From ac176d3d2464068ad19adb01a74f54d61c442562 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 1 Sep 2019 22:03:21 +1000 Subject: [PATCH] [spec] reorganise notes for sac --- spec/dynamics.pdf | Bin 178283 -> 163504 bytes spec/dynamics.tex | 152 +++++++++++++--------------- spec/spec-collated.pdf | Bin 377933 -> 449594 bytes spec/spec-collated.tex | 75 +++++++++----- spec/statistics.pdf | Bin 203428 -> 212441 bytes spec/statistics.tex | 218 +++++++++++++++++++++++++---------------- 6 files changed, 257 insertions(+), 188 deletions(-) diff --git a/spec/dynamics.pdf b/spec/dynamics.pdf index ca06d3747805d82297d78caf4cf3f3f531790f1d..dc59835e128918c05f5f7faeb3a93f4cea6c7a8f 100644 GIT binary patch delta 99341 zcmZU4Q*hu<&~3P}ZQHhO+qSjgFSc!OY}?7k8*c24Z728t)vd4Y!|jKzu6dlAo^$5( zne!XihEJG8GH?zy_KZO*P->u7cP$>T6T^Q+Ye|oMq_~p_=0}v(=^(si?y93XbHi~_ zP2E*`$!PkLVIc8N{;(-V486)(+W}O&EV{S%idzP2ts|}OC->`<>)p$cW<2+dhljTT zjY11rngv+F@}b6~>)ZQuw2j)er5nje>g~B<+_*LluF 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-\psset{dimen=monkey,fillstyle=solid,opacity=.5} -\def\object{% - \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1) - \psaxes[linecolor=gray,labels=none,ticks=none]{->}(0,0)(-3,-3)(3,2)[$x$,0][$y$,90] - \rput{*0}{% - \psline{->}(0,-2)% - \uput[-90]{*0}(0,-2){$\vec{w}$}} -} - -\usepackage{tabularx} -\usetikzlibrary{angles} -\usepackage{keystroke} -\usepackage{listings} -\usepackage{xcolor} % used only to show the phantomed stuff -\definecolor{cas}{HTML}{e6f0fe} - -\pagestyle{fancy} -\fancyhead[LO,LE]{Year 12 Specialist - Dynamics} -\fancyhead[CO,CE]{Andrew Lorimer} - -\setlength\parindent{0pt} - +\documentclass[spec-collated.tex]{subfiles} \begin{document} -\title{Dynamics} -\author{} -\date{} -\maketitle +\section{Dynamics} -\section{Resolution of forces} +\subsection*{Resolution of forces} \textbf{Resultant force} is sum of force vectors -\subsection*{In angle-magnitude form} +\subsubsection*{In angle-magnitude form} \makebox[3cm]{Cosine rule:} \(c^2=a^2+b^2-2ab\cos\theta\) \makebox[3cm]{Sine rule:} \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) -\subsection*{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form} +\subsubsection*{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form} Vector of \(a\) N at \(\theta\) to \(x\) axis is equal to \(a \cos \theta \boldsymbol{i} + a \sin \theta \boldsymbol{j}\). Convert all force vectors then add. To find angle of an \(a\boldsymbol{i} + b\boldsymbol{j}\) vector, use \(\theta = \tan^{-1} \frac{b}{a}\) -\subsection*{Resolving in a given direction} +\subsubsection*{Resolving in a given direction} The resolved part of a force \(P\) at angle \(\theta\) is has magnitude \(P \cos \theta\) To convert force \(||\vec{OA}\) to angle-magnitude form, find component \(\perp\vec{OA}\) then \(|\boldsymbol{r}|=\sqrt{\left(||\vec{OA}\right)^2 + \left(\perp\vec{OA}\right)^2},\quad \theta = \tan^{-1}\dfrac{\perp\vec{OA}}{||\vec{OA}}\) -\section{Newton's laws} +\subsection*{Newton's laws} \begin{tcolorbox} - \begin{enumerate} - \item Velocity is constant without a net external velocity + \begin{enumerate}[leftmargin=1mm] + \item Velocity is constant without a net external force \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\) \item Equal and opposite forces \end{enumerate} \end{tcolorbox} -\subsection*{Weight} +\subsubsection*{Weight} A mass of \(m\) kg has force of \(mg\) acting on it -\subsection*{Momentum \(\rho\)} +\subsubsection*{Momentum \(\rho\)} \[ \rho = mv \tag{units kg m/s or Ns} \] -\subsection*{Reaction force \(R\)} +\subsubsection*{Reaction force \(R\)} \begin{itemize} \item With no vertical velocity, \(R=mg\) - \item With upwards acceleration, \(R-mg=ma\) + \item With vertical acceleration, \(|R|=m|a|-mg\) \item With force \(F\) at angle \(\theta\), then \(R=mg-F\sin\theta\) \end{itemize} -\subsection*{Friction} +\subsubsection*{Friction} \[ F_R = \mu R \tag{friction coefficient} \] -\section{Inclined planes} +\subsection*{Inclined planes} \[ \boldsymbol{F} = |\boldsymbol{F}| \cos \theta \boldsymbol{i} + |\boldsymbol{F}| \sin \theta \boldsymbol{j} \] \begin{itemize} @@ -122,9 +75,7 @@ A mass of \(m\) kg has force of \(mg\) acting on it pulley/.style={thick} } -\begin{figure}[!htb] - \centering - \begin{tikzpicture} + \begin{center}\begin{tikzpicture} \pgfmathsetmacro{\Fnorme}{2} \pgfmathsetmacro{\Fangle}{30} @@ -153,17 +104,9 @@ A mass of \(m\) kg has force of \(mg\) acting on it \end{scope} \draw[force,->] (M.center) -- ++(0,-1) node[below] {$mg$}; \draw (M.center)+(-90:\arcr) arc [start angle=-90,end angle=\iangle-90,radius=\arcr] node [below, pos=.5] {\footnotesize\(\theta\)}; - \end{tikzpicture} -\end{figure} + \end{tikzpicture}\end{center} -\section{Connected particles} - -\begin{itemize} - \item \textbf{Suspended pulley:} tension in both sections of rope are equal - \item \textbf{Linear connection:} find acceleration of system first - \item \textbf{Pulley on right angle:} \(a = \frac{m_2g}{m_1+m_2}\) where \(m_2\) is suspended (frictionless on both surfaces) - \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane -\end{itemize} +\subsection*{Connected particles} \def\boxwidth{0.5} \tikzset{ @@ -172,8 +115,7 @@ A mass of \(m\) kg has force of \(mg\) acting on it } -\begin{figure}[!htb] - \centering +\begin{center} \begin{tikzpicture} \matrix[column sep=1cm] { @@ -231,22 +173,68 @@ A mass of \(m\) kg has force of \(mg\) acting on it \\ }; \end{tikzpicture} -\end{figure} + \end{center} + +\begin{itemize} + \item \textbf{Suspended pulley:} tension in both sections of rope are equal \\ + \(|a| = g \frac{m_1 - m_2}{m_1 + m_2}\) where \(m_1\) accelerates down \\ + With tension: \\ + \[ \begin{cases}m_1 g - T = m_1 a\\ T - m_2 g = m_2 a\end{cases} \\ \implies m_1 g - m_2 g = m_1 a + m_2 a \] + \item \textbf{String pulling mass on inclined pane:} Resolve parallel to plane \\ + \[ T-mg \sin \theta = ma \] + \item \textbf{Linear connection:} find acceleration of system first + \item \textbf{Pulley on right angle:} \(a = \frac{m_2g}{m_1+m_2}\) where \(m_2\) is suspended (frictionless on both surfaces) + \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane +\end{itemize} + +\hspace{2em}\parbox{8em}{In this example, note \(T_1 \ne T_2\):} + \begin{tikzpicture} + + \begin{scope} -\section{Equilibrium} + \coordinate (O) at (0,0); + \coordinate (A) at ($({3*cos(\iangle)},{3*sin(\iangle)})$); + \coordinate (B) at ($({3*cos(\iangle)},0)$); + \coordinate (C) at ($({(1-0.25*\boxwidth)*cos(\iangle)},{(1-0.25*\boxwidth)*sin(\iangle)})$); % centre of box + \coordinate (D) at ($(C)+(\iangle:\boxwidth)$); + \coordinate (E) at ($(D)+(90+\iangle:0.5*\boxwidth)$); + \coordinate (F) at ($(B)+(0,{1.5*sin(\iangle)})$); + \coordinate (G) at ($(A)+(\iangle:-2*\boxwidth)$); + \coordinate (H) at ($(G)+(90+\iangle:0.5*\boxwidth)$); + \coordinate (I) at ($(H)+(\iangle:-0.5*\boxwidth)$); + \coordinate (J) at ($(H)+(\iangle:\boxwidth)$); + \coordinate (X) at ($(A)+(\iangle:0.5*\boxwidth)$); % centre of pulley + \coordinate (Y) at ($(X)+(90+\iangle:0.5*\boxwidth)$); % chord of pulley + + \draw[plane] (O) -- (A) -- (B) -- (O); + \draw (O)+(\arcr,0) arc [start angle=0,end angle=\iangle,radius=\arcr] node [right, pos=.75] {\footnotesize\(\theta\)}; + + \draw [rotate=\iangle, m] (C) rectangle ++(\boxwidth,\boxwidth) node (z) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)}; + \draw [rotate=\iangle, m] (G) rectangle ++(\boxwidth,\boxwidth) node (l) [rotate=\iangle, midway, font=\footnotesize] {\(m_2\)}; + \draw [pulley] (A) -- (X) ++(0.5*\boxwidth, 0) arc[rotate=\iangle, start angle=0, delta angle=360, x radius=0.25, y radius=0.25] node(r) [midway, rotate=\iangle] {}; + \draw [string] (E) -- (H) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_2\)}; + \draw [string] (J) -- (Y) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_1\)} arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)})$) node [midway, above right, font=\footnotesize] {\(T_1\)} node[m] {\(m_3\)}; + + \end{scope} + + \end{tikzpicture} +\subsection*{Equilibrium} \[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\] +\[ c^2 = a^2 + b^2 - 2ab \cos \theta \tag{cosine rule} \] Three methods: \begin{enumerate} \item Lami's theorem (sine rule) - \item Triangle of forces or CAS (use to verify) + \item Triangle of forces (cosine rule) \item Resolution of forces (\(\Sigma F = 0\) - simultaneous) \end{enumerate} -\colorbox{cas}{On CAS:} use Geometry, lock known constants. + \begin{cas} + \textbf{To verify:} Geometry tab, then select points with normal cursor. Click right arrow at end of toolbar and input point, then lock known constants. + \end{cas} -\section{Variable forces (DEs)} +\subsection*{Variable forces (DEs)} \[ a = \dfrac{d^2x}{dt^2} = \dfrac{dv}{dt} = v\dfrac{dv}{dx} = \dfrac{d}{dx} \left( \frac{1}{2} v^2 \right) \] diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 2af5693473797d195b23dfaba30384137b9a4b4f..7aa2159ebf421d89567fbb2924c33b3875b71986 100644 GIT binary patch delta 385608 zcmZU)V{j!*^Zy;&wr$&Xvf;+I?VQ+7Ha0f4ZQI${KC$^+_w#%CpI2QqRbACxHT|9W zbk9sAc4s?23Z;sK6eBAWCmiMc^2jsCN1+}#RbsQuDvU)*PLoSGM8^(+gfEP zFwX6{`2@7_y&Lj0{JEnqOxA8M%B6r(abj2T%0hAZ{;b>m24*w=ObLGx|GWXQ4SW5< zl=J=Hw*M(xV)V<3lSf00k{4<3z3<{G(xG7jN6ZuI!b%z__&2dL5;?8;2doYX{?zmw;ci$1+Q7E37kL zOMrh2jzo2uMDfvXca6m&aRI;|5{Woy@a0G(631(w4Q2<2+c=)SzAkUS_JY&Pu9+|5 zPc7Swx-P4*z_zxS%O{Hle_>gEMhVLk>oE|;Du4vVPNW(J19-q%tox)@!gI? zHqk;p5%#8rGNW9|SPRs_p_iWQ&#fbo9Cu|?7JeI}2;@d&{9=IedWL#E*-$nY=zBFA zVees3LCl7NLIoka^&VpG%wyT4LaJNxC{WPo7j8FzQV#mNW43 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