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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}
- $F = m g \sin\theta - F_{frict} = m a$
+ $F = m g \sin\theta - F_{frict} = m a$
+% -----------------------
\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
+ \includegraphics[height=4cm]{/mnt/andrew/graphics/banked-track.png}
- $\Sigma F = {{mv^2} \over r} = mg \tan \theta$
+ $$\theta = \tan^{-1} {{v^2} \over rg}$$
- Design speed $v = \sqrt{gr\tan\theta}$
+ $\Sigma F$ always acts towards centre, but not necessarily horizontally
- \subsection*{Work and energy}
- $W=Fx=\Delta \Sigma E$ (work)
+ $\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$
- $E_K = {1 \over 2}mv^2$ (kinetic)
+ Design speed $v = \sqrt{gr\tan\theta}$
- $E_G = mgh$ (potential)
-
- $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)
+% -----------------------
+ \subsection*{Work and energy}
- \subsection*{Horizontal motion}
+ $W=Fx=\Delta \Sigma E$ (work)
- $v = {{2 \pi r} \over T}$
+ $E_K = {1 \over 2}mv^2$ (kinetic)
- $f = {1 \over T}, \quad T = {1 \over f}$
+ $E_G = mgh$ (potential)
- $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$
+ $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer)
- $\Sigma F$ towards centre, $v$ tangential
+% -----------------------
+ \subsection*{Horizontal circular motion}
- $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
+ $v = {{2 \pi r} \over T}$
- \includegraphics[height=4cm]{/mnt/andrew/graphics/circ-forces.png}
+ $f = {1 \over T}, \quad T = {1 \over f}$
- \subsection*{Vertical circular motion}
- $T =$ tension, e.g. circular pendulum
+ $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$
- $T+mg = {{mv^2}\over r}$ at highest point
- $T-mg = {{mv^2} \over r}$ at lowest point
+ $\Sigma F, a$ towards centre, $v$ tangential
- \subsection*{Projectile motion}
- \begin{itemize}
- \item{horizontal component of velocity is constant if no air resistance}
+ $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$
- \item{vertical component affected by gravity: $a_y = -g$}
-\end{itemize}
+ \includegraphics[height=4cm]{/mnt/andrew/graphics/circ-forces.png}
-$v=\sqrt{v^2_x + v^2_y}$ (vector addition)
+% -----------------------
+ \subsection*{Vertical circular motion}
-$h={{u^2\sin \theta ^2}\over 2g}$ (max height)
+ $T =$ tension, e.g. circular pendulum
-$y=ut \sin \theta-{1 \over 2}gt^2$ (time of flight)
+ $T+mg = {{mv^2}\over r}$ at highest point
-$d={v^2 \over g}sin \theta$ (horizontal range)
- \includegraphics[height=4cm]{/mnt/andrew/graphics/projectile-motion.png}
+ $T-mg = {{mv^2} \over r}$ 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]{/mnt/andrew/graphics/projectile-motion.png}
+
+% -----------------------
\subsection*{Pulley-mass system}
- $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended
+ $a = {{m_2g} \over {m_1 + m_2}}$ where $m_2$ is suspended
- \subsection*{Graphs}
- \begin{itemize}
- \item{Force-time: $A=\Delta \rho$}
- \item{Force-disp: $A=W$}
- \item{Force-ext: $m=k,\quad A=E_{spr}$}
- \end{itemize}
+ $\Sigma F = m_2g-m_1g=\Sigma ma$ (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}
-\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}
-\subsection*{Momentum}
+ $\rho = mv$
-$\rho = mv$
+ $\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
-$\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$
+ $\Sigma mv_0=\Sigma mv_1$ (conservation)
-Momentum is conserved.
+ $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elastic
-$\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ 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]{/mnt/andrew/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}
+
+ $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$
+
+ e- accelerated 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]{/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}