b6a1af8cd718e464c8be211f60add75fc9525b33
1\documentclass[spec-collated.tex]{subfiles}
2\begin{document}
3
4\section{Dynamics}
5
6\subsection*{Resolution of forces}
7
8\textbf{Resultant force} is sum of force vectors
9
10\subsubsection*{In angle-magnitude form}
11
12\makebox[3cm]{Cosine rule:} \(c^2=a^2+b^2-2ab\cos\theta\)
13\makebox[3cm]{Sine rule:} \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
14
15\subsubsection*{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form}
16
17Vector 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.
18
19To find angle of an \(a\boldsymbol{i} + b\boldsymbol{j}\) vector, use \(\theta = \tan^{-1} \frac{b}{a}\)
20
21\subsubsection*{Resolving in a given direction}
22
23The resolved part of a force \(P\) at angle \(\theta\) is has magnitude \(P \cos \theta\)
24
25To convert force \(||\vec{OA}\) to angle-magnitude form, find component \(\perp\vec{OA}\) then \(|\boldsymbol{r}|=\sqrt{\left(||\vec{OA}\right)^2 + \left(\perp\vec{OA}\right)^2},\quad \theta = \tan^{-1}\dfrac{\perp\vec{OA}}{||\vec{OA}}\)
26
27\subsection*{Newton's laws}
28
29\begin{tcolorbox}
30 \begin{enumerate}[leftmargin=1mm]
31 \item Velocity is constant without a net external force
32 \item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\)
33 \item Equal and opposite forces
34 \end{enumerate}
35\end{tcolorbox}
36
37\subsubsection*{Weight}
38A mass of \(m\) kg has force of \(mg\) acting on it
39
40\subsubsection*{Momentum \(\rho\)}
41\[ \rho = mv \tag{units kg m/s or Ns} \]
42
43\subsubsection*{Reaction force \(R\)}
44
45\begin{itemize}
46 \item With no vertical velocity, \(R=mg\)
47 \item With vertical acceleration, \(|R|=m|a|-mg\)
48 \item With force \(F\) at angle \(\theta\), then \(R=mg-F\sin\theta\)
49\end{itemize}
50
51\subsubsection*{Friction}
52
53\[ F_R = \mu R \tag{friction coefficient} \]
54
55\subsection*{Inclined planes}
56
57\[ \boldsymbol{F} = |\boldsymbol{F}| \cos \theta \boldsymbol{i} + |\boldsymbol{F}| \sin \theta \boldsymbol{j} \]
58\begin{itemize}
59 \item Normal force \(R\) is at right angles to plane
60 \item Let direction up the plane be \(\boldsymbol{i}\) and perpendicular to plane \(\boldsymbol{j}\)
61\end{itemize}
62
63\def\iangle{30} % Angle of the inclined plane
64
65\def\down{-90}
66\def\arcr{0.5cm} % Radius of the arc used to indicate angles
67
68\tikzset{
69 force/.style={->,draw=blue,fill=blue},
70 axis/.style={densely dashed,gray,font=\small},
71 M/.style={rectangle,draw,fill=lightgray,minimum size=0.5cm,thin},
72 m/.style={rectangle,draw=black,fill=lightgray,minimum size=0.3cm,thin},
73 plane/.style={draw=black,fill=blue!10},
74 string/.style={draw=red, thick},
75 pulley/.style={thick}
76}
77
78 \begin{center}\begin{tikzpicture}
79
80 \pgfmathsetmacro{\Fnorme}{2}
81 \pgfmathsetmacro{\Fangle}{30}
82
83 \begin{scope}[rotate=\iangle]
84 \node[M,transform shape] (M) {};
85 \coordinate (xmin) at ($(M.south west)-({abs(1.1*\Fnorme*sin(-\Fangle))},0)$);
86 \coordinate (xmax) at ($(M.south east)+({abs(1.1*\Fnorme*sin(-\Fangle))},0)$);
87 \coordinate (ymax) at ($(M.north)+(0, {abs(1.1*\Fnorme*cos(-\Fangle))})$);
88 \coordinate (ymin) at ($(M.south)-(0, 1cm)$);
89 \coordinate (axiscentre) at ($(M.south)+(0.5cm, 0.5cm)$);
90 \draw[postaction={decorate, decoration={border, segment length=2pt, angle=-45},draw,red}] (xmin) -- (xmax);
91 \coordinate (N) at ($(M.center)+(0,{\Fnorme*cos(-\Fangle)})$);
92 \coordinate (fr) at ($(M.center)+({\Fnorme*sin(-\Fangle)}, 0)$);
93 {[axis,-]
94 \draw (ymin) -- (M.center);
95 }
96 {[axis,->]
97 \draw ($(M)+(1,0)$) -- ($(M)+(2,0)$) node[above right] {\(\boldsymbol{i}\)};
98 \draw ($(M)+(1,0)$) -- ($(M)+(1,1)$) node[above right] {\(\boldsymbol{j}\)};
99 }
100 {[force,->]
101 \draw (M.center) -- (N) node [right] {\(R\)};
102 \draw (M.center) -- (fr) node [left] {\(\mu R\)};
103 }
104 \end{scope}
105 \draw[force,->] (M.center) -- ++(0,-1) node[below] {$mg$};
106 \draw (M.center)+(-90:\arcr) arc [start angle=-90,end angle=\iangle-90,radius=\arcr] node [below, pos=.5] {\footnotesize\(\theta\)};
107 \end{tikzpicture}\end{center}
108
109\subsection*{Connected particles}
110
111\def\boxwidth{0.5}
112\tikzset{
113 box/.style={rectangle,draw,fill=lightgray,minimum width=\boxwidth,thin},
114 m/.style={rectangle,draw=black,fill=lightgray,minimum size=\boxwidth, font=\footnotesize, thin}
115}
116
117
118\begin{center}
119 \begin{tikzpicture}
120
121 \matrix[column sep=1cm] {
122 \begin{scope}
123
124 \coordinate (O) at (0,0);
125 \coordinate (A) at ($({3*cos(\iangle)},{3*sin(\iangle)})$);
126 \coordinate (B) at ($({3*cos(\iangle)},0)$);
127 \coordinate (C) at ($({(1.5-0.5*\boxwidth)*cos(\iangle)},{(1.5-0.5*\boxwidth)*sin(\iangle)})$); % centre of box
128 \coordinate (D) at ($(C)+(\iangle:\boxwidth)$);
129 \coordinate (E) at ($(D)+(90+\iangle:0.5*\boxwidth)$);
130 \coordinate (F) at ($(B)+(0,{1.5*sin(\iangle)})$);
131 \coordinate (X) at ($(A)+(\iangle:0.5*\boxwidth)$); % centre of pulley
132 \coordinate (Y) at ($(X)+(90+\iangle:0.5*\boxwidth)$); % chord of pulley
133
134 \draw[plane] (O) -- (A) -- (B) -- (O);
135 \draw (O)+(\arcr,0) arc [start angle=0,end angle=\iangle,radius=\arcr] node [right, pos=.75] {\footnotesize\(\theta\)};
136
137 \draw [rotate=\iangle, m] (C) rectangle ++(\boxwidth,\boxwidth) node (z) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
138 \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] {};
139 \draw [string] (E) -- (Y) arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)})$) node[m] {\(m_2\)};
140
141 \end{scope}
142
143 &
144
145 \begin{scope}[rotate=\iangle]
146
147 \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node (m1) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
148
149 {[axis,-]
150 \draw (0,-1) -- (0,0);
151 \draw[solid,shorten >=0.5pt] (\down-\iangle:\arcr) arc(\down-\iangle:\down:\arcr);
152 \node at (\down-0.5*\iangle:1.3*\arcr) {\(\theta\)};
153 }
154
155 {[force,->]
156 \draw (M.center) -- ++(0,{cos(\iangle)}) node[above right] {\(R_1\)};
157 \draw (M.west) -- ++(-0.5,0) node[left] {\(\mu R_1\)};
158 \draw (M.east) -- ++(1,0) node[above] {\(T_1\)};
159 }
160
161 \draw[force,->, rotate=-\iangle] (M.center) -- ++(0,-1) node[below] {\(m_1 g\)};
162
163 \end{scope}
164
165 &
166
167 \draw [m] ++(-0.5*\boxwidth,-0.5*\boxwidth) rectangle ++(\boxwidth,\boxwidth) node [midway, font=\footnotesize] {\(m_2\)};
168
169 {[force,->]
170 \draw (0,0.5*\boxwidth) -- ++(0,1) node[above] {\(T_2\)};
171 \draw (0,-0.5*\boxwidth) -- ++(0,-1) node[right] {\(m_2 g\)};
172 }
173 \\
174 };
175 \end{tikzpicture}
176 \end{center}
177
178\begin{itemize}
179 \item \textbf{Suspended pulley:} tension in both sections of rope are equal \\
180 \(|a| = g \frac{m_1 - m_2}{m_1 + m_2}\) where \(m_1\) accelerates down \\
181 With tension: \\
182 \[ \begin{cases}m_1 g - T = m_1 a\\ T - m_2 g = m_2 a\end{cases} \\ \implies m_1 g - m_2 g = m_1 a + m_2 a \]
183 \item \textbf{String pulling mass on inclined pane:} Resolve parallel to plane \\
184 \[ T-mg \sin \theta = ma \]
185 \item \textbf{Linear connection:} find acceleration of system first
186 \item \textbf{Pulley on right angle:} \(a = \frac{m_2g}{m_1+m_2}\) where \(m_2\) is suspended (frictionless on both surfaces)
187 \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane
188\end{itemize}
189
190\hspace{2em}\parbox{8em}{In this example, note \(T_1 \ne T_2\):}
191 \begin{tikzpicture}
192
193 \begin{scope}
194
195 \coordinate (O) at (0,0);
196 \coordinate (A) at ($({3*cos(\iangle)},{3*sin(\iangle)})$);
197 \coordinate (B) at ($({3*cos(\iangle)},0)$);
198 \coordinate (C) at ($({(1-0.25*\boxwidth)*cos(\iangle)},{(1-0.25*\boxwidth)*sin(\iangle)})$); % centre of box
199 \coordinate (D) at ($(C)+(\iangle:\boxwidth)$);
200 \coordinate (E) at ($(D)+(90+\iangle:0.5*\boxwidth)$);
201 \coordinate (F) at ($(B)+(0,{1.5*sin(\iangle)})$);
202 \coordinate (G) at ($(A)+(\iangle:-2*\boxwidth)$);
203 \coordinate (H) at ($(G)+(90+\iangle:0.5*\boxwidth)$);
204 \coordinate (I) at ($(H)+(\iangle:-0.5*\boxwidth)$);
205 \coordinate (J) at ($(H)+(\iangle:\boxwidth)$);
206 \coordinate (X) at ($(A)+(\iangle:0.5*\boxwidth)$); % centre of pulley
207 \coordinate (Y) at ($(X)+(90+\iangle:0.5*\boxwidth)$); % chord of pulley
208
209 \draw[plane] (O) -- (A) -- (B) -- (O);
210 \draw (O)+(\arcr,0) arc [start angle=0,end angle=\iangle,radius=\arcr] node [right, pos=.75] {\footnotesize\(\theta\)};
211
212 \draw [rotate=\iangle, m] (C) rectangle ++(\boxwidth,\boxwidth) node (z) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
213 \draw [rotate=\iangle, m] (G) rectangle ++(\boxwidth,\boxwidth) node (l) [rotate=\iangle, midway, font=\footnotesize] {\(m_2\)};
214 \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] {};
215 \draw [string] (E) -- (H) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_2\)};
216 \draw [string] (J) -- (Y) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_1\)} arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)})$) node [midway, above right, font=\footnotesize] {\(T_1\)} node[m] {\(m_3\)};
217
218 \end{scope}
219
220 \end{tikzpicture}
221\subsection*{Equilibrium}
222
223\[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\]
224\[ c^2 = a^2 + b^2 - 2ab \cos \theta \tag{cosine rule} \]
225
226Three methods:
227\begin{enumerate}
228 \item Lami's theorem (sine rule)
229 \item Triangle of forces (cosine rule)
230 \item Resolution of forces (\(\Sigma F = 0\) - simultaneous)
231\end{enumerate}
232
233 \begin{cas}
234 \textbf{To verify:} Geometry tab, then select points with normal cursor. Click right arrow at end of toolbar and input point, then lock known constants.
235 \end{cas}
236
237\subsection*{Variable forces (DEs)}
238
239\[ a = \dfrac{d^2x}{dt^2} = \dfrac{dv}{dt} = v\dfrac{dv}{dx} = \dfrac{d}{dx} \left( \frac{1}{2} v^2 \right) \]
240
241\end{document}