From b53e3ff8b51532dc526dffc6ca7f3f29acbd5711 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Mon, 2 Sep 2019 22:06:12 +1000 Subject: [PATCH] [spec] tidy up tension diagrams in notes --- spec/dynamics.pdf | Bin 163504 -> 164884 bytes spec/dynamics.tex | 93 +++++++++++++++++++++++++++-------------- spec/spec-collated.pdf | Bin 449594 -> 450195 bytes spec/spec-collated.tex | 1 + 4 files changed, 62 insertions(+), 32 deletions(-) diff --git a/spec/dynamics.pdf b/spec/dynamics.pdf index dc59835e128918c05f5f7faeb3a93f4cea6c7a8f..fb01f97dc4745ffa501ceeb161f88704272cbbde 100644 GIT binary patch delta 40523 zcmV)7K*ztZ{Rxza3XmiMF))`f7y&4M#ai2r8#fYt*H`r0N?42L{SqM9b(~C)An~rp zgAD>Z4{eQRMqXWPNn^*rV!wS4o6T;rMM>ieun!)oDYD2aR-HOkZCc${wEA)M-PP#v zg{>TMn(ON7rc#<{Co0PXKC7$i>gD+Ni525{y`4;j5M#!}=Zm+KDYxVKY*8KecOTu~yNtv)m}SCzDkSck96>c!~!h_=eo zRB5IwCW(-mCu$p`2p6(H72|4u{Qbvuy;{t6K@Y!<5vNqMj#g7aiLoXuxt#Ad%lrLe zvxWtX8gFjG=gnq2n@0gIF$C6goI>~O@0a^}JrR5ilL{EVSsGTR_hcNg+C~E|cERGi 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zl3NLB8O~Z^>-FZ#qoV?6Yn20PUjs5a{7ztR078zqCExWC#DD)K$1PNAMU|shdx!68 zR4cTkXNG5-ekf3_h};GRKrJ|H1&6o$A3uGhisWFGikzC+oXWprbMxd9@+}c^8>#m> zS0($1lT?))4$crA-y;KE70}FU> zFJ3bvaes9Z&UCt>BUVi2AqZKmZpPh&lU4*7taW`n+7T!Iuc7)rmaGP7g^+I@DUwHR0*b(l# z!gXj0*9+t}uP?lNr0NKdOTtqNc~~I#>kIFkV}Clbb60jUE#M8=_prClX`?MW=-YAA zw}u(%_t1FfSkRGeZtxdJOTrE^?)B4go zm2pS-lDUOdK7J23YnO#>$#FThHNy)9>=|gczT|cZ*O6>*m%PAi?-5kng;q!MlASqw zGhJqP56szveVgb8;d)sx{f2cy zKVrRx7%1x1;B+Fsl%K?!^7By3;h^)|*X=%viP2fFEYSGv>VE+R5281h9D)LDQZ_|G zMm0h(G&C|sLq#?@G&eUjG(Jr|9n8Bj15gx&;kAw=h!8g!2 zYLKO(r~^2H4QRj?9Ktc|!3k``rqfL|Zcm{Pl9j)Rw|&@w7VJV3vXHTA=dm{kc^HBM z48sU#6k!xfFb3l=0cDtk3QSq`$N02rU0p{_TMwU6GuHD{)U4ILjjCC{mr-+ew^Lw3o|e>F$yImMNdWwe`,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}, @@ -78,8 +78,30 @@ A mass of \(m\) kg has force of \(mg\) acting on it 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} @@ -88,26 +110,26 @@ A mass of \(m\) kg has force of \(mg\) acting on it \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} @@ -115,15 +137,15 @@ A mass of \(m\) kg has force of \(mg\) acting on it \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 ($({3*cos(\iangle)},{3*sin(\iangle)})$); @@ -138,17 +160,18 @@ A mass of \(m\) kg has force of \(mg\) acting on it \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); @@ -157,9 +180,9 @@ A mass of \(m\) kg has force of \(mg\) acting on it } {[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] {\(m_1 g\)}; @@ -167,13 +190,15 @@ A mass of \(m\) kg has force of \(mg\) acting on it \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} @@ -182,16 +207,18 @@ A mass of \(m\) kg has force of \(mg\) acting on it \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} @@ -221,7 +248,9 @@ A mass of \(m\) kg has force of \(mg\) acting on it \end{scope} - \end{tikzpicture} + \end{tikzpicture}} +\end{tabular} + \subsection*{Equilibrium} \[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\] diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 7aa2159ebf421d89567fbb2924c33b3875b71986..def810118956fe37581f55bf2ea46a7adc6fdc83 100644 GIT binary patch delta 26772 zcmV(xK|hZbb9)?00X^ zzWOFKv5QW*=Iy)2y3R%21gY`ayuE6EIREQK(C6#b?xNLNpG%cK-+Z`eRXAT?ZdQLc zoBjPoE0uTp{Oj%2Zv82Ne0R6o+^%;QKfe9;t8a`OjbfziympuZkKpjO5aRqlF3{s& zzyJDAfBgL$dh6^DZ_j={lK>Nq?58Hsk1);U?b#1MislNxecOmm$IyIY$lJyQ*@cM5 zo94~gt1~f1R{2A0y|U^DBuQt2& z<^ASvn~)q+QZ%iyo%23*`}X}BuXuC5xQ;>afGmxnK&&W}kmLe81YK z`QEOsTY9HyWw0H{i3i_BtrPrqbN}&bc4X1TXpi6#JJXMPdAFaVg+Ukb_~ct_I|Ke= 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decorations.text, + decorations.pathreplacing, scopes } \usetikzlibrary{datavisualization.formats.functions} -- 2.47.1