From: Andrew Lorimer Date: Thu, 19 Jul 2018 08:10:09 +0000 (+1000) Subject: finalise midyear reference & tidy tex code X-Git-Tag: yr11~86 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/e34d8eeb7921f4c11413fd02f18775c7f28f4595 finalise midyear reference & tidy tex code --- diff --git a/physics/midyear.fdb_latexmk b/physics/midyear.fdb_latexmk index 6f9b2aa..8d8598f 100644 --- a/physics/midyear.fdb_latexmk +++ b/physics/midyear.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1531545115 "midyear.tex" "midyear.pdf" "midyear" 1531896370 +["pdflatex"] 1531987725 "midyear.tex" "midyear.pdf" "midyear" 1531987726 "/mnt/andrew/graphics/ac-generator.png" 1529320281 28991 f74a831edcaed66632e709a6b2e2ebfd "" "/mnt/andrew/graphics/ac-motor.png" 1529313736 18626 d0f4943be26fdce44a4ce45ccb01633c "" "/mnt/andrew/graphics/banked-track.png" 1531025364 51228 cc5d8f3f8efb06a64aca4afabfd63eed "" @@ -70,9 +70,9 @@ "/usr/share/texmf-dist/web2c/texmf.cnf" 1525040116 33301 a3134070eacafb10b1f371612ce2650d "" "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1530607795 2586890 ddabbb88c498d59d5daa7919afb0a27f "" "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1530607774 7906415 c2bc2a1e85cbebe301ae391211242f7e "" - "midyear.aux" 1531545115 223 c9c8cc73e6c5d1c4bae220d8eb29b53c "" - "midyear.tex" 1531896360 7760 df61705a0ad9f9a0549b9ac5cd9a1da8 "" + "midyear.aux" 1531987726 223 c9c8cc73e6c5d1c4bae220d8eb29b53c "" + "midyear.tex" 1531987725 9117 4a1990d0b0ea8d63b7cd19f496ac4706 "" (generated) - "midyear.pdf" "midyear.log" + "midyear.pdf" "midyear.aux" diff --git a/physics/midyear.log b/physics/midyear.log index 8d34683..60f4e25 100644 --- a/physics/midyear.log +++ b/physics/midyear.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018/Arch Linux) (preloaded format=pdflatex 2018.7.3) 14 JUL 2018 15:11 +This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018/Arch Linux) (preloaded format=pdflatex 2018.7.3) 19 JUL 2018 18:08 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -239,54 +239,44 @@ Package grfext Info: Graphics extension search list: (/usr/share/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live )) -LaTeX Font Info: Try loading font information for U+msa on input line 23. +LaTeX Font Info: Try loading font information for U+msa on input line 28. 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PDF statistics: 69 PDF objects out of 1000 (max. 8388607) 43 compressed objects within 1 object stream diff --git a/physics/midyear.pdf b/physics/midyear.pdf index ceb2f03..e86882b 100644 Binary files a/physics/midyear.pdf and b/physics/midyear.pdf differ diff --git a/physics/midyear.synctex.gz b/physics/midyear.synctex.gz index 529cc9f..353f42d 100644 Binary files a/physics/midyear.synctex.gz and b/physics/midyear.synctex.gz differ diff --git a/physics/midyear.tex b/physics/midyear.tex index 46ae5ab..865a0ab 100644 --- a/physics/midyear.tex +++ b/physics/midyear.tex @@ -16,292 +16,312 @@ \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} + + $\theta = \tan^{-1} {{v^2} \over rg}$ (also for objects on string) + + $\Sigma F$ always acts towards centre, but not necessarily horizontally - $\Sigma F = {{mv^2} \over r} = mg \tan \theta$ + $\Sigma F = {{mv^2} \over r} = mg \tan \theta$ - Design speed $v = \sqrt{gr\tan\theta}$ + 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) + $W=Fx=\Delta \Sigma E$ (work) - $E_G = mgh$ (potential) + $E_K = {1 \over 2}mv^2$ (kinetic) - $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer) + $E_G = mgh$ (potential) + $\Sigma E = {1 \over 2} mv^2 + mgh$ (energy transfer) + +% ----------------------- \subsection*{Horizontal motion} - $v = {{2 \pi r} \over T}$ + $\operatorname{m/s} \times 3.6 = \operatorname{km/h}$ + + $v = {{2 \pi r} \over T}$ - $f = {1 \over T}, \quad T = {1 \over f}$ + $f = {1 \over T}, \quad T = {1 \over f}$ - $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$ + $a_{centrip} = {v^2 \over r} = {{4 \pi^2 r} \over T^2}$ - $\Sigma F$ towards centre, $v$ tangential + $\Sigma F$ towards centre, $v$ tangential - $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$ + $F_{centrip} = {{mv^2} \over r} = {{4 \pi^2 rm} \over T^2}$ - \includegraphics[height=4cm]{/mnt/andrew/graphics/circ-forces.png} + \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 + $T =$ tension, e.g. circular pendulum - \subsection*{Projectile motion} - \begin{itemize} - \item{horizontal component of velocity is constant if no air resistance} + $T+mg = {{mv^2}\over r}$ at highest point - \item{vertical component affected by gravity: $a_y = -g$} -\end{itemize} + $T-mg = {{mv^2} \over r}$ at lowest point -$v=\sqrt{v^2_x + v^2_y}$ (vector addition) - -$h={{u^2\sin \theta ^2}\over 2g}$ (max height) +% ----------------------- + \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} -$y=ut \sin \theta-{1 \over 2}gt^2$ (time of flight) + \begin{align*} + v=\sqrt{v^2_x + v^2_y} \tag{vectors} \\ + h={{u^2\sin \theta ^2}\over 2g} \tag{max height}\\ + y=ut \sin \theta-{1 \over 2}gt^2 \tag{time of flight} \\ + d={v^2 \over g}\sin \theta \tag{horiz. range} \\ + \end{align*} -$d={v^2 \over g}sin \theta$ (horizontal range) - \includegraphics[height=3.2cm]{/mnt/andrew/graphics/projectile-motion.png} + \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: $m=k,\quad A=E_{spr}$} - \item{Force-dist: $A=\Delta \operatorname{gpe}$} - \item{Field-dist: $A=\Delta \operatorname{gpe} / \operatorname{kg}$} - \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} + $\rho = mv$ -$\rho = mv$ + $\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$ -$\operatorname{impulse} = \Delta \rho, \quad F \Delta t = m \Delta v$ + Momentum is conserved. -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 + \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) -2. Speed of light $c$ is the same to all observers (Michelson-Morley) + $\therefore , t$ must dilate as speed changes -$\therefore , t$ must dilate as speed changes + {\bf Inertial reference frame} - $a=0$ -{\bf Inertial reference frame} - $a=0$ + {\bf Proper time $t_0$ $\vert$ length $l_0$} - measured by observer in same frame as events -{\bf Proper time $t_0$ $\vert$ length $l_0$} - measured by observer in same frame as events +% ----------------------- + \subsection*{Lorentz factor} -\subsection*{Lorentz factor} + $$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}$$ -$$\gamma = {1 \over {\sqrt{1-{v^2 \over c^2}}}}$$ + $t=t_0 \gamma$ ($t$ longer in moving frame) -$t=t_0 \gamma$ ($t$ longer in moving frame) + $l={l_0 \over \gamma}$ ($l$ contracts $\parallel v$: shorter in moving frame) -$l={l_0 \over \gamma}$ ($l$ contracts $\parallel v$: shorter in moving frame) + $m=m_0 \gamma$ (mass dilation) -$m=m_0 \gamma$ (mass dilation) + $$v = c\sqrt{1-{1 \over \gamma^2}}$$ -$$v = c\sqrt{1-{1 \over \gamma^2}}$$ +% ----------------------- + \subsection*{Energy and work} -\subsection*{Energy and work} + $E_0 = mc^2$ (rest) -$E_0 = mc^2$ (rest) + $E_{total} = E_K + E_{rest} = \gamma mc^2$ -$E_{total} = E_K + E_{rest} = \gamma mc^2$ + $E_K = (\gamma - 1)mc^2$ -$E_K = (\gamma - 1)mc^2$ + $W = \Delta E = \Delta mc^2$ -$W = \Delta E = \Delta mc^2$ +% ----------------------- + \subsection*{Relativistic momentum} -\subsection*{Relativistic momentum} + $$\rho = {mv \over \sqrt{1-{v^2 \over c^2}}}= {\gamma mv} = {\gamma \rho_0}$$ -$$\rho = {mv \over \sqrt{1-{v^2 \over c^2}}}= {\gamma mv} = {\gamma \rho_0}$$ + $\rho \rightarrow \infty$ as $v \rightarrow c$ -$\rho \rightarrow \infty$ as $v \rightarrow c$ + $v=c$ is impossible (requires $E=\infty$) -$v=c$ is impossible (requires $E=\infty$) + $$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$ -$$v={\rho \over {m\sqrt{1+{p^2 \over {m^2 c^2}}}}}$$ +% ----------------------- + \subsection*{Fusion and fission} -\subsection*{Fusion and fission} + $1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$ -$1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$ + e- accelerated with $x$ V is given $x$ eV -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} +% ----------------------- + \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} -\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: 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} + \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 means out of page, cross means into page + \item +ve corresponds to N pole + \end{itemize} -\subsection*{Gravity} -\[ -F_g=G{{m_1m_2}\over r^2}\tag{grav. force} -\] +% ----------------------- + \subsection*{Gravity} -\[ -g={F_g \over m}=G{M_{\operatorname{planet}} \over r^2}\tag{grav. acc.} -\] + \[F_g=G{{m_1m_2}\over r^2}\tag{grav. force}\] -\[ -E_g = mg \Delta h\tag{gpe} -\] + \[g={F_g \over m}=G{M_{\operatorname{planet}} \over r^2}\tag{grav. acc.}\] -\[ -W = \Delta E_g = Fx\tag{work} -\] + \[E_g = mg \Delta h\tag{gpe}\] -\subsection*{Satellites} -\[ -v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T} -\] + \[W = \Delta E_g = Fx\tag{work}\] -\[ -T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period} -\] + \[w=m(g-a) \tag{app. weight}\] -\[ -\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius} -\] +% ----------------------- + \subsection*{Satellites} + \[v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T}\] + \[T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period}\] -\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} + \[\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius}\] -% \[ -% {E_1 \over E_2}={r_1 \over r_2}^2 -% \] +% ----------------------- + \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} -\[ -F=qvB\tag{force on moving charged particles} -\] + \[{E_1 \over E_2}={r_1 \over r_2}^2\] -if $B {\not \perp} A, \Phi \rightarrow 0$ \hspace{1em}, \hspace{1em} if $B \parallel A, \Phi = 0$ + \[F=qvB\tag{force on moving charged particles}\] + 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} + \includegraphics[height=2cm]{/mnt/andrew/graphics/field-lines.png} -\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*} +% ----------------------- + \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{Lenz's law:} ``$-n$'' in Faraday - emf opposes $\Delta \Phi$ -\textbf{Eddy currents:} counter movement within a field + \textbf{Eddy currents:} counter movement within a field -\textbf{Right hand grip:} thumb points to north or $I$ + \textbf{Right hand grip:} thumb points to north or $I$ -\textbf{Right hand slap:} field, current, force are $\perp$ + \textbf{Right hand slap:} field, current, force are $\perp$ -\textbf{Flux-time graphs:} gradient $\times n = \operatorname{emf}$ + \textbf{Flux-time graphs:} gradient $\times n = \operatorname{emf}$ -\textbf{Transformers:} core strengthens \& focuses $\Phi$ + \textbf{Transformers:} core strengthens \& focuses $\Phi$ -% \columnbreak +% ----------------------- + \subsection*{Power transmission} -\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} \\ + \end{align*} -\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*} + Use high-$V$ side for correct $|V_{drop}|$ -\begin{itemize} - {\item Parallel - voltage is constant} - {\item Series - voltage is shared within branch} -\end{itemize} + \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} + \includegraphics[height=4cm]{/mnt/andrew/graphics/ac-generator.png} -\subsection*{Motors} +% ----------------------- + \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} \\ + \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) + \textbf{DC:} split ring (two halves) % \begin{wrapfigure}{r}{0.3\textwidth} % \end{wrapfigure} -\textbf{AC:} slip ring (separate rings with constant contact) + \textbf{AC:} slip ring (separate rings with constant contact) \end{multicols}