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\begin{document}

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\begin{multicols}{3}

% +++++++++++++++++++++++
  
{\huge Physics}\hfill Andrew Lorimer\hspace{2em}

% +++++++++++++++++++++++
\section{Motion}

  \subsection*{Inclined planes}
    $F = m g \sin\theta - F_{frict} = m a$

% -----------------------
  \subsection*{Banked tracks}

    \includegraphics[height=4cm]{/mnt/andrew/graphics/banked-track.png}

    $\theta = \tan^{-1} {{v^2} \over rg}$ (also for objects on string)

    $\Sigma F$ always acts towards centre, but not necessarily horizontally

    $\Sigma F = {{mv^2} \over r} = mg \tan \theta$

    Design speed $v = \sqrt{gr\tan\theta}$

% -----------------------
  \subsection*{Work and energy}

    $W=Fx=\Delta \Sigma E$ (work)

    $E_K = {1 \over 2}mv^2$ (kinetic)

    $E_G = mgh$ (potential)

    $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)

% -----------------------
  \subsection*{Horizontal motion}

    $\operatorname{m/s} \times 3.6 = \operatorname{km/h}$

    $v = {{2 \pi r} \over T}$

    $f = {1 \over T}, \quad T = {1 \over f}$

    $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$

    $\Sigma F$ towards centre, $v$ tangential

    $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$

    \includegraphics[height=4cm]{/mnt/andrew/graphics/circ-forces.png}

% -----------------------
  \subsection*{Vertical circular motion}

    $T =$ tension, e.g. circular pendulum

    $T+mg = {{mv^2}\over r}$ at highest point

    $T-mg = {{mv^2} \over r}$ at lowest point

% -----------------------
  \subsection*{Projectile motion}
    \begin{itemize}
      \item{horizontal component of velocity is constant if no air resistance}
      \item{vertical component affected by gravity: $a_y = -g$}
    \end{itemize}

    \begin{align*}
      v=\sqrt{v^2_x + v^2_y} \tag{vectors} \\
      h={{u^2\sin \theta ^2}\over 2g} \tag{max height}\\
      y=ut \sin \theta-{1 \over 2}gt^2 \tag{time of flight} \\
      d={v^2 \over g}\sin \theta \tag{horiz. range} \\
    \end{align*}

    \includegraphics[height=3.2cm]{/mnt/andrew/graphics/projectile-motion.png}

% -----------------------
  \subsection*{Pulley-mass system}

    $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended

    $\Sigma F = m_2g-m_1g=\Sigma ma$ (solve)

% -----------------------
  \subsection*{Graphs}
    \begin{itemize}
      \item{Force-time: $A=\Delta \rho$}
      \item{Force-disp: $A=W$}
      \item{Force-ext: $m=k,\quad A=E_{spr}$}
      \item{Force-dist: $A=\Delta \operatorname{gpe}$}
      \item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$}
    \end{itemize}

% -----------------------
  \subsection*{Hooke's law}

  $F=-kx$

  $E_{elastic} = {1 \over 2}kx^2$

% -----------------------
  \subsection*{Motion equations}

    \begin{tabular}{ l r }
      $v=u+at$ & $x$ \\
      $x = {1 \over 2}(v+u)t$ & $a$ \\
      $x=ut+{1 \over 2}at^2$ & $v$ \\
      $x=vt-{1 \over 2}at^2$ & $u$ \\
      $v^2=u^2+2ax$ & $t$ \\
    \end{tabular}

% -----------------------
  \subsection*{Momentum}

    $\rho = mv$

    $\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$

    Momentum is conserved.

    $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic

    $n$-body collisions: $\rho$ of each body is independent

% ++++++++++++++++++++++
\section{Relativity}

  \subsection*{Postulates}
    1. Laws of physics are constant in all intertial reference frames

    2. Speed of light $c$ is the same to all observers (Michelson-Morley)

    $\therefore , t$ must dilate as speed changes

    {\bf Inertial reference frame} - $a=0$

    {\bf Proper time $t_0$ $\vert$ length $l_0$} - measured by observer in same frame as events

% -----------------------
  \subsection*{Lorentz factor}

    $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}$$

    $t=t_0 \gamma$ ($t$ longer in moving frame)

    $l={l_0 \over \gamma}$ ($l$ contracts $\parallel v$: shorter in moving frame)

    $m=m_0 \gamma$ (mass dilation)

    $$v = c\sqrt{1-{1 \over \gamma^2}}$$

% -----------------------
  \subsection*{Energy and work}

    $E_0 = mc^2$ (rest)

    $E_{total} = E_K + E_{rest} = \gamma mc^2$

    $E_K = (\gamma - 1)mc^2$

    $W = \Delta E = \Delta mc^2$

% -----------------------
  \subsection*{Relativistic momentum}

    $$\rho = {mv \over \sqrt{1-{v^2 \over c^2}}}= {\gamma mv} = {\gamma \rho_0}$$

    $\rho \rightarrow \infty$ as $v \rightarrow c$

    $v=c$ is impossible (requires $E=\infty$)

    $$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$

% -----------------------
  \subsection*{Fusion and fission}

    $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$

    e- accelerated with $x$ V is given $x$ eV

% -----------------------
  \subsection*{High-altitude muons}
    \begin{itemize}
      {\item $t$ dilation - more muons reach Earth than expected}
      {\item normal half-life is $2.2 \operatorname{\mu s}$ in stationary frame}
      {\item at $v \approx c$, muons observed from Earth have halflife $> 2.2 \operatorname{\mu s}$}
      {\item slower time - more time to travel, so muons reach surface}
    \end{itemize}

% +++++++++++++++++++++++
\section{Fields and power}

  \subsection*{Non-contact forces}
    \begin{itemize}
      {\item electric fields (dipoles \& monopoles)}
      {\item magnetic fields (dipoles only)}
      {\item gravitational fields (monopoles only)}
    \end{itemize}

    \vspace{1em}

    \begin{itemize}
      \item monopoles: lines towards centre
      \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (or perpendicular to wire)
      \item closer field lines means larger force
      \item dot means out of page, cross means into page
      \item +ve corresponds to N pole
    \end{itemize}

% -----------------------
  \subsection*{Gravity}

    \[F_g=G{{m_1m_2}\over r^2}\tag{grav. force}\]

    \[g={F_g \over m}=G{M_{\operatorname{planet}} \over r^2}\tag{grav. acc.}\]

    \[E_g = mg \Delta h\tag{gpe}\]

    \[W = \Delta E_g = Fx\tag{work}\]

    \[w=m(g-a) \tag{app. weight}\]

% -----------------------
  \subsection*{Satellites}

    \[v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]

    \[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\]

    \[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]

% -----------------------
  \subsection*{Magnetic fields}
    \begin{itemize}
      \item field strength $B$ measured in tesla
      \item magnetic flux $\Phi$ measured in weber
      \item charge $q$ measured in coulombs
      \item emf $\mathcal{E}$ measured in volts
    \end{itemize}

    \[{E_1 \over E_2}={r_1 \over r_2}^2\]

    \[F=qvB\tag{force on moving charged particles}\]

    if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$


    \includegraphics[height=2cm]{/mnt/andrew/graphics/field-lines.png}

% -----------------------
  \subsection*{Electric fields}

    \begin{align*}
      F=qE \tag{$E$ = strength} \\
      W=q_{\operatorname{point}}\Delta V \tag{in field or points} \\
      F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \\
      E=k{Q \over r^2} \tag{$r=||EQ||$} \\
      F=BInl \tag{force on a coil} \\
      \Phi = B_{\perp}A\tag{magnetic flux} \\
      \mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \\
      {V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \\
    \end{align*}

    \textbf{Lenz's law:}  ``$-n$'' in Faraday - emf opposes $\Delta \Phi$

    \textbf{Eddy currents:} counter movement within a field

    \textbf{Right hand grip:} thumb points to north or $I$

    \textbf{Right hand slap:} field, current, force are $\perp$

    \textbf{Flux-time graphs:} gradient $\times n = \operatorname{emf}$

    \textbf{Transformers:} core strengthens \& focuses $\Phi$

% -----------------------
  \subsection*{Power transmission}

    \begin{align*}
      V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}} \\
      P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \\
    \end{align*}

    Use high-$V$ side for correct $|V_{drop}|$

    \begin{itemize}
      {\item Parallel - voltage is constant}
      {\item Series - voltage is shared within branch}
    \end{itemize}

    \includegraphics[height=4cm]{/mnt/andrew/graphics/ac-generator.png}

% -----------------------
  \subsection*{Motors}
% \begin{wrapfigure}{r}{-0.1\textwidth}

    \includegraphics[height=4cm]{/mnt/andrew/graphics/dc-motor-2.png}
      \includegraphics[height=3cm]{/mnt/andrew/graphics/ac-motor.png} \\
% \end{wrapfigure}
    \textbf{DC:} split ring (two halves)

% \begin{wrapfigure}{r}{0.3\textwidth}

% \end{wrapfigure}
    \textbf{AC:} slip ring (separate rings with constant contact)


\end{multicols}
\end{document}