--- /dev/null
+\documentclass[a4paper]{article}
+\usepackage{multicol}
+\usepackage[cm]{fullpage}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\setlength{\parindent}{0cm}
+\usepackage[nodisplayskipstretch]{setspace}
+\setstretch{1.3}
+\usepackage{graphicx}
+\usepackage{wrapfig}
+\usepackage{enumitem}
+\setitemize{noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=5pt}
+
+
+\begin{document}
+
+\pagenumbering{gobble}
+\begin{multicols}{3}
+
+% +++++++++++++++++++++++
+
+{\huge Physics}\hfill Andrew Lorimer\hspace{2em}
+
+% +++++++++++++++++++++++
+\section{Motion}
+
+ $\operatorname{m/s} \times 3.6 = \operatorname{km/h}$
+
+ \subsection*{Inclined planes}
+ $F = m g \sin\theta F_{frict} = m a$
+
+% -----------------------
+ \subsection*{Banked tracks}
+
+ \includegraphics[height=4cm]{graphics/banked-track.png}
+
+ $$\theta = \tan^{-1} {{v^2} \over rg}$$
+
+ $\Sigma F$ always acts towards centre, but not necessarily horizontally
+
+ $\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$
+
+ 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 circular motion}
+
+ $v = {{2 \pi r} \over T}$
+
+ $f = {1 \over T}, \quad T = {1 \over f}$
+
+ $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$
+
+ $\Sigma F, a$ towards centre, $v$ tangential
+
+ $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
+
+ \includegraphics[height=4cm]{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}\\
+ x=ut\cos\theta \tag{$\Delta x$ at $t$} \\
+ y=ut \sin \theta-{1 \over 2}gt^2 \tag{height at $t$} \\
+ t={{2u\sin\theta}\over g} \tag{time of flight}\\
+ d={v^2 \over g}\sin \theta \tag{horiz. range} \\
+ \end{align*}
+
+ \includegraphics[height=3.2cm]{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}
+
+ $\rho = mv$
+
+ $\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
+
+ $\Sigma mv_0=\Sigma mv_1$ (conservation)
+
+ $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ 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*{High-altitude muons}
+ \begin{itemize}
+ {\item $t$ dilation more muons reach Earth than expected}
+ {\item normal half-life $2.2 \operatorname{\mu s}$ in stationary frame, $> 2.2 \operatorname{\mu s}$ observed from Earth}
+ \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: out of page, cross: into page
+ \item +ve corresponds to N pole
+ \end{itemize}
+
+ \includegraphics[height=2cm]{graphics/field-lines.png}
+
+% -----------------------
+ \subsection*{Gravity}
+
+ \[F_g=G{{m_1m_2}\over r^2}\tag{grav. force}\]
+ \[g={F_g \over m_2}=G{m_{1} \over r^2}\tag{field of $m_1$}\]
+ \[E_g = mg \Delta h\tag{gpe}\]
+ \[W = \Delta E_g = Fx\tag{work}\]
+ \[w=m(g-a) \tag{app. weight}\]
+
+ % \columnbreak
+
+% -----------------------
+ \subsection*{Satellites}
+
+ \[v=\sqrt{Gm_{\operatorname{planet}} \over r} = \sqrt{gr} = {{2 \pi r} \over T}\]
+
+ \[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\]
+
+ \[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\]
+
+% -----------------------
+ \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}
+
+ % \[{E_1 \over E_2}={r_1 \over r_2}^2\]
+
+ \[F=qvB\tag{$F$ on moving $q$}\]
+ \[F=IlB\tag{$F$ of $B$ on $I$}\]
+ \[r={mv \over qB} \tag{radius of $q$ in $B$}\]
+
+ if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$
+
+% -----------------------
+ \subsection*{Electric fields}
+
+ \[F=qE \tag{$E$ = strength} \]
+ \[F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \]
+ \[E=k{q \over r^2} \tag{field on point charge} \]
+ \[E={V \over d} \tag{field between plates}\]
+ \[F=BInl \tag{force on a coil} \]
+ \[\Phi = B_{\perp}A\tag{magnetic flux} \]
+ \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} \tag{induced emf} \]
+ \[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]
+
+ \textbf{Lenz's law:} $I_{\operatorname{emf}}$ opposes $\Delta \Phi$
+
+ \textbf{Eddy currents:} counter movement within a field
+
+ \textbf{Right hand grip:} thumb points to $I$ (single wire) or N (solenoid / coil)
+
+ \textbf{Right hand slap:} $B \perp I \perp F$
+
+ \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}$
+
+ \textbf{Transformers:} core strengthens \& focuses $\Phi$
+
+% -----------------------
+ \subsection*{Particle acceleration}
+
+ $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
+
+ eaccelerated with $x$ V is given $x$ eV
+
+ \[W={1\over2}mv^2=qV \tag{field or points}\]
+ \[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]
+
+
+% -----------------------
+ \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} \\
+ V_{\operatorname{loss}}=IR
+ % \end{align*}
+
+ Use high-$V$ side for correct $|V_{drop}|$
+
+ \begin{itemize}
+ {\item Parallel $V$ is constant}
+ {\item Series $V$ shared within branch}
+ \end{itemize}
+
+ \includegraphics[height=4cm]{graphics/ac-generator.png}
+
+% -----------------------
+ \subsection*{Motors}
+% \begin{wrapfigure}{r}{-0.1\textwidth}
+
+ \includegraphics[height=4cm]{graphics/dc-motor-2.png}
+ \includegraphics[height=3cm]{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)
+
+% +++++++++++++++++++++++
+\section{Waves}
+
+ \textbf{nodes:} fixed on graph
+
+ \textbf{Longitudinal (motion $||$ wave)}
+ \includegraphics[height=4cm]{graphics/longitudinal-waves.png}
+
+ \textbf{Transverse (motion $\perp$ wave)}
+ \includegraphics[height=4cm]{graphics/transverse-waves.png}
+
+ % -----------------------
+ \subsection*{Motors}
+ $T={1 \over f}\quad$(period: time for one cycle)
+ $v=f \lambda \quad$(speed: displacement per second)
+
+ % -----------------------
+ \subsection*{Doppler effect}
+ When $P_1$ approaches $P_2$, each wave $w_n$ has slightly less distance to travel than $w_{n-1}$. Hence, $w_n$ reaches the observer sooner than $w_{n-1}$, increasing "apparent" wavelength.
+
+ % -----------------------
+ \subsection*{Interference}
+ When a medium changes character, energy is reflected, absorbed, and transmitted
+
+ % -----------------------
+ \subsection*{Polarisation}
+ \includegraphics[height=4cm]{graphics/polarisation.png}
+
+ % -----------------------
+ \subsection*{Refraction}
+ \includegraphics[height=4cm]{graphics/refraction.png}
+
+ Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
+
+ Critical angle $\theta_c = \sin^-1{n_2 \over n_1}$
+
+ Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$
+
+% +++++++++++++++++++++++
+\section{Light and Matter}
+
+ % -----------------------
+ \subsection*{Planck's equation}
+
+ f={c \over \lambda},\quad E=hf={hc \over \lambda}=\rho c
+
+ h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s}
+
+ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}
+
+ \subsection*{Force of electrons}
+ F={2P_{\text{in}}\over c}
+
+ \text{photons per second}={\text{total energy} \over \text{energy per photon}}={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
+
+ \subsection*{Photoelectric effect}
+
+ \begin{itemize}
+ \item $V_{\operatorname{supply}}$ does not affect photocurrent
+ \item $V_{\operatorname{sup}} > 0$: eattracted to collector anode
+ \item $V_{\operatorname{sup}} < 0$: attracted to illuminated cathode, $I\rightarrow 0$
+ \item $v$ of edepends on ionisation energy (shell)
+ \item max current depends on intensity
+ \end{itemize}
+
+ \textbf{Threshold frequency $f_0$}
+
+ Minimum $f$ for photoelectrons to be ejected. $x$-intercept of frequency vs $E_K$ graph. if $f < f_0$, no photoelectrons are detected.
+
+ \textbf{Work function $\phi$}
+
+ Minimum $E$ required to release photoelectrons. Magnitude of $y$-intercept of frequency vs $E_K$ graph. $\phi$ is determined by strength of bonding.
+
+ $\phi=hf_0$
+
+ \textbf{Kinetic energy}
+
+ E_{\operatorname{k-max}}=hf - \phi
+
+ voltage in circuit or stopping voltage = max $E_K$ in eV
+ equal to $x$-intercept of volts vs current graph (in eV)
+
+ \textbf{Stopping potential $V$ for min $I$}
+
+ $V=h_{\text{eV}}(f-f_0)$
+
+ \subsection*{De Broglie's theory}
+
+ $\lambda = {h \over \rho} = {h \over mv}$
+ $\rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c$
+ \begin{itemize}
+ \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
+ \item confirmed by similar e- and x-ray diff patterns
+ \end{itemize}
+
+ \subsection*{X-ray electron interaction}
+
+ \begin{itemize}
+ \item e- is only stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$
+ \item rearranging this, $2\pi r = n{h \over mv} = n \lambda$ (circumference)
+ \item if $2\pi r \ne n{h \over mv}$, no standing wave
+ \item if e- = x-ray diff patterns, $E_{\text{e-}}={\rho^2 \over 2m}={({h \over \lambda})^2 \div 2m}$
+ \item calculating $h$: $\lambda = {h \over \rho}$
+ \end{itemize}
+
+ \subsection*{Spectral analysis}
+
+ \begin{itemize}
+ $n\item $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
+ $n\item $E$ and $f$ of photon: $E_2 - E_1 = hf = {hc \over \lambda}$
+ $n\item Ionisation energy - min $E$ required to remove e-
+ $n\item EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
+ \end{itemize}
+
+ \subsection{Indeterminancy principle}
+
+ measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it.
+
+ \subsection{Wave-particle duaity}
+
+ wave model:
+
+ \item cannot explain photoelectric effect
+ \item $f$ is irrelevant to photocurrent
+ \item predicts delay between incidence and ejection
+ \item speed depends on medium
+
+ particle model:
+
+ \item explains photoelectric effect
+ \item rate of photoelectron release $\propto$ intensity
+ \item no time delay - one photon releases one electron
+ \item double slit: photons interact. interference pattern still appears when a dim light source is used so that only one photon can pass at a time
+ \item light exerts force
+ \item light bent by gravity
+
+
+
+
+
+
+
+
+\end{multicols}
+\end{document}