\def\arcr{0.5cm} % Radius of the arc used to indicate angles
\tikzset{
- force/.style={->,draw=blue,fill=blue},
+ force/.style={->,draw=blue,fill=blue, thick},
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},
string/.style={draw=red, thick},
pulley/.style={thick}
}
-
- \begin{center}\begin{tikzpicture}
+\tikzset{
+ % style to apply some styles to each segment of a path
+ on each segment/.style={
+ decorate,
+ decoration={
+ show path construction,
+ moveto code={},
+ lineto code={
+ \path [#1]
+ (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast);
+ },
+ closepath code={
+ \path [#1]
+ (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast);
+ },
+ },
+ },
+ % style to add an arrow in the middle of a path
+ mid arrow/.style={postaction={decorate,decoration={
+ markings,
+ mark=at position .5 with {\arrow[#1]{stealth}}
+ }}},
+}
+ \begin{center}\begin{tikzpicture}[scale=1.8]
\pgfmathsetmacro{\Fnorme}{2}
\pgfmathsetmacro{\Fangle}{30}
\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 (ymax) at ($(M.center)+(0,{cos(\Fangle)})$);
+ \coordinate (ymin) at ($(M.center)-(0,{cos(\Fangle)})$);
\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 ($(M.center)+(0,{\Fnorme*cos(-\Fangle)})$);
+ \draw[postaction={decorate, decoration={border, segment length=4pt, angle=-45},draw,red}] (xmin) -- (xmax);
\coordinate (fr) at ($(M.center)+({\Fnorme*sin(-\Fangle)}, 0)$);
- {[axis,-]
- \draw (ymin) -- (M.center);
- }
{[axis,->]
- \draw ($(M)+(1,0)$) -- ($(M)+(2,0)$) node[above right] {\(\boldsymbol{i}\)};
- \draw ($(M)+(1,0)$) -- ($(M)+(1,1)$) node[above right] {\(\boldsymbol{j}\)};
+ \draw ($(M)+(1,0)$) -- ($(M)+(1.5,0)$) node[above right] {\(\boldsymbol{i}\)};
+ \draw ($(M)+(1,0)$) -- ($(M)+(1,0.5)$) node[above right] {\(\boldsymbol{j}\)};
}
{[force,->]
- \draw (M.center) -- (N) node [right] {\(R\)};
+ \draw (M.center) -- (ymax) node [right] {\(R\)};
\draw (M.center) -- (fr) node [left] {\(\mu R\)};
}
\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\(\theta\)};
+ \draw (xmin)+(0:\arcr) arc [start angle=0, end angle=\iangle, radius=\arcr] node [right, midway] {\footnotesize\(\theta\)};
+ \coordinate (xbottom) at ($(1, {4*\Fnorme*-cos(\Fangle)})$);
+ \draw [->] (xmin) -- ++($({1.35*\Fnorme*cos(\Fangle)}, 0)$);
+ \begin{scope}[darkgray, rotate=\iangle] \path [draw=darkgray, postaction={on each segment={mid arrow}}] (M.center) -- (ymin) node [pos=0.5, right] {\(mg \cos \theta\)} -- ++(-0.5,0) node[pos=0.5, below right] {\(mg \sin \theta\)};
+ \end{scope}
\end{tikzpicture}\end{center}
\subsection*{Connected particles}
\def\boxwidth{0.5}
\tikzset{
box/.style={rectangle,draw,fill=lightgray,minimum width=\boxwidth,thin},
- m/.style={rectangle,draw=black,fill=lightgray,minimum size=\boxwidth, font=\footnotesize, thin}
+ m/.style={rectangle,draw=black,fill=lightgray,minimum size=\boxwidth, thin}
}
\begin{center}
- \begin{tikzpicture}
+ \begin{tikzpicture}[scale=1.5]
- \matrix[column sep=1cm] {
- \begin{scope}
+ \matrix {
+ \begin{scope}[scale=1.5]
\coordinate (O) at (0,0);
\coordinate (A) at ($({3*cos(\iangle)},{3*sin(\iangle)})$);
\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] (C) rectangle ++(\boxwidth,\boxwidth) node (z) [rotate=\iangle, midway] {\(m_1\)};
\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) -- (Y) arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)})$) node[m] {\(m_2\)};
+ \draw [string] (E) -- (Y) arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)-\boxwidth})$) node (p) {};
+ \coordinate (Z) at ($(p.center)+({-0.5*\boxwidth},0)$);
+ \draw [m] (Z) rectangle ++(\boxwidth, \boxwidth) node [midway] {\(m_2\)};
\end{scope}
+\\
- &
-
- \begin{scope}[rotate=\iangle]
+ \begin{scope}[rotate=\iangle, scale=1.5]
- \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node (m1) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
+ \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node (n) [rotate=\iangle, midway] {\(m_1\)};
{[axis,-]
\draw (0,-1) -- (0,0);
}
{[force,->]
- \draw (M.center) -- ++(0,{cos(\iangle)}) node[above right] {\(R_1\)};
- \draw (M.west) -- ++(-0.5,0) node[left] {\(\mu R_1\)};
- \draw (M.east) -- ++(1,0) node[above] {\(T_1\)};
+ \draw (n.center) -- ++(0,{cos(\iangle)}) node[above right] {\(R_1\)};
+ \draw (n.west) -- ++(-0.5,0) node[left] {\(\mu R_1\)};
+ \draw (n.east) -- ++(1,0) node[above] {\(T_1\)};
}
\draw[force,->, rotate=-\iangle] (M.center) -- ++(0,-1) node[below] {\(m_1 g\)};
\end{scope}
&
+ \begin{scope}[scale=1.5]
- \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node [midway, font=\footnotesize] {\(m_2\)};
+ \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node [midway] {\(m_2\)};
{[force,->]
\draw (0,0.5*\boxwidth) -- ++(0,1) node[above] {\(T_2\)};
\draw (0,-0.5*\boxwidth) -- ++(0,-1) node[right] {\(m_2 g\)};
}
+ \end{scope}
\\
};
\end{tikzpicture}
\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: \\
+ 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 \\
+ \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{tabular}{rl}
+ \parbox[t][][t]{8em}{In this example, note \(T_1 \ne T_2\):} &
+ \parbox{12em}{
\begin{tikzpicture}
\begin{scope}
\end{scope}
- \end{tikzpicture}
+ \end{tikzpicture}}
+\end{tabular}
+
\subsection*{Equilibrium}
\[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\]