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\begin{document}

\pagenumbering{gobble}
\begin{multicols}{3}
{\huge Physics}\hfill Andrew Lorimer\hspace{2em}

\section{Motion}
  \subsection*{Unit conversion}
  $\operatorname{m/s} \times 3.6 = \operatorname{km/h}$

  \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}

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

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

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

  $\Sigma F$ 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}

$v=\sqrt{v^2_x + v^2_y}$ (vector addition)

$h={{u^2\sin \theta ^2}\over 2g}$ (max height)

$y=ut \sin \theta-{1 \over 2}gt^2$ (time of flight)

$d={v^2 \over g}sin \theta$ (horizontal range)
  \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

  \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

\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}

\begin{itemize}
\item monopoles: field lines radiate towards central object
\item dipoles - field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (opposite in solenoid)
\item closer field lines means larger force
\item dot means out of page, cross means into page
\end{itemize}

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

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

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

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

\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 $\Phi$

% \columnbreak

\subsection*{Power transmission}

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

\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 (one ring split into two halves)

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

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


\end{multicols}
\end{document}