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b/spec/dynamics.tex index e22575b..c219f62 100644 --- a/spec/dynamics.tex +++ b/spec/dynamics.tex @@ -1,4 +1,4 @@ -\documentclass[a4paper, tikz]{article} +\documentclass[a4paper, tikz, pstricks]{article} \usepackage[a4paper,margin=2cm]{geometry} \usepackage{array} \usepackage{amsmath} @@ -12,7 +12,18 @@ decorations, scopes, } +\usepackage{pst-plot} +\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 @@ -50,6 +61,8 @@ 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} \begin{enumerate} @@ -128,12 +141,100 @@ \draw (M.center)+(-90:\arcr) arc [start angle=-90,end angle=\iangle-90,radius=\arcr] node [below, pos=.5] {\tiny\(\theta\)}; \end{tikzpicture} - \subsection{Connected particles} + \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 edge of incline:} find downwards force \(W_2\) and components of mass on plane \end{itemize} +\def\iangle{25} % Angle of the inclined plane + +\def\down{-90} +\def\arcr{0.5cm} % Radius of the arc used to indicate angles + +{\begin{centering} {\begin{tikzpicture}[ + force/.style={>=latex,draw=blue,fill=blue}, + axis/.style={densely dashed,gray,font=\small}, + M/.style={rectangle,draw,fill=lightgray,minimum size=0.6cm,thin}, + m/.style={rectangle,draw=black,fill=lightgray,minimum size=0.3cm,thin}, + plane/.style={draw=black,fill=blue!10}, + string/.style={draw=red, thick}, + pulley/.style={thick}, + scale=1.5 +] + +\matrix[column sep=1cm] { + %% Sketch + \draw[plane] (0,-1) coordinate (base) + -- coordinate[pos=0.5] (mid) ++(\iangle:3) coordinate (top) + |- (base) -- cycle; + \path (mid) node[M,rotate=\iangle,yshift=0.3cm,font=\footnotesize] (M) {\(m_1\)}; + \draw[pulley] (top) -- ++(\iangle:0.25) circle (0.25cm) + ++ (90-\iangle:0.5) coordinate (pulley); + \draw[string] (M.east) -- ++(\iangle:1.4cm) arc (90+\iangle:0:0.25) + -- ++(0,-1) node[m,font=\scriptsize] {\(m_2\)}; + + \draw[->] (base)++(\arcr,0) arc (0:\iangle:\arcr); + \path (base)++(\iangle*0.5:\arcr+5pt) node {\(\theta\)}; + %% + +& + %% Free body diagram of m1 + \begin{scope}[rotate=\iangle] + \node[M,transform shape] (M) {}; + % Draw axes and help lines + + {[axis,->] + \draw (0,-1) -- (0,2) node[right] {\(+\boldsymbol{i}\)}; + \draw (M) -- ++(2,0) node[right] {\(+\boldsymbol{j}\)}; + % Indicate angle. The code is a bit awkward. + + \draw[solid,shorten >=0.5pt] (\down-\iangle:\arcr) + arc(\down-\iangle:\down:\arcr); + \node at (\down-0.5*\iangle:1.3*\arcr) {\(\theta\)}; + } + + % Forces + {[force,->] + % Assuming that Mg = 1. The normal force will therefore be cos(alpha) + \draw (M.center) -- ++(0,{cos(\iangle)}) node[above right] {$N$}; + \draw (M.west) -- ++(-1,0) node[left] {\(F_R\)}; + \draw (M.east) -- ++(1,0) node[above] {\(T_1\)}; + } + + \end{scope} + % Draw gravity force. The code is put outside the rotated + % scope for simplicity. No need to do any angle calculations. + \draw[force,->] (M.center) -- ++(0,-1) node[below] {\(m_1g\)}; + %% + +& + %%% + % Free body diagram of m2 + \node[m] (m) {}; + \draw[axis,->] (m) -- ++(0,-2) node[left] {$+$}; + {[force,->] + \draw (m.north) -- ++(0,1) node[above] {\(T_2\)}; + \draw (m.south) -- ++(0,-1) node[right] {\(m_2g\)}; + } + +\\ +}; +\end{tikzpicture}}\end{centering} } + \section{Equilibrium} + + \[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\] + + Three methods: + \begin{enumerate} + \item Lami's theorem (sine rule) + \item Triangle of forces or CAS (use to verify) + \item Resolution of forces (\(\Sigma F = 0\) - simultaneous) + \end{enumerate} + + + \colorbox{cas}{On CAS:} use Geometry, lock known constants. + \end{document} -- 2.47.1