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+\usetikzlibrary{arrows, + calc, + decorations, + scopes, +} \usepackage{tabularx} \usepackage{keystroke} \usepackage{listings} @@ -25,10 +31,109 @@ \date{} \maketitle - \section{Forces} + \section{Resolution of forces} - \subsection{Resolution of forces} + \textbf{Resultant force} is sum of force vectors - The resolved part of a force \(P\) at angle \(\theta\) is equal to \(P \cos \Theta\) + \subsection{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} + + 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} + + The resolved part of a force \(P\) at angle \(\theta\) is has magnitude \(P \cos \theta\) + + \section{Newton's laws} + + \begin{enumerate} + \item Velocity is constant without a net external velocity + \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\) + \item Equal and opposite forces + \end{enumerate} + + \subsection{Weight} + A mass of \(m\) kg has force of \(mg\) acting on it + + \subsection{Momentum \(\rho\)} + \[ \rho = mv \tag{units kg m/s or Ns} \] + + \subsection{Reaction force \(R\)} + + \begin{itemize} + \item With no vertical velocity, \(R=mg\) + \item With upwards acceleration, \(R-mg=ma\) + \item With force \(F\) at angle \(\theta\), then \(R=mg-F\sin\theta\) + \end{itemize} + + \subsection{Friction} + + \[ F_R = \mu R \tag{friction coefficient} \] + + \section{Inclined planes} + + \[ \boldsymbol{F} = |\boldsymbol{F}| \cos \theta \boldsymbol{i} + |\boldsymbol{F}| \sin \theta \boldsymbol{j} \] + \def\iangle{30} % Angle of the inclined plane + + \def\down{-90} + \def\arcr{0.5cm} % Radius of the arc used to indicate angles + +\begin{tikzpicture}[ + >=latex', + scale=1, + force/.style={->,draw=blue,fill=blue}, + axis/.style={densely dashed,gray,font=\small}, + M/.style={rectangle,draw,fill=lightgray,minimum size=0.5cm,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}, + ] + \pgfmathsetmacro{\Fnorme}{2} + \pgfmathsetmacro{\Fangle}{30} + \begin{scope}[rotate=\iangle] + \node[M,transform shape] (M) {}; + \coordinate (xmin) at ($(M.south west)-({abs(1.1*\Fnorme*sin(-\Fangle))},0)$); + \coordinate (xmax) at ($(M.south east)+({abs(1.1*\Fnorme*sin(-\Fangle))},0)$); + \coordinate (ymax) at ($(M.north)+(0, {abs(1.1*\Fnorme*cos(-\Fangle))})$); + \coordinate (ymin) at ($(M.south)-(0, 1cm)$); + \coordinate (axiscentre) at ($(M.south)+(0.5cm, 0.5cm)$); + \draw[postaction={decorate, decoration={border, segment length=2pt, angle=-45},draw,red}] (xmin) -- (xmax); + \coordinate (N) at ($(M.center)+(0,{\Fnorme*cos(-\Fangle)})$); + \coordinate (fr) at ($(M.center)+({\Fnorme*sin(-\Fangle)}, 0)$); + % Draw axes and help lines + + {[axis,->] + \draw (ymin) -- (ymax) node[right] {\(\boldsymbol{j}\)}; + \draw (M) --(M-|xmax) node[right] {\(\boldsymbol{i}\)}; % mental note for me: change "right" to "above" + } + + % Forces + {[force,->] + % Assuming that Mg = 1. The normal force will therefore be cos(alpha) + \draw (M.center) -- (N) node [right] {\(R\)}; + \draw (M.center) -- (fr) node [left] {\(\mu R\)}; + } +% \draw [densely dotted, gray] (fr) |- (N) node [pos=.25, left] {\tiny$\lVert \vec F\rVert\cos\theta$} node [pos=.75, above] {\tiny$\lVert \vec F\rVert\sin\theta$}; + \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] {$mg$}; + \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} + + \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} \end{document} -- 2.47.1