From 7fa21b6df0e195d9d0db3bd0c0cd751806ebb6aa Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 9 Sep 2018 11:15:02 +1000 Subject: [PATCH] start final cheatsheet --- physics/final.tex | 472 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 472 insertions(+) create mode 100644 physics/final.tex diff --git a/physics/final.tex b/physics/final.tex new file mode 100644 index 0000000..f809976 --- /dev/null +++ b/physics/final.tex @@ -0,0 +1,472 @@ +\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} -- 2.47.1