Andrew's git / notes.git /
28bcdfea4afe3f1377e045286bab3a5f6fda7f97
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  14\begin{document}
  15
  16\pagenumbering{gobble}
  17\begin{multicols}{3}
  18{\huge Physics}\hfill Andrew Lorimer\hspace{2em}
  19
  20\section{Motion}
  21  \subsection*{Unit conversion}
  22  $\operatorname{m/s} \times 3.6 = \operatorname{km/h}$
  23
  24  \subsection*{Inclined planes}
  25  $F = m g \sin\theta - F_{frict} = m a$
  26
  27  \subsection*{Banked tracks}
  28  \includegraphics[height=4cm]{/mnt/andrew/graphics/banked-track.png}
  29  $\theta = \tan^{-1} {{v^2} \over rg}$ (also for objects on string)
  30
  31  $\Sigma F$ always acts towards centre, but not necessarily horizontally
  32
  33  $\Sigma F = {{mv^2} \over r} = mg \tan \theta$
  34
  35  Design speed $v = \sqrt{gr\tan\theta}$
  36
  37  \subsection*{Work and energy}
  38  $W=Fx=\Delta \Sigma E$ (work)
  39
  40  $E_K = {1 \over 2}mv^2$ (kinetic)
  41
  42  $E_G = mgh$ (potential)
  43
  44  $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)
  45
  46  \subsection*{Horizontal motion}
  47
  48  $v = {{2 \pi r} \over T}$
  49
  50  $f = {1 \over T}, \quad T = {1 \over f}$
  51
  52  $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$
  53
  54  $\Sigma F$ towards centre, $v$ tangential
  55
  56  $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
  57
  58  \includegraphics[height=4cm]{/mnt/andrew/graphics/circ-forces.png}
  59
  60  \subsection*{Vertical circular motion}
  61  $T =$ tension, e.g. circular pendulum
  62
  63  $T+mg = {{mv^2}\over r}$ at highest point
  64  $T-mg = {{mv^2} \over r}$ at lowest point
  65
  66  \subsection*{Projectile motion}
  67  \begin{itemize}
  68  \item{horizontal component of velocity is constant if no air resistance}
  69
  70  \item{vertical component affected by gravity: $a_y = -g$}
  71\end{itemize}
  72
  73$v=\sqrt{v^2_x + v^2_y}$ (vector addition)
  74
  75$h={{u^2\sin \theta ^2}\over 2g}$ (max height)
  76
  77$y=ut \sin \theta-{1 \over 2}gt^2$ (time of flight)
  78
  79$d={v^2 \over g}sin \theta$ (horizontal range)
  80  \includegraphics[height=4cm]{/mnt/andrew/graphics/projectile-motion.png}
  81
  82  \subsection*{Pulley-mass system}
  83
  84  $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended
  85
  86  \subsection*{Graphs}
  87  \begin{itemize}
  88    \item{Force-time: $A=\Delta \rho$}
  89    \item{Force-disp: $A=W$}
  90    \item{Force-ext: $m=k,\quad A=E_{spr}$}
  91  \end{itemize}
  92
  93  \subsection*{Hooke's law}
  94
  95  $F=-kx$
  96
  97  $E_{elastic} = {1 \over 2}kx^2$
  98
  99  \subsection*{Motion equations}
 100
 101
 102\begin{tabular}{ l r }
 103  $v=u+at$ & $x$ \\
 104  $x = {1 \over 2}(v+u)t$ & $a$ \\
 105  $x=ut+{1 \over 2}at^2$ & $v$ \\
 106  $x=vt-{1 \over 2}at^2$ & $u$ \\
 107  $v^2=u^2+2ax$ & $t$ \\
 108\end{tabular}
 109
 110\subsection*{Momentum}
 111
 112$\rho = mv$
 113
 114$\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
 115
 116Momentum is conserved.
 117
 118$\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
 119
 120\section{Relativity}
 121
 122
 123
 124
 125
 126
 127
 128\end{multicols}
 129\end{document}