course.tex

1% !TEX TS-program = lualatex
2\documentclass{book}
3\usepackage{amsmath}
4\usepackage{graphicx}
5\input{shorthands}
6
7\newcommand{\vect}[1]{\overrightarrow{#1}}
8\newcommand{\unit}[2]{#1\,\mathrm{#2}}
9\newenvironment{remark}{\par\medskip\noindent\textbf{Remark.}\ }{\par\medskip}
10
11\begin{document}
12
13\chapter{Point mechanics}
14
15\section{Positions and velocities}
16
17A point is given by $\vect{r}\overrightarrow{r}(t)$, its velocity by $\vect{v}\overrightarrow{v} = \dot{\vect{r}\overrightarrow{r}}$.
18The speed of a free fall after $t = \unit{2}{s}2\,\mathrm{s}$ is about $\unit{20}{m.s^{-1}}20\,\mathrm{m.s^{-1}}$.
19
20\begin{itemize}
21\item the position, $\vect{r}\overrightarrow{r}$;
22\item the velocity, $\vect{v}\overrightarrow{v}$;
23\item the acceleration, $\vect{a}\overrightarrow{a}$.
24\end{itemize}
25
26\begin{figure}
27\centering
28\includegraphics[width=0.6\textwidth]{figures/trajectory}
29\caption{A trajectory in the plane.}
30\end{figure}
31
32\section{Energy}
33
34The kinetic energy of a point of mass $m$ is
35\[
36    E_c = \frac{1}{2} m v^2 .
37\]
38
39\begin{center}
40Energy is conserved when no non-conservative force does any work.
41\end{center}
42
43\begin{remark}\begin{remark}\par\medskip\noindent\textbf{Remark.}\ 
44The radius runs over $r\in\[0;\infty\[$, written with escaped brackets: inside
45$\dots$ those are commands, not the display math of \verb|\[ \]|.
46\end{remark}\par\medskip\end{remark}
47
48\begin{figure}
49\centering
50\includegraphics[width=0.5\textwidth]{figures/energy}
51\caption{Potential energy against the radius.}
52\end{figure}
53
54\section{Exercises}
55
56\begin{enumerate}
57\item Show that $\vect{a}\overrightarrow{a}$ is normal to $\vect{v}\overrightarrow{v}$ for a uniform circular motion.
58\item Give $E_c$ for $v = \unit{3}{m.s^{-1}}3\,\mathrm{m.s^{-1}}$ and $m = \unit{2}{kg}2\,\mathrm{kg}$.
59\end{enumerate}
60
61\begin{center}
62\includegraphics[width=0.3\textwidth]{figures/circle}
63\end{center}
64
65\end{document}