- \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}
-
- \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