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\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}
    $F = m g \sin\theta - F_{\text{frict}} = m a$

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

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

    $\theta = \tan^{-1} {{v^2} \over rg}$

    $\Sigma F$ always acts towards centre (horizontally)

    $\Sigma F = F_{\operatorname{norm}} + F_{\operatorname{g}}={{mv^2} \over r} = mg \tan \theta$

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

    $n\sin \theta = {mv^2 \div r}, \quad n\cos \theta = mg$

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

    $W=Fs=Fs \cos \theta=\Delta \Sigma E$

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

    $\Sigma F = F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}=T \sin \theta = mg \tan \theta$

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

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

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

    $T+mg = {{mv^2}\over r}, v = \sqrt{rg}$ (top)

    $T-mg = {{mv^2} \over r}, v = \sqrt{2rg}$ (bottom)

    $E_K_{\text{bottom}}=E_K_{\text{top}}+mgh$

% -----------------------
  \subsection*{Projectile motion}
    \begin{itemize}
      \item $v_x$ is constant: $v_x = {s \over t}$
      \item use suvat to find $t$ from $y$-component
      \item vertical component gravity: $a_y = -g$
    \end{itemize}

    % \begin{align*}
      $v=\sqrt{v^2_x + v^2_y}$ \hfill vectors \\
      $h={{u^2\sin \theta ^2}\over 2g}$ \hfill max height \\
      $x=ut\cos\theta$ \hfill $\Delta x$ at $t$ \\
      $y=ut \sin \theta-{1 \over 2}gt^2$ \hfill height at $t$ \\
      $t={{2u\sin\theta}\over g}$ \hfill time of flight \\
      $d={v^2 \over g}\sin \theta$ \hfill 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{$F_g$-dist: $A=\Delta \operatorname{gpe}$}
      \item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$}
    \end{itemize}

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

  $F=-kx$ (intercepts origin)

  $\text{elastic potential energy} = {1 \over 2}kx^2$

  $x={2mg \over k}$

  Vertical: $\Delta E = {1 \over 2}kx^2 + mgh

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

    \begin{tabular}{ l r }
      & no \\
      $v=u+at$ & $x$ \\
      $x = {1 \over 2}(v+u)t$ & $a$ \\
      $x=ut+{1 \over 2}at^2$ & $v$ \\
      $x=vt-{1 \over 2}at^2$ & $u$ \\
      $v^2=u^2+2ax$ & $t$ \\
    \end{tabular}

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

    $\rho = mv$

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

    $\Sigma (mv_0)=(\Sigma m)v_1$ (conservation)

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

    % $\Sigma E_K = \Sigma ({1 \over 2} m v^2) = {1 \over 2} (\Sigma m)v_f$

    if elastic:
    $$\sum _{i{\mathop {=}}1}^{n}E_K (i)=\sum _{i{\mathop {=}}1}^{n}({1 \over 2}m_i v_{i0}^2)={1 \over 2}\sum _{i{\mathop {=}}1}^{n}(m_i) v_f^2$$

    % $n$-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 high-altitude particles:} $t$ dilation means more particles reach Earth than expected (half-life greater when obs. from Earth)

    {\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}

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

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

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

    $m=m_0 \gamma$ (mass dilation)

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

  Total energy = mass energy

    $E_{\text{rest}} = mc^2, \quad E_K = (\gamma-1)mc^2$

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

    $W = \Delta E = \Delta mc^2=(\gamma-1)m_{\text{rest}} c^2$

% -----------------------
  \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 $E=\infty$)

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

% -----------------------

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

  \subsection*{Non-contact forces}
    \begin{itemize}
      {\item electric (dipoles \& monopoles)}
      {\item magnetic (dipoles only)}
      {\item gravitational (monopoles only, $F_g=0$ at mid, attractive only)}
    \end{itemize}

    \vspace{1em}

    \begin{itemize}
      \item monopoles: lines towards centre
      \item dipoles: field lines $+ \rightarrow -$ or $\operatorname{N} \rightarrow \operatorname{S}$ (two magnets) or $\rightarrow$ N (single)
      \item closer field lines means larger force
      \item dot: out of page, cross: into page
      \item +ve corresponds to N pole
      \item Inv. sq. ${E_1 \over E_2} = ({r_2 \over r_1})^2$
    \end{itemize}

    \includegraphics[height=2cm]{graphics/field-lines.png}
    % \includegraphics[height=2cm]{graphics/bar-magnet-fields-rotated.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}

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

    \[T={\sqrt{4 \pi^2 r^3 \over {GM}}}=2 \pi \sqrt{r_{\text{alt}} \over g_{\text{alt}}}\tag{period}\]

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

% -----------------------
  \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}

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

    \[F=qvB\tag{$F$ on moving $q$}\]
    \[F=IlB\tag{$F$ of $B$ on $I$}\]
    \[B={mv \over qr}\tag{field strength on e-}\]
    \[r={mv \over qB} \tag{radius of $q$ in $B$}\]

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

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

    \[F=qE(=ma) \tag{strength} \]
    \[F=k{{q_1q_2}\over r^2}\tag{force between $q_{1,2}$} \]
    \[E=k{q \over r^2} \tag{field on point charge} \]
    \[E={V \over d} \tag{field between plates}\]
    \[F=BInl \tag{force on a coil} \]
    \[\Phi = B_{\perp}A\tag{magnetic flux} \]
    \[\mathcal{E} = -N{{\Delta \Phi}\over{\Delta t}} = Blv\tag{induced emf} \]
    \[{V_p \over V_s}={N_p \over N_s}={I_s \over I_p} \tag{xfmr coil ratios} \]

    \textbf{Lenz's law:}  $I_{\operatorname{emf}}$ opposes $\Delta \Phi$ \\
    (emf creates $I$ with associated field that 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{Magnet through ring:} consider $g$

    \includegraphics[height=2cm]{graphics/slap-2.jpeg}
    \includegraphics[height=3cm]{graphics/grip.png}

    % \textbf{Right hand slap:} $B \perp I \perp F$ \\
    % ($I$ = thumb)

    \includegraphics[width=\columnwidth]{graphics/lenz.png}

    \textbf{Flux-time graphs:} $m \times n = \operatorname{emf}.$
    If $f$ increases, ampl. \& $f$ of $\mathcal{E}$ increase

    \textbf{Xfmr} core strengthens \& focuses $\Phi$

    \columnbreak

% -----------------------
  \subsection*{Particle acceleration}

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

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

    \[W={1\over2}mv^2=qV \tag{field or points}\]
    \[V_{\text{point}} = (V_1 - V_2) \div 2 \tag{if midpoint} \]
    \[v=\sqrt{{2qV} \over {m}}\tag{velocity of particle}\]

    Circular path: $F\perp B \perp v$

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

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

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

    \includegraphics[height=4cm]{graphics/dc-motor-2.png}
    % \includegraphics[height=3cm]{graphics/ac-motor.png} \\
    \includegraphics[height=4cm]{graphics/ac-generator.png} \\

    Force on I-carying wire, not Cu \\
    $F=0$ for front & back of coil (parallel) \\
    Any angle $> 0$ will produce force \\
% \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)

% \pagebreak

% +++++++++++++++++++++++
\section{Waves}

  \textbf{nodes:} fixed on graph \\
  \textbf{amplitude:} max disp. from $y=0$ \\
  \textbf{rarefactions} and \textbf{compressions} \\
  \textbf{mechanical:} transfer of energy without net transfer of matter \\


  \textbf{Longitudinal (motion $||$ wave)}
  \includegraphics[width=6cm]{graphics/longitudinal-waves.png}

  \textbf{Transverse (motion $\perp$ wave)}
  \includegraphics[width=6cm]{graphics/transverse-waves.png}

  % -----------------------
  $T={1 \over f}\quad$(period: time for one cycle)
  $v=f \lambda \quad$(speed: displacement / sec)
  $f={c \over \lambda}\quad\hspace{0.7em}$(for $v=c$)

  % -----------------------
  \subsection*{Doppler effect}

  When $P_1$ approaches $P_2$, each wave $w_n$ has slightly less distance to travel than $w_{n-1}$. $w_n$ reaches observer sooner than $w_{n-1}$ ("apparent" $\lambda$).

  % -----------------------
  \subsection*{Interference}

  \includegraphics[width=4.5cm]{graphics/poissons-spot.png} \\
  Poissons's spot supports wave theory (circular diffraction)

  \textbf{Standing waves} - constructive int. at resonant freq. Rebound from ends.

  \textbf{Coherent } - identical frequency, phase, direction (ie strong & directional). e.g. laser

  \textbf{Incoherent} - e.g. incandescent/LED





  % -----------------------
  \subsection*{Harmonics}

  1st harmonic = fundamental

  \textbf{for nodes at both ends:} \\
  \(\hspace{2em} \lambda = {{2l} \div n}\)
  \(\hspace{2em} f = {nv \div 2l} \)

  \textbf{for node at one end ($n$ is odd):} \\
  \(\hspace{2em} \lambda = {{4l} \div n}\)
  \(\hspace{2em} f = {nv \div 4l} \) \\
  alternatively, $\lambda = {4l \over {2n-1}}$ where $n\in \mathbb{Z}$ and $n+1$ is the next possible harmonic


  % \(a=2\) for nodes at both ends, \\ \(a=4\) for node at one end

  % -----------------------
  \subsection*{Polarisation}
  \includegraphics[height=3.5cm]{graphics/polarisation.png} \\
  Transverse only. Reduces total $A$.

  % -----------------------
  \subsection*{Diffraction}
  \includegraphics[width=6cm]{graphics/diffraction.jpg}
  \includegraphics[width=6cm]{graphics/diffraction-2.png}
  \begin{itemize}
    % \item \(pd = |S_1P-S_2P|\) for \(p\) on screen
    \item Constructive: \(pd = n\lambda, n \in \mathbb{Z}\)
    \item Destructive: \(pd = (n-{1 \over 2})\lambda, n \in \mathbb{Z}\)
    \item Path difference: \(\Delta x = {{\lambda l }\over d}\) where \\
    % \(\Delta x\) = fringe spacing \\
    \(l\) = distance from source to observer\\
    \(d\) = separation between each wave source (e.g. slit) \(=S_1-S_2\)
    \item diffraction $\propto {\lambda \over d} \propto$ fringe spacing
    \item $d(|\overrightarrow{S_1W}|-|\overrightarrow{S_2W}|)=d \Delta x = \lambda l$
    \item significant diffraction when ${\lambda \over \Delta x} \ge 1$
    \item diffraction creates distortion (electron $\gt$ optical microscopes)
  \end{itemize}


  % -----------------------
  \subsection*{Refraction}
  \includegraphics[height=3.5cm]{graphics/refraction.png}

  When a medium changes character, light is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}. $\lambda$ changes, not $f$. $n$ changes slightly with $f$ (dispersion)

  angle of incidence $\theta_i =$ angle of reflection $\theta_r$

  Critical angle $\theta_c = \sin^{-1}{n_2 \over n_1}$

  Snell's law $n_1 \sin \theta_1=n_2 \sin \theta_2$

  ${v_1 \div v_2} = {\sin\theta_1 \div \sin\theta_2}$

  $n_1 v_1 = n_2 v_2$

  $n={c \over v}$


% +++++++++++++++++++++++
\section{Light and Matter}

  % -----------------------
  \subsection*{Planck's equation}

  \[ \quad E=hf={hc \over \lambda}=\rho c = qV\]
  \[ h=6.63 \times 10^{-34}\operatorname{J s}=4.14 \times 10^{-15} \operatorname{eV s} \]
  \[ 1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J} \]

  \subsection*{De Broglie's theory}

  \[ \lambda = {h \over \rho} = {h \over mv} = {h \over {m \sqrt{2W \over m}}}\]
  \[ \rho = {hf \over c} = {h \over \lambda} = mv, \quad E = \rho c \]
  \[ v = \sqrt{2E_K \div m} \]

  \begin{itemize}
    \item cannot confirm with double-slit (slit $< r_{\operatorname{proton}}$)
    \item confirmed by e- and x-ray patterns
  \end{itemize}

  \subsection*{Force of electrons}
  \[ F={2P_{\text{in}}\over c} \]
  % \begin{align*}
  \[ \text{photons / sec} = {\text{total energy} \over \text{energy / photon}} \]
  \[ ={{P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf} \]
  % ={P_{\text{in}} \lambda} \over hc}={P_{\text{in}} \over hf}
  % \end{align*}

  \subsection*{X-ray electron interaction}

  \begin{itemize}
    \item e- stable if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$ and $r$ is radius of orbit
    \item $\therefore 2\pi r = n{h \over mv} = n \lambda$ (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*{Photoelectric effect}

  \begin{itemize}
    \item $V_{\operatorname{supply}}$ does not affect photocurrent
    \item $V_{\operatorname{sup}} > 0$: attracted to +ve
    \item $V_{\operatorname{sup}} < 0$: attracted to -ve, $I\rightarrow 0$
    \item $v$ of e- depends on shell
    \item max $I$ (not $V$) depends on intensity
  \end{itemize}

  \subsubsection*{Threshold frequency $f_0$}

  min $f$ for photoelectron release. if $f < f_0$, no photoelectrons.

  \subsubsection*{Work function $\phi=hf_0$}

  min $E$ for photoelectron release. determined by strength of bonding. Units: eV or J.

  \subsubsection*{Kinetic energy E_K=hf - \phi = qV_0}


  $V_0 = E_K$ in eV \\
  % $E_K = x$-int of $V\cdot I$ graph (in eV) \\
  dashed line below $E_K=0$


  \subsubsection*{Stopping potential $V_0$ for min $I$}

  $$V_0=h_{\text{eV}}(f-f_0)$$
  Opposes induced photocurrent

  \subsubsection*{Graph features}

  \newcolumntype{b}{>{\hsize=.75\hsize}X}
\newcolumntype{s}{>{\hsize=.3\hsize}X}

  \begin{tabularx}{\columnwidth}{bbbb}
\hline
&$m$&$x$-int&$y$-int \\
\hline
\hline
$f \cdot E_K$ & $h$ & $f_0$ & $-\phi$ \\
$V \cdot I$ &  & $V_0$ & intensity\\
$f \cdot V$ & ${h \over q}$ & $f_0$ & $-\phi \over q$ &
\hline
\end{tabularx}



  \subsection*{Spectral analysis}

  \begin{itemize}
    \item $\Delta E = hf = {hc \over \lambda}$ between ground / excited state
    \item $E$ and $f$ of photon: $E_2 - E_1 = hf = {hc \over \lambda}$
    \item Ionisation energy - min $E$ required to remove e-
    \item EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$)
    \item No. of lines - include all possible states. \Delta E \ne |\Delta E|
  \end{itemize}

  \subsection*{Uncertainty principle}

  $\Delta x \approx {\text{slit width} \over 2$}

  measurement: $\rho$ transferred to e-\\ slit: possibility of diff. before slit

  \subsection*{Wave-particle duality}

  \subsubsection*{wave model}
  \begin{itemize}
    % \item cannot explain photoelectric effect
    \item any $f$ works, given $t$
    \item predicts delay between incidence and ejection
    \item speed depends on medium
    \item supported by bright spot in centre
    \item $\lambda = {hc \over E}$
  \end{itemize}

  \subsubsection*{particle model}

  \begin{itemize}
    % \item explains photoelectric effect
    \item rate of photoelectron release $\propto$ intensity
    \item no time delay - one photon releases one electron
    \item threshold frequency
    \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
    \item quantised energy
    \item $\lambda = {h \over \rho}$
  \end{itemize}

  % +++++++++++++++++++++++
  \section{Experimental \\ design}

  \textbf{Absolute uncertainty} $\Delta$ \\
  (same units as quantity)
  \[ \Delta(m) = {{\mathcal{E}(m)} \over 100} \cdot m \]
  \[ (A \pm \Delta A) + (B \pm \Delta A) = (A+B) \pm (\Delta A + \Delta B) \]
  \[ (A \pm \Delta A) - (B \pm \Delta A) = (A-B) \pm (\Delta A + \Delta B) \]
  \[ c(A \pm \Delta A) = cA \pm c \Delta A \]

  \textbf{Relative uncertainty} $\mathcal{E}$ (unitless)
  \[ \mathcal{E}(m) = {{\Delta(m)} \over m} \cdot 100 \]
  \[ (A \pm \mathcal{E} A) \cdot (B \pm \mathcal{E} B) = (A \cdot B) \pm (\mathcal{E} A + \mathcal{E} B) \]
  \[ (A \pm \mathcal{E} A) \div (B \pm \mathcal{E} B) = (A \div B) \pm (\mathcal{E} A + \mathcal{E} B) \]
  \[ (A \pm \mathcal{E} A)^n = (A^n \pm n \mathcal{E} A) \]
  \[ c(A \pm \mathcal{E} A)=cA \pm \mathcal{E} A \]

  Uncertainty of a measurement is $1 \over 2$ the smallest division

  \textbf{Precision} - concordance of values \\
  \textbf{Accuracy} - closeness to actual value\\
  \textbf{Random errors} - unpredictable, reduced by more tests \\
  \textbf{Systematic errors} - not reduced by more tests \\
  \textbf{Uncertainty} - margin of potential error \\
  \textbf{Error} - actual difference \\
  \textbf{Hypothesis} - can be tested experimentally \\
  \textbf{Model} - evidence-based but indirect representation

\end{multicols}

\begin{center}
  \includegraphics[height=2.95cm]{graphics/spectrum.png}
\end{center}

\end{document}
