm]{Cosine rule:} \(c^2=a^2+b^2-2ab\cos\theta\)
  13\makebox[3cm]{Sine rule:} \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
  14
  15\subsubsection*{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form}
  16
  17Vector 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.
  18
  19To find angle of an \(a\boldsymbol{i} + b\boldsymbol{j}\) vector, use \(\theta = \tan^{-1} \frac{b}{a}\)
  20
  21\subsubsection*{Resolving in a given direction}
  22
  23The resolved part of a force \(P\) at angle \(\theta\) is has magnitude \(P \cos \theta\)
  24
  25To convert force \(||\vec{OA}\) to angle-magnitude form, find component \(\perp\vec{OA}\) then:
  26\begin{align*}
  27  |\boldsymbol{r}| &= \sqrt{\left(||\vec{OA}\right)^2 + \left(\perp\vec{OA}\right)^2} \\
  28  \theta &= \tan^{-1}\dfrac{\perp\vec{OA}}{||\vec{OA}}
  29\end{align*}
  30
  31\subsection*{Newton's laws}
  32
  33\begin{tcolorbox}
  34  \begin{enumerate}[leftmargin=1mm]
  35    \item Velocity is constant without \(\Sigma F\)
  36    \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\)
  37    \item Equal and opposite forces
  38  \end{enumerate}
  39\end{tcolorbox}
  40
  41\subsubsection*{Weight}
  42A mass of \(m\) kg has force of \(mg\) acting on it
  43
  44\subsubsection*{Momentum \(\rho\)}
  45\[ \rho = mv \tag{units kg m/s or Ns} \]
  46
  47\subsubsection*{Reaction force \(R\)}
  48
  49\begin{itemize}
  50  \item With no vertical velocity, \(R=mg\)
  51  \item With vertical acceleration, \(|R|=m|a|-mg\)
  52  \item With force \(F\) at angle \(\theta\), then \(R=mg-F\sin\theta\)
  53\end{itemize}
  54
  55\subsubsection*{Friction}
  56
  57\[ F_R = \mu R \tag{friction coefficient} \]
  58
  59\subsection*{Inclined planes}
  60
  61\[ \boldsymbol{F} = |\boldsymbol{F}| \cos \theta \boldsymbol{i} + |\boldsymbol{F}| \sin \theta \boldsymbol{j} \]
  62\begin{itemize}
  63  \item Normal force \(R\) is at right angles to plane
  64  \item Let direction up the plane be \(\boldsymbol{i}\) and perpendicular to plane \(\boldsymbol{j}\)
  65\end{itemize}
  66
  67\def\iangle{30} % Angle of the inclined plane
  68
  69\def\down{-90}
  70\def\arcr{0.5cm} % Radius of the arc used to indicate angles
  71
  72\tikzset{
  73  force/.style={->,draw=blue,fill=blue},
  74  axis/.style={densely dashed,gray,font=\small},
  75  M/.style={rectangle,draw,fill=lightgray,minimum size=0.5cm,thin},
  76  m/.style={rectangle,draw=black,fill=lightgray,minimum size=0.3cm,thin},
  77  plane/.style={draw=black,fill=blue!10},
  78  string/.style={draw=red, thick},
  79  pulley/.style={thick}
  80}
  81
  82  \begin{center}\begin{tikzpicture}
  83
  84    \pgfmathsetmacro{\Fnorme}{2}
  85    \pgfmathsetmacro{\Fangle}{30}
  86
  87    \begin{scope}[rotate=\iangle]
  88      \node[M,transform shape] (M) {};
  89      \coordinate (xmin) at ($(M.south west)-({abs(1.1*\Fnorme*sin(-\Fangle))},0)$);
  90      \coordinate (xmax) at ($(M.south east)+({abs(1.1*\Fnorme*sin(-\Fangle))},0)$);
  91      \coordinate (ymax) at ($(M.north)+(0, {abs(1.1*\Fnorme*cos(-\Fangle))})$);
  92      \coordinate (ymin) at ($(M.south)-(0, 1cm)$);
  93      \coordinate (axiscentre) at ($(M.south)+(0.5cm, 0.5cm)$);
  94      \draw[postaction={decorate, decoration={border, segment length=2pt, angle=-45},draw,red}] (xmin) -- (xmax);
  95      \coordinate (N) at ($(M.center)+(0,{\Fnorme*cos(-\Fangle)})$);
  96      \coordinate (fr) at ($(M.center)+({\Fnorme*sin(-\Fangle)}, 0)$);
  97      {[axis,-]
  98      \draw (ymin) -- (M.center);
  99      }
 100      {[axis,->]
 101      \draw ($(M)+(1,0)$) -- ($(M)+(2,0)$) node[above right] {\(\boldsymbol{i}\)};
 102      \draw ($(M)+(1,0)$) -- ($(M)+(1,1)$) node[above right] {\(\boldsymbol{j}\)};
 103      }
 104      {[force,->]
 105        \draw (M.center) -- (N) node [right] {\(R\)};
 106        \draw (M.center) -- (fr) node [left] {\(\mu R\)};
 107      }
 108    \end{scope}
 109    \draw[force,->] (M.center) -- ++(0,-1) node[below] {$mg$};
 110    \draw (M.center)+(-90:\arcr) arc [start angle=-90,end angle=\iangle-90,radius=\arcr] node [below, pos=.5] {\footnotesize\(\theta\)};
 111  \end{tikzpicture}\end{center}
 112
 113\subsection*{Connected particles}
 114
 115\def\boxwidth{0.5}
 116\tikzset{
 117  box/.style={rectangle,draw,fill=lightgray,minimum width=\boxwidth,thin},
 118  m/.style={rectangle,draw=black,fill=lightgray,minimum size=\boxwidth, font=\footnotesize, thin}
 119}
 120
 121
 122\begin{center}
 123  \begin{tikzpicture}
 124
 125    \matrix[column sep=1cm] {
 126      \begin{scope}
 127
 128        \coordinate (O) at (0,0);
 129        \coordinate (A) at ($({3*cos(\iangle)},{3*sin(\iangle)})$);
 130        \coordinate (B) at ($({3*cos(\iangle)},0)$);
 131        \coordinate (C) at ($({(1.5-0.5*\boxwidth)*cos(\iangle)},{(1.5-0.5*\boxwidth)*sin(\iangle)})$); % centre of box
 132        \coordinate (D) at ($(C)+(\iangle:\boxwidth)$);
 133        \coordinate (E) at ($(D)+(90+\iangle:0.5*\boxwidth)$);
 134        \coordinate (F) at ($(B)+(0,{1.5*sin(\iangle)})$);
 135        \coordinate (X) at ($(A)+(\iangle:0.5*\boxwidth)$); % centre of pulley
 136        \coordinate (Y) at ($(X)+(90+\iangle:0.5*\boxwidth)$); % chord of pulley
 137
 138        \draw[plane] (O) -- (A) -- (B) -- (O);
 139        \draw (O)+(\arcr,0) arc [start angle=0,end angle=\iangle,radius=\arcr] node [right, pos=.75] {\footnotesize\(\theta\)};
 140
 141        \draw [rotate=\iangle, m] (C) rectangle ++(\boxwidth,\boxwidth) node (z) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
 142        \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] {};
 143        \draw [string] (E) -- (Y) arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)})$) node[m] {\(m_2\)};
 144
 145      \end{scope}
 146
 147      &
 148
 149      \begin{scope}[rotate=\iangle]
 150
 151        \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node (m1) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
 152
 153        {[axis,-]
 154          \draw (0,-1) -- (0,0);
 155          \draw[solid,shorten >=0.5pt] (\down-\iangle:\arcr) arc(\down-\iangle:\down:\arcr);
 156          \node at (\down-0.5*\iangle:1.3*\arcr) {\(\theta\)};
 157        }
 158
 159        {[force,->]
 160          \draw (M.center) -- ++(0,{cos(\iangle)}) node[above right] {\(R_1\)};
 161          \draw (M.west) -- ++(-0.5,0) node[left] {\(\mu R_1\)};
 162          \draw (M.east) -- ++(1,0) node[above] {\(T_1\)};
 163        }
 164
 165        \draw[force,->, rotate=-\iangle] (M.center) -- ++(0,-1) node[below] {\(m_1 g\)};
 166
 167      \end{scope}
 168
 169      &
 170
 171      \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node [midway, font=\footnotesize] {\(m_2\)};
 172
 173      {[force,->]
 174        \draw (0,0.5*\boxwidth) -- ++(0,1) node[above] {\(T_2\)};
 175        \draw (0,-0.5*\boxwidth) -- ++(0,-1) node[right] {\(m_2 g\)};
 176      }
 177      \\
 178    };
 179  \end{tikzpicture}
 180  \end{center}
 181
 182\begin{itemize}
 183  \item \textbf{Suspended pulley:} tension in both sections of rope are equal \\
 184    \(|a| = g \frac{m_1 - m_2}{m_1 + m_2}\) where \(m_1\) accelerates down \\
 185    With tension: \\
 186    \[ \begin{cases}m_1 g - T = m_1 a\\ T - m_2 g = m_2 a\end{cases} \\ \implies m_1 g - m_2 g = m_1 a + m_2 a \]
 187  \item \textbf{String pulling mass on inclined pane:} Resolve parallel to plane \\
 188    \[ T-mg \sin \theta = ma \]
 189  \item \textbf{Linear connection:} find acceleration of system first
 190  \item \textbf{Pulley on right angle:} \(a = \frac{m_2g}{m_1+m_2}\) where \(m_2\) is suspended (frictionless on both surfaces)
 191  \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane
 192\end{itemize}
 193
 194\hspace{2em}\parbox{8em}{In this example, note \(T_1 \ne T_2\):}
 195  \begin{tikzpicture}
 196
 197      \begin{scope}
 198
 199        \coordinate (O) at (0,0);
 200        \coordinate (A) at ($({3*cos(\iangle)},{3*sin(\iangle)})$);
 201        \coordinate (B) at ($({3*cos(\iangle)},0)$);
 202        \coordinate (C) at ($({(1-0.25*\boxwidth)*cos(\iangle)},{(1-0.25*\boxwidth)*sin(\iangle)})$); % centre of box
 203        \coordinate (D) at ($(C)+(\iangle:\boxwidth)$);
 204        \coordinate (E) at ($(D)+(90+\iangle:0.5*\boxwidth)$);
 205        \coordinate (F) at ($(B)+(0,{1.5*sin(\iangle)})$);
 206        \coordinate (G) at ($(A)+(\iangle:-2*\boxwidth)$);
 207        \coordinate (H) at ($(G)+(90+\iangle:0.5*\boxwidth)$);
 208        \coordinate (I) at ($(H)+(\iangle:-0.5*\boxwidth)$);
 209        \coordinate (J) at ($(H)+(\iangle:\boxwidth)$);
 210        \coordinate (X) at ($(A)+(\iangle:0.5*\boxwidth)$); % centre of pulley
 211        \coordinate (Y) at ($(X)+(90+\iangle:0.5*\boxwidth)$); % chord of pulley
 212
 213        \draw[plane] (O) -- (A) -- (B) -- (O);
 214        \draw (O)+(\arcr,0) arc [start angle=0,end angle=\iangle,radius=\arcr] node [right, pos=.75] {\footnotesize\(\theta\)};
 215
 216        \draw [rotate=\iangle, m] (C) rectangle ++(\boxwidth,\boxwidth) node (z) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
 217        \draw [rotate=\iangle, m] (G) rectangle ++(\boxwidth,\boxwidth) node (l) [rotate=\iangle, midway, font=\footnotesize] {\(m_2\)};
 218        \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] {};
 219        \draw [string] (E) -- (H) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_2\)};
 220        \draw [string] (J) -- (Y) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_1\)} arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)})$) node [midway, above right, font=\footnotesize] {\(T_1\)} node[m] {\(m_3\)};
 221
 222      \end{scope}
 223
 224  \end{tikzpicture}
 225\subsection*{Equilibrium}
 226
 227\[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\]
 228\[ c^2 = a^2 + b^2 - 2ab \cos \theta \tag{cosine rule} \]
 229
 230Three methods:
 231\begin{enumerate}
 232  \item Lami's theorem (sine rule)
 233  \item Triangle of forces (cosine rule)
 234  \item Resolution of forces (\(\Sigma F = 0\) - simultaneous)
 235\end{enumerate}
 236
 237  \begin{cas}
 238    \textbf{To verify:} Geometry tab, then select points with normal cursor. Click right arrow at end of toolbar and input point, then lock known constants.
 239  \end{cas}
 240
 241\subsection*{Variable forces (DEs)}
 242
 243\[ a = \dfrac{d^2x}{dt^2} = \dfrac{dv}{dt} = v\dfrac{dv}{dx} = \dfrac{d}{dx} \left( \frac{1}{2} v^2 \right) \]
 244
 245\end{document}