From: Andrew Lorimer Date: Mon, 16 Jul 2018 05:43:13 +0000 (+1000) Subject: add midyear cheatsheet, add ch8-9 notes for waves X-Git-Tag: yr11~88 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/e57f9b6f9ecb90556c314815b6c1afe4db7dc8cf add midyear cheatsheet, add ch8-9 notes for waves --- diff --git a/physics/midyear.aux b/physics/midyear.aux index 0ed01dd..894cdc5 100644 --- a/physics/midyear.aux +++ b/physics/midyear.aux @@ -1,3 +1,4 @@ \relax \@writefile{toc}{\contentsline {section}{\numberline {1}Motion}{}} \@writefile{toc}{\contentsline {section}{\numberline {2}Relativity}{}} +\@writefile{toc}{\contentsline {section}{\numberline {3}Fields and power}{}} diff --git a/physics/midyear.fdb_latexmk b/physics/midyear.fdb_latexmk index 520331f..3b95bb6 100644 --- a/physics/midyear.fdb_latexmk +++ b/physics/midyear.fdb_latexmk @@ -1,13 +1,25 @@ # Fdb version 3 -["pdflatex"] 1531051766 "midyear.tex" "midyear.pdf" "midyear" 1531051767 +["pdflatex"] 1531545115 "midyear.tex" 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5a97061616e0c8b2aa79c6615ff769f4 "" @@ -55,9 +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" 1531051767 146 a59c3471c01dfdc38f374ef57c657d3c "" - "midyear.tex" 1531051765 2909 a6a51cef65d7d592a8a6d071f6f9bbf7 "" + "midyear.aux" 1531545115 223 c9c8cc73e6c5d1c4bae220d8eb29b53c "" + "midyear.tex" 1531545115 7760 df61705a0ad9f9a0549b9ac5cd9a1da8 "" (generated) "midyear.aux" - "midyear.log" "midyear.pdf" + "midyear.log" diff --git a/physics/midyear.fls b/physics/midyear.fls index a8fd227..fd1ed4e 100644 --- a/physics/midyear.fls +++ b/physics/midyear.fls @@ -7,16 +7,6 @@ INPUT /usr/share/texmf-dist/tex/latex/base/article.cls INPUT /usr/share/texmf-dist/tex/latex/base/article.cls INPUT /usr/share/texmf-dist/tex/latex/base/size10.clo INPUT /usr/share/texmf-dist/tex/latex/base/size10.clo -INPUT /usr/share/texmf-dist/tex/latex/geometry/geometry.sty -INPUT /usr/share/texmf-dist/tex/latex/geometry/geometry.sty -INPUT /usr/share/texmf-dist/tex/latex/graphics/keyval.sty -INPUT /usr/share/texmf-dist/tex/latex/graphics/keyval.sty -INPUT /usr/share/texmf-dist/tex/generic/oberdiek/ifpdf.sty -INPUT /usr/share/texmf-dist/tex/generic/oberdiek/ifpdf.sty -INPUT /usr/share/texmf-dist/tex/generic/oberdiek/ifvtex.sty -INPUT /usr/share/texmf-dist/tex/generic/oberdiek/ifvtex.sty -INPUT /usr/share/texmf-dist/tex/generic/ifxetex/ifxetex.sty -INPUT /usr/share/texmf-dist/tex/generic/ifxetex/ifxetex.sty INPUT /usr/share/texmf-dist/tex/latex/tools/multicol.sty INPUT /usr/share/texmf-dist/tex/latex/tools/multicol.sty INPUT /usr/share/texmf-dist/tex/latex/preprint/fullpage.sty @@ -31,10 +21,16 @@ INPUT /usr/share/texmf-dist/tex/latex/amsmath/amsbsy.sty INPUT 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entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -23,50 +23,25 @@ File: size10.clo 2014/09/29 v1.4h Standard LaTeX file (size option) \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 -) (/usr/share/texmf-dist/tex/latex/geometry/geometry.sty -Package: geometry 2018/03/24 v5.7 Page Geometry - (/usr/share/texmf-dist/tex/latex/graphics/keyval.sty -Package: keyval 2014/10/28 v1.15 key=value parser (DPC) -\KV@toks@=\toks14 -) (/usr/share/texmf-dist/tex/generic/oberdiek/ifpdf.sty -Package: ifpdf 2017/03/15 v3.2 Provides the ifpdf switch -) (/usr/share/texmf-dist/tex/generic/oberdiek/ifvtex.sty -Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) -Package ifvtex Info: VTeX not detected. -) (/usr/share/texmf-dist/tex/generic/ifxetex/ifxetex.sty -Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional -) -\Gm@cnth=\count88 -\Gm@cntv=\count89 -\c@Gm@tempcnt=\count90 -\Gm@bindingoffset=\dimen103 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PDF statistics: - 51 PDF objects out of 1000 (max. 8388607) - 34 compressed objects within 1 object stream + 69 PDF objects out of 1000 (max. 8388607) + 43 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) - 16 words of extra memory for PDF output out of 10000 (max. 10000000) + 36 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/physics/midyear.pdf b/physics/midyear.pdf index 1c2fd27..ceb2f03 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 c9a4e05..529cc9f 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 28bcdfe..46ae5ab 100644 --- a/physics/midyear.tex +++ b/physics/midyear.tex @@ -1,14 +1,15 @@ \documentclass[a4paper]{article} - -\usepackage[a4paper]{geometry} \usepackage{multicol} \usepackage[cm]{fullpage} \usepackage{amsmath} +\usepackage{amssymb} \setlength{\parindent}{0cm} \usepackage[nodisplayskipstretch]{setspace} -\setstretch{1.5} +\setstretch{1.3} \usepackage{graphicx} \usepackage{wrapfig} +\usepackage{enumitem} +\setitemize{noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt,leftmargin=5pt} \begin{document} @@ -77,7 +78,7 @@ $h={{u^2\sin \theta ^2}\over 2g}$ (max height) $y=ut \sin \theta-{1 \over 2}gt^2$ (time of flight) $d={v^2 \over g}sin \theta$ (horizontal range) - \includegraphics[height=4cm]{/mnt/andrew/graphics/projectile-motion.png} + \includegraphics[height=3.2cm]{/mnt/andrew/graphics/projectile-motion.png} \subsection*{Pulley-mass system} @@ -88,6 +89,8 @@ $d={v^2 \over g}sin \theta$ (horizontal range) \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} @@ -119,10 +122,186 @@ $\Sigma E_{K \operatorname{before}} = \Sigma E_{K \operatorname{after}}$ if elas \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*{Fusion and fission} + +$1 \operatorname{eV} = 1.6 \times 10^{-19} \operatorname{J}$ + +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} + +\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} + +\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} + +\subsection*{Gravity} +\[ +F_g=G{{m_1m_2}\over r^2}\tag{grav. force} +\] + +\[ +g={F_g \over m}=G{M_{\operatorname{planet}} \over r^2}\tag{grav. acc.} +\] + +\[ +E_g = mg \Delta h\tag{gpe} +\] + +\[ +W = \Delta E_g = Fx\tag{work} +\] + +\subsection*{Satellites} +\[ +v=\sqrt{GM \over r} = \sqrt{gr} = {{2 \pi r} \over T} +\] + +\[ +T={\sqrt{4 \pi^2 r^2} \over {GM}}\tag{period} +\] + +\[ +\sqrt[3]{{GMT^2}\over{4\pi^2}}\tag{radius} +\] + + + +\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} + +% \[ +% {E_1 \over E_2}={r_1 \over r_2}^2 +% \] + +\[ +F=qvB\tag{force on moving charged particles} +\] + +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} + +\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{Eddy currents:} counter movement within a field + +\textbf{Right hand grip:} thumb points to north or $I$ + +\textbf{Right hand slap:} field, current, force are $\perp$ + +\textbf{Flux-time graphs:} gradient $\times n = \operatorname{emf}$ + +\textbf{Transformers:} core strengthens \& focuses $\Phi$ + +% \columnbreak + +\subsection*{Power transmission} + +\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*} + +\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} +\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 (one ring split into two halves) +% \begin{wrapfigure}{r}{0.3\textwidth} +% \end{wrapfigure} +\textbf{AC:} slip ring (separate rings with constant contact) \end{multicols} diff --git a/physics/waves.md b/physics/waves.md index 92cf75d..0b085c5 100644 --- a/physics/waves.md +++ b/physics/waves.md @@ -1,24 +1,124 @@ # Waves -## Longitudinal and Transverse Waves +## Mechanical waves +- need a medium to travel through +- cannot transfer energy through vacuum +- individual particles have little movement regardless of the distance of wave +- **transfer of energy without net transfer of matter** -Longitudinal - direction of motion parallel to wave -Transverse - direction of motion perpendicular to wave +**Nodes** - point of no motion (fixed point on graph) +**Antinodes** - point of maximum motion (peaks) +**Crests** (peaks) & **troughs** (azimuths) -## Wavelength & frequency +### Longitudinal waves +**Direction of motion is parallel to wave** -$$v=f \times \lambda$$ +![](/mnt/andrew/graphics/longitudinal-waves.png) + +### Transverse waves +**Direction of motion is perpendicular to wave** +- rarefactions (expansions) +- compressions +![](/mnt/andrew/graphics/transverse-waves.png) + +### Measuring mechanical waves + +**Amplitude $A$** - max displacement from rest position (0) +**Wavelength $\lambda$** - distance between two points of same y-value (points are in phase) +**Frequency $f$** - number of cycles (wavelengths) per second + +$T={1 \over f}\quad$(period: time for one cycle) +$v=f \lambda \quad$(speed: displacement per second) + +### Doppler effect +- occurs when there is relative movement between source and observer +- inverse relationship between frequency and distance: $f \propto {1 \over d}$ +- applies to all types of wave +- only affects apparent $f$; actual $f$ is constant + +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. -Nodes - point of no motion -Antinodes - point of maximum motion (peaks) -Crests (peaks) & troughs (azimuths) ## Interference patterns -Standing wave - constructive interference at resonant frequency +When a medium changes character: +- some energy is *reflected* +- some energy is *absorbed* by new medium +- some energy is *transmitted* + +**Superposition** - stimuli add together at a given point (vector addition) +**Standing wave** - constructive interference at resonant frequency + +### Reflection + +**Diffuse** reflection - irregular surface reflects each ray in a different angle. + +#### Rays +Two- or three-dimensional *wave fronts* can be reflected, e.g. waves at a beach. + +Direction of motion of wave fronts can be shown by arrows, called *rays*, which are perpendicular to the wave front: + +![](/mnt/andrew/graphics/rays.png) + +Angle of incidence $\theta_i =$ angle of reflection $\theta_r$ +- Normal: $\perp$ to wall +- Incident wave front: $\perp$ to incident ray +- Incident ray: $ + +#### Transverse +- sign of reflected transverse wave is inverted when endpoint is fixed in y-axis (equivalent to $180^\circ=\pi={\lambda \over 2}$ phase change) +- phase is constant if endpoint is free to move in y-axis (**reflected** is same as **incident**) + +## Harmonics + +**Harmonic** - fundamental (lowest) frequency to produce a certain number of wavelengths +**Overtone** - a multiple of the fundamental harmonic which produces the same no. of wavelengths at a different frequency (due to constructive interference) + +#### Wave has antinodes at both ends: +$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic) +$f = {nv \div 2l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$) + +#### Wave has antinode at one end: +$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic) +$f = {nv \div 4l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$) + +## Light + +Newton - light as a particle +- light speeds up as it travels through a solid medium + +Hooke - light as a wave +- light slows down through solid medium + +### Huygen's principle +**Each point on a wavefront can be considered a source of secondary wavelets** +![](/mnt/andrew/graphics/huygen.png) + +### Refraction +**Change in direction caused by change in speed** e.g. prism +$\Delta v$ depends on $\lambda$, so wavelengths become "split" +![](/mnt/andrew/graphics/refraction.png) + +Refractive index of a medium depends $\Delta v$ from $c$ +$n={c \over v}\quad$ (refractive index of a medium) +$n_1v_1=n_2v_2$ (equivalence between media) + +### Snell's law +$n$ can be used to determine how much a ray will refract going between two media. + +$$n_1 \sin \theta_1=n_2 \sin \theta_2$$ + +### Total internal reflection +When $n_1 < n_2$, light is refracted *towards* normal ($90^\circ$ to medium border - "vertical" line in case of air/water). +When $n_1 > n_2$, light is reflected *away* from normal. +**Critical angle $\theta_c$** - angle of incidence $\theta_1$ at which $\theta_2 \gt 90^\circ$ to normal +$n_1 sin \theta_c = n_2 \sin 90^\circ$ +$\therefore \theta_c = {n_2 \over n_1}$ + +### Dispersion