From 36cc59bb94575fae05f0918471cc3ca2b35d1005 Mon Sep 17 00:00:00 2001 From: JBR Date: Sun, 15 Mar 2026 23:26:36 +0100 Subject: [PATCH] Aggiunti libreria di base, loghi. LaTeX: git, meccanica --- .gitignore | 276 +++++++ git/favicon.png | Bin 0 -> 3129 bytes git/git.pdf | Bin 0 -> 205657 bytes git/git.tex | 368 +++++++++ jacbr.sty | 230 ++++++ loghi/PDF.svg | 3 + loghi/idrogeno.svg | 9 + loghi/informatica.svg | 11 + loghi/meccanica.svg | 18 + loghi/original.svg | 3 + meccanica/favicon.png | Bin 0 -> 2035 bytes meccanica/meccanicaBello.pdf | Bin 0 -> 370728 bytes meccanica/meccanicaBello.tex | 1500 ++++++++++++++++++++++++++++++++++ 13 files changed, 2418 insertions(+) create mode 100644 .gitignore create mode 100644 git/favicon.png create mode 100644 git/git.pdf create mode 100644 git/git.tex create mode 100644 jacbr.sty create mode 100644 loghi/PDF.svg create mode 100644 loghi/idrogeno.svg create mode 100644 loghi/informatica.svg create mode 100644 loghi/meccanica.svg create mode 100644 loghi/original.svg create mode 100644 meccanica/favicon.png create mode 100644 meccanica/meccanicaBello.pdf create mode 100644 meccanica/meccanicaBello.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..790cdeb --- /dev/null +++ b/.gitignore @@ -0,0 +1,276 @@ +## Core latex/pdflatex auxiliary files: +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb + +## Intermediate documents: +*.dvi +*.xdv +*-converted-to.* + +## Bibliography auxiliary files (bibtex/biblatex/biber): +*.bbl +*.bcf +*.blg +*-blx.aux +*-blx.bib +*.run.xml + +## Build tool auxiliary files: +*.fdb_latexmk +*.synctex +*.synctex(busy) +*.synctex.gz +*.synctex.gz(busy) +*.pdfsync + +## Build tool directories for auxiliary files +# latexrun +latex.out/ + +## Auxiliary and intermediate files from other packages: +# algorithms +*.alg +*.loa + +# achemso +acs-*.bib + +# amsthm +*.thm + +# beamer +*.nav +*.pre +*.snm +*.vrb + +# changes +*.soc + +# comment +*.cut + +# cprotect +*.cpt + +# elsarticle (documentclass of Elsevier journals) +*.spl + +# endnotes +*.ent + +# fixme +*.lox + +# feynmf/feynmp +*.mf +*.mp +*.t[1-9] +*.t[1-9][0-9] +*.tfm + +#(r)(e)ledmac/(r)(e)ledpar +*.end +*.?end +*.[1-9] +*.[1-9][0-9] +*.[1-9][0-9][0-9] +*.[1-9]R +*.[1-9][0-9]R +*.[1-9][0-9][0-9]R +*.eledsec[1-9] +*.eledsec[1-9]R +*.eledsec[1-9][0-9] +*.eledsec[1-9][0-9]R +*.eledsec[1-9][0-9][0-9] +*.eledsec[1-9][0-9][0-9]R + +# glossaries +*.acn +*.acr +*.glg +*.glo +*.gls +*.glsdefs +*.lzo +*.lzs + +# uncomment this for glossaries-extra (will ignore makeindex's style files!) +# *.ist + +# gnuplottex +*-gnuplottex-* + +# gregoriotex +*.gaux +*.glog +*.gtex + +# htlatex +*.4ct +*.4tc +*.idv +*.lg +*.trc +*.xref + +# hyperref +*.brf + +# knitr +*-concordance.tex +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files +# *.tikz +*-tikzDictionary + +# listings +*.lol + +# luatexja-ruby +*.ltjruby + +# makeidx +*.idx +*.ilg +*.ind + +# minitoc +*.maf +*.mlf +*.mlt +*.mtc[0-9]* +*.slf[0-9]* +*.slt[0-9]* +*.stc[0-9]* + +# minted +_minted* +*.pyg + +# morewrites +*.mw + +# newpax +*.newpax + +# nomencl +*.nlg +*.nlo +*.nls + +# pax +*.pax + +# pdfpcnotes +*.pdfpc + +# sagetex +*.sagetex.sage +*.sagetex.py +*.sagetex.scmd + +# scrwfile +*.wrt + +# sympy +*.sout +*.sympy +sympy-plots-for-*.tex/ + +# pdfcomment +*.upa +*.upb + +# pythontex +*.pytxcode +pythontex-files-*/ + +# tcolorbox +*.listing + +# thmtools +*.loe + +# TikZ & PGF +*.dpth +*.md5 +*.auxlock + +# todonotes +*.tdo + +# vhistory +*.hst +*.ver + +# easy-todo +*.lod + +# xcolor +*.xcp + +# xmpincl +*.xmpi + +# xindy +*.xdy + +# xypic precompiled matrices and outlines +*.xyc +*.xyd + +# endfloat +*.ttt +*.fff + +# Latexian +TSWLatexianTemp* + +## Editors: +# WinEdt +*.bak +*.sav + +# Texpad +.texpadtmp + +# LyX +*.lyx~ + +# Kile +*.backup + +# gummi +.*.swp + +# KBibTeX +*~[0-9]* + +# TeXnicCenter +*.tps + +# auto folder when using emacs and auctex +./auto/* +*.el + +# expex forward references with \gathertags +*-tags.tex + +# standalone packages +*.sta + +# Makeindex log files +*.lpz + +# xwatermark package +*.xwm diff --git 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aggiunti da git) e scrivere quello che si vuole mantenere. Potremmo voler tenere la versione che già avevamo e quindi eliminare tutto quello che riguarda il branch che stà venendo legato, oppure volere la versione in merging. Non è però necessario tenere una delle versioni presenti ed è possibile scrivere una soluzione custum. + +Risolto il conflitto, come consiglia il comando di merge andato male, si deve fare commit della soluzione. +Sarà quindi necessario fare ad esempio: +\begin{lstlisting} + git add -A + git commit -m "Merge Conflit risolto" +\end{lstlisting} + + +\subsection{Muoversi nel Tempo} +Ora che abbiamo le fondamenta di quello che possiamo fare con il branching usiamolo per spostarci tra commit. +Sappiamo che con \lstinline{git log --oneline} possiamo ottenere gli hash ridotti dei nostri commit, che sono i nomi dati da git a quelle versioni. Ora con: +\begin{lstlisting} + git checkout hash +\end{lstlisting} +possiamo muoverci a quel specifico commit come se avessimo riportato il tempo indietro nella nostra Working Directory. + +In questo stato noi possiamo creare nuovi commit, ma saremo in uno stato di \textit{detached HEAD}. +Essenzialmente quelle modifiche non fanno parte di nessun ramo, in gergo si dice che il commit è orfano. +Per risolvere il problema basta modernamente si usa: +\begin{lstlisting} + git switch -c nuovoNomeBranch +\end{lstlisting} +In un colpo solo viene creata una testa per il nostro commit e ci si sposta in essa. + + +\section{Mostrare le differenze} +Per capire cosa è stato effettivamente modificato tra un commit ed un altro, o tra 2 branch è possibile usare: +\begin{lstlisting} + git diff base modificato +\end{lstlisting} +base e modificato non sono altro che placeholder. La base è ciò che viene considerata il punto di partenza, mentre il modificato è quello che "importa" le modifiche. + + + +\section{Finalmente online} +Ora che abbiamo capito come funzione git in locale è il momento di spingerci al di fuori della nostra macchina. +Con tutto quello che già sappiamo sarà molto facile entrare nell'ottica di comunicazione con il server online.\\ +Qui non discuteremo di come connettere la repository online a quella locale perché dipende leggermente da fornitore a fornitore.\\ +Ci sono 3 concetti fondamentali: +\begin{itemize} + \item push + \item fetch + \item Pull Request (inutili nel nostro caso) +\end{itemize} + +\section{Push} +Push è il comando che ci permette di spingere codice sul server. Per poterlo fare è però necessario che la repository locale sia aggiornata e che abbia avuto delle modifiche. Inoltre si deve essere nel branch di cui si vuole fare push. Per essere nel branch di cui si vuole inviare copia al server basta \lstinline{git checkout nomeBranch}. \\ +Il comando completo: +\begin{lstlisting} + git push origin nomeBranch +\end{lstlisting} + + +\section{Fetch} +Fetch serve a richiamare le modifiche dal server. +Dopo un fetch la versione più aggiornata del server vive sulla propria macchina in locale, ma non è ancora stato inserito dentro la nostra Working Directory. +\begin{lstlisting} + git fetch +\end{lstlisting} +Fetch serve solo a fare download dell'albero sul server, ma non lo installa. +In generale fetch lanciato in questo modo: senza specificare un origine specifica scarica tutti i branch remoti. +Per vederli basta: +\begin{lstlisting} + git branch -r +\end{lstlisting} +Come prima il comando serve a vedere i branch e la flag \lstinline{r} mostra quelli remoti. +Per portare la versione più recente sul nostro sistema bisogna fare: +\begin{lstlisting} + git merge +\end{lstlisting} +In questo caso non è necessario specificare quale branch va unito, ma basta lanciare il questo modo il comando e git saprà automaticamente portare le modifiche remote nel proprio sistema. +Esiste anche un comando che unisce i 2 passaggi in uno unico: +\begin{lstlisting} + git pull +\end{lstlisting} +Sotto la coscocca quello che fa è chiamare in sequenza \lstinline{fetch} e \lstinline{merge}. + +Va fatto notare che come prima abbiamo visto i merge conflict, anche in questo caso possiamo ottenere la stessa situazione se 2 persone lavorano contemporaneamente alla stessa sezione dello stesso file, una lo carica sul server e l'altra poi tenta di effettuare un \lstinline{pull}. +I conflitti si risolvono esattamente come discusso prima. + + +\section{Quality of life} +In questa sezione spiegherò delle piccole cose che rendono l'utilizzare git estremamente comodo. + +\subsection{Stash} +Immaginiamo che tu stia lavorando ad un tuo commit, e qualcuno ti chieda di andare a visionare un altro branch. +Prima proponevo la soluzione di fare un commit, ma quella non ha tanto senso per quanto sia efficace. +Tendenzialmente ogni commit dovrebbe essere giustificato e quella di andare a visionare un altro branch non è una giustificazione molte forte. +Per superare i limiti di \lstinline{git checkout} noi vogliamo che le nostre modifiche non ancora dentro un commit vengano messe da parte fino al nostro ritorno. +\begin{lstlisting} + git stash +\end{lstlisting} +Mette tutte le modifiche correnti dentro lo stash, che è una zona di memoria separata da quelle che abbiamo visto prima, riportando la nostra Working Directory allo stato del precedente commit. +In questo modo è possibile muoversi liberamente tra i vari branch. +Quando però torniamo al nostro ramo noteremo che i file sono ancora alla versione dell'ultimo commit. +Per riportare il nostro lavoro indietro dallo stash basta fare: +\begin{lstlisting} + git stash pop +\end{lstlisting} + +Note: lo stash è molto più potente di questo singolo caso ed è possibile usarlo come "cestino sicuro" delle proprie modifiche. +Infatti è possibile mettere nello stash diverse modifiche e non solo una, ma la trattazione di git che stò facendo mi sembra già abbastanza esaustiva per quello che dobbiamo fare. + +Sfortunatamente non credo che esista un comando build-in per effetture stash automatico al cambio e poi riportarlo nella Working Directory. + + +\subsection{.gitignore} +Immaginiamo che stiate facendo un progetto con \LaTeX, la compilazione genera molti file inutili, come: \lstinline{ *.pdf, *.log, *.out, ...} . +Poiché non vogliamo caricare tutte queste cose che a gli altri daranno solo fastidio, con le conoscenze a nostra disposizione, dovremmo stare molto attenti a usare: +\begin{lstlisting} + git add nomeFile.ext +\end{lstlisting} +Questa cosa non è molto pratica e ci farebbe perdere un sacco di tempo soprattutto se lavoriamo con file diversi. + +Per ovviare a questo problema basta creare un file chiamato \lstinline{.gitignore}: +In questo file si può specificare riga per riga i file da ignorare. +Esistono generatori online per crearli in modo procedurale a seconda di cosa bisogna fare. + +\end{document} diff --git a/jacbr.sty b/jacbr.sty new file mode 100644 index 0000000..ab4984d --- /dev/null +++ b/jacbr.sty @@ -0,0 +1,230 @@ +%!! ffmpeg -i logo.svg -vf scale=64:64 favicon.png +\ProvidesPackage{jacbr} + +\RequirePackage[most]{tcolorbox} +\RequirePackage{subcaption} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage[a4paper,margin=2cm,nohead,foot=0.5cm]{geometry} +\usepackage{xcolor} +\usepackage{amsmath} +\usepackage{graphicx} +\usepackage{tabularx} +\usepackage{caption} +\usepackage{subcaption} +\usepackage{booktabs} +\usepackage{fancyhdr} +\usepackage{svg} +\usepackage{siunitx-v2} %unità di misura +\usepackage{listings} + +% Rende il titolo e l'autore delle variabili +\makeatletter +\let\thetitle\@title +\let\theauthor\@author +\makeatother +\newcommand{\theemail}{bassoriccijacopo@gmail.com} + +% Un po di colori + \definecolor{pastelgreen}{RGB}{225,245,225} + \definecolor{darkgreenline}{RGB}{0,120,0} + \definecolor{headgreen}{RGB}{0,90,30} + + \definecolor{azurelight}{RGB}{20,70,130} + + \definecolor{pastelblue}{RGB}{225,235,245} + \definecolor{darkblueline}{RGB}{0,80,160} + \definecolor{headblue}{RGB}{0,60,120} + + \definecolor{pastelblack}{RGB}{230,230,230} + \definecolor{darkblackline}{RGB}{20,20,20} + \definecolor{headblack}{RGB}{10,10,10} + + \definecolor{pastelviolet}{RGB}{240,230,250} + \definecolor{darkvioletline}{RGB}{120,60,160} + \definecolor{headviolet}{RGB}{90,40,140} + + \definecolor{pastelorange}{RGB}{250,238,225} + \definecolor{darkorangeline}{RGB}{200,120,40} + \definecolor{headorange}{RGB}{160,90,30} + + \definecolor{pastelred}{RGB}{250,225,225} + \definecolor{darkredline}{RGB}{170,40,40} + \definecolor{headred}{RGB}{120,0,0} + + \definecolor{pastelyellow}{RGB}{250,248,225} + \definecolor{darkyellowline}{RGB}{190,170,0} + \definecolor{headyellow}{RGB}{150,135,0} + +\usepackage[ + colorlinks=true, + linkcolor=azurelight, % link interni + urlcolor=azurelight, % link esterni + citecolor=azurelight, % citazioni +]{hyperref} + +% Utilità +\newcommand{\monthname}[1]{% + \ifcase#1\or + Gennaio\or Febbraio\or Marzo\or Aprile\or Maggio\or Giugno\or + Luglio\or Agosto\or Settembre\or Ottobre\or Novembre\or Dicembre + \fi +} + +\captionsetup{tableposition=top, figureposition=bottom, font=small} +%fare siusetup + +% Header +\setlength{\headheight}{14pt} +\setlength{\headsep}{10pt} +\addtolength{\topmargin}{-5pt} + +\fancyhf{} +\fancyhead[L]{\textbf{\thetitle} --- \monthname{\the\month} \the\year} +\fancyhead[C]{\smash{\raisebox{-0.2cm}{\includegraphics[height=1cm]{favicon.png}}}} +\fancyhead[R]{\textbf{\href{mailto:\theemail}{\theauthor}}} +\renewcommand{\headrulewidth}{1pt} +\fancyfoot[C]{\thepage} + +\pagestyle{fancy} + +% table of contents +\usepackage{tocloft} +\renewcommand{\contentsname}{Indice} +\renewcommand{\cftdotsep}{5} +\renewcommand{\cftsecfont}{\bfseries} +\renewcommand{\cftsecpagefont}{\bfseries} +\renewcommand{\cftsubsecfont}{\normalsize} +\renewcommand{\cftsubsecpagefont}{\normalsize} +\setlength{\cftbeforesecskip}{4pt} +\setlength{\cftbeforesubsecskip}{2pt} + +% Valore assoluto +\newcommand{\abs}[1]{\left\lvert #1 \right\rvert} +% Implicazione e non +\newcommand{\imp}{\Rightarrow} +\newcommand{\nimp}{\not\Rightarrow} +% inversa +\newcommand{\inv}{^{-1}} +% Momento di inerzia +\newcommand{\I}{\text{I}} +% x nelle equazioni +\newcommand{\x}{\times} +%cappellino (unità versore) +\newcommand{\un}[1]{\hat{U_{ #1}}} +%derivata +\newcommand{\dert}[1]{\frac{d #1}{dt}} +\newcommand{\dertt}[1]{\frac{d^2 #1}{dt^2}} +\newcommand{\der}[2]{\frac{d #1}{d #2}} +%per le parentesi +\newcommand{\8}{\left(} +\newcommand{\9}{\right)} + +% Massimo Comune Divisore +\newcommand{\mcd}{\text{MCD}} +%minimo comune multiplo +\newcommand{\mcm}{\text{mcm}} + +%teorema +\tcbset{ + teoboxstyle/.style={ + enhanced, + unbreakable, + colback=pastelgreen, + colframe=black!0, + boxrule=0pt, + left=8pt, + right=12pt, + top=8pt, + bottom=8pt, + sharp corners, + fonttitle=\sffamily\bfseries\color{headgreen}, + coltitle=headgreen, + before skip=8pt, + after skip=8pt, + overlay={ + \draw[line width=3pt,color=darkgreenline] + (frame.south west) -- (frame.north west); + }, + } +} + +\newtcbtheorem[number within=section]{teoJ}{Teorema} +{teoboxstyle, coltitle=headgreen, fonttitle=\sffamily\bfseries, separator sign={\ ---\ } }{alg} +\newtcbtheorem[number within=section]{greenJ}{} +{teoboxstyle, coltitle=headgreen, fonttitle=\sffamily\bfseries, separator sign={\ ---\ } }{alg} + +%definizioni +\tcbset{ + defboxstyle/.style={ + enhanced, + unbreakable, + colback=pastelblue, + colframe=black!0, + boxrule=0pt, + left=8pt, + right=12pt, + top=8pt, + bottom=8pt, + sharp corners, + fonttitle=\sffamily\bfseries\color{headblue}, + coltitle=headblue, + before skip=11pt, + after skip=11pt, + overlay={ + \draw[line 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0000000..2efe50c --- /dev/null +++ b/meccanica/meccanicaBello.tex @@ -0,0 +1,1500 @@ +\documentclass[a4paper,12pt]{article} +\title{Meccanica} +\author{Jacopo Basso Ricci} +\date{\today} +\usepackage{./../jacbr} +%\usepackage[italian]{babel} + + +\begin{document} +\maketitle +\tableofcontents + +\pagebreak +\section{Introduzione} +Storicamente si passa da uno studio qualitativo della fisica (si descrive a parole un fenomeno) a uno studio quantitativo che usa la matematica ed i numeri. +Sempre per ragioni storiche più che fisiche, si è divisa la materia in 4 macro categorie: +\begin{itemize} + \item {\large Meccanica}: Moti, Gravitazione + \item {\large Onde e oscillazioni}: Processi oscillatori + \item {\large Termodinamica}: Temperatura, Calore, Entropia + \item {\large Elettomagnetismo}: Campi elettrici e magnetici, Circuiti +\end{itemize} +\begin{blueJ*}{Limiti di applicabilità?} + Fenomeni a livello atomico e subatomico $d \sim 10^{-10} m$ \\ + Fenomeni a velocità prossime a quelle della luce $c \sim 3 * 10^{8} m/s $ +\end{blueJ*} + +\subsection{Meccanica} +Lo studio si dividerà in 2 parti: +\begin{itemize} + \item{\large Cinematica}: Descrizione del moto a prescindere dalle sue cause. +\item{\large Dinamica}: Studio del moto a partire della cause. +\end{itemize} + +\section{Un po' di definizioni e assiomi} +\begin{greenJ*}{Assioma di Tempo Assolto} + Il tempo assoluto per sua natura scorre in modo separato e del tutto indipendente da ogni cosa esterna. +\end{greenJ*} +\begin{greenJ*}{Assioma di Spazio Assolto} + Lospazio assoluto è per sua natura indipendente e immutabile. +\end{greenJ*} +\begin{defJ}{Punto Materiale}{def:pMat} + Corpo di dimensioni trascurabili rispetto allo spazio nel quale si nuove. +\end{defJ} + +\subsection{Matematica} +\begin{defJ}{Grandezze Scalari}{def:gSca} + Grandezze caratterizzate da un solo numero chiamato modulo o intensità. +\end{defJ} +\begin{defJ}{Grandezze Vettoriali}{def:gVet} + Grandezze caratterizzate da un vettore: modulo e orientamento(direzione e verso). +\end{defJ} +\begin{defJ}{Momento di un vettire}{def:momento} + Il momento di un vettore riespetto un polo O. (Lettere casualmente scelte) + \[ + \vec M = \vec r \x \vec F + \] + \[ + \abs{\vec M}= r F \sin \theta + \] +\end{defJ} +\begin{defJ}{Versori}{def:ver} + Vettori di modulo unitario $\abs{\hat{U_a}} = 1 $ +\end{defJ} +\begin{defJ}{Coordinate cartesiane}{def:coCar} + Coordinate espresse come somma di versiori perpendicolari. + \[ + \vert \un{x} \vert= \vert \hat{U_y} \vert=\vert \hat{U_z} \vert=1 + \] + \[ + \hat{U_x} \times \hat{U_y} = \hat{U_z}\text{, } + \hat{U_y} \times \hat{U_z} = \hat{U_x}\text{, } + \hat{U_z} \times \hat{U_x} = \hat{U_y} + \] +\end{defJ} +\begin{defJ}{Coordinate polari}{def:coPo} + Coordinate espresse come modulo di un vettore e gli angoli ad esso associati in senso antioriario rispetto gli assi. +\end{defJ} + +\subsubsection{Moto 2d} +Coordinate cartesiane: +\[ + \vec{r} = x \un{x} + y \un{y} +\] +Coordinate polari: +\[ + \vec{r}: r\text{, }\theta +\] +\begin{greyJ*}{Conversione coordinate cartesiane $\imp$ polari} + \[ + \vec{r} = x \un{x} + y \un{y} \imp r = \sqrt{x^2 + y^2}\text{, }\theta = \arctan \left( \frac x y \right) + \] +\end{greyJ*} +\begin{greyJ*}{Conversione coordinate polari $\imp$ cartesiane} + \[ + r\text{, }\theta \imp \vec{r} = r ( \cos \theta \un{x} + \sin \theta \un{y} ) + \] +\end{greyJ*} +\begin{greyJ*}{La dirivata di un versore} + \[ + \dert{\un{x}} = \dert{\omega} \un{\theta} + \] +\end{greyJ*} + + +\subsubsection{Moto 3d} +Coordinate cartesiane: +\[ + \vec{r} = x\un{x} + y\un{y} + z\un{z} +\] +Coordinate polari(in ordine modulo, angolo polare [asse Z], angolo azimutale [piano xOy]): +\[ + \vec{r}: r \text{, } \theta \text{, } \varphi +\] +\begin{greyJ*}{Conversione coordinate cartesiane $\imp$ polari} + \[ + \vec{r} = x\un{x} + y\un{y} + z\un{z} \imp r = \sqrt{x^2 + y^2 + z^2}\text{, }\theta = \arccos \frac z r \text{, } \varphi = \arctan \frac y x + \] +\end{greyJ*} +\begin{greyJ*}{Conversione coordinate polari $\imp$ cartesiane} + \[ + \vec{r}: r \text{, } \theta \text{, } \varphi \imp \vec{r} = r(\sin \theta \cos \varphi \un{x} + \sin \theta \sin \varphi \un{y} + \cos \theta \un{z}) + \] +\end{greyJ*} + + + + + +\section{Cinematica} +\begin{defJ}{Distanza}{def:dist} + Grandezza Scalare + \[ + \abs{\Delta \vec x} = \abs{\vec x_f - \vec x_i} + \] +\end{defJ} +\begin{defJ}{Spostamento}{def:spo} + Grandezza Vettoriale + \[ + \Delta \vec x = \vec x_f - \vec x_i + \] +\end{defJ} +\begin{defJ}{Velocità}{def:vel} + \[ + \vec v (t) = \dert {\vec x} + \] +\end{defJ} +\begin{defJ}{Accelerazione}{def:acc} + \[ + \vec a (t) = \dert{ \vec v} = \dertt{ \vec x} + \] +\end{defJ} + +\subsection{Moto rettilineo uniforme} +\begin{dimJ}{Moto rettilineo uniforme}{dim:mru} +Nel moto rettilineo uniforme il corpo si muove lungo una retta, quindi per conoscere la sua posizione basta identificare la distanza dall'origine. +\[ + a = 0 +\] +\[ + v = const. +\] +\[ + x(t) = x_0 + \int_{t_0}^{t_f} v dt = v \Delta t +\] +\end{dimJ} + +\subsection{Moto rettilineo uniformemente accelerato} +In questo caso il corpoè sempre in un moto rettilineo, vale ciò detto sopra per i moti rettilinei. +\begin{dimJ}{Moto rettilineo uniformemente accelerato}{dim:mrua} +\[ + a = const. +\] +\[ + v(t) = v_0 + \int_{t_0}^{t_f} a dt = v_0 + a\Delta t +\] +\[ + x(t) = x_0 + \int_{t_0}^{t_f} v(t) dt = x_0 + \int_{t_0}^{t_f}(v_0 + a\Delta t) dt = x_0 + v_0 \Delta t + \frac{1}{2} a \left(\Delta t\right)^2 +\] +\end{dimJ} +Inoltre è utile tenere a mente una comoda formula facilmente ricavabile che semplifica l'equazioni togliendo la dipendenza dal tempo. +\[ + \Delta x = \frac{v_f^2 - v_i^2}{2a} +\] + +\subsection{Moto rettilineo smorzato} +In questo caso si assume $a(t) = -kv$. La decelerazione avviene proporzionalmente alla velocità. +\begin{dimJ}{Moto rettilineo smorzato}{dim:mrs}dimJdimJPer semplificare i conti si assuma: +\[ + t_0 = 0 \text{, } x_0 = 0 \text{, } v_0 > 0 +\] +\[ + a = \dert{v} = -kv +\] +Questa è una equazione differenziale del primo ordine che si risolve "portando tutti i termini con la v da un lato". +\[ + \frac 1 v dv = - k dt +\] +Integriamo a destra e sinistra. +\[ + \int_{v_0}^{v} \frac 1 v dv = -k \int_0^t dt +\] +\[ + \ln(v) - \ln(v_0) = \ln \frac{v}{v_0} = -kt +\] +\[ + v(t) = v_0 e^{-kt} +\] +Per conoscere la posizione dobbiamo integrare nuovamente. Per mostrare l'integrale notevole moltiplichiamo e dividiamo per k. +\[ + x(t) = \frac{-v_o}{k}\int_0^t -ke^{-tk} dt +\] +\[ + x(t) = \frac{v_0}{k} ( 1 - e^{-kt} ) +\] +\end{dimJ} +Si noti che il punto si ferma a $x = \frac{v_0}{k}$ dopo un tempo teoricamente infinito. + +\subsection{Moto armonico} +Il moto armonico è un moto rettilineo che avviene con uno spostamento sinosuidale.\\ +In tale moto $A \text{, } \omega_p \text{, } \phi$ sono costanti. In ordine Ampiezza, Pulsazione, Fase Iniziale. +\begin{defJ}{Fase}{def:fase} +\[ + \omega = \omega_p t + \phi +\] +\end{defJ} +\begin{dimJ}{Moto armonico}{dim:ma} + \[ + x(t) = A \sin \omega + \] + \[ + v(t) = \dert{x} = A \omega_p \cos \omega + \] + \[ + a(t) = \dert{v} = - A \omega_p^2 \sin \omega = - \omega_p^2 x(t) + \] +\end{dimJ} +Si noti che come detto il moto è sinosuidale, inoltre è limitato nello spazio: +\[ + -1 \leq \sin \omega \leq 1 \imp -A \leq x(t) \leq A +\] +Il periodo di questo moto periodico risulta essere: +\[ + T = \frac{2\pi}{\omega_p} +\] +Per i conti è utile risordare che esiste una relazione tra la posizione e la velocità iniziali e $A \text{, } \phi$. Questo perché osservando la relazione dell'accelerazione vediamo un equazione differenziale del primo ordine che caratterezza il moto. +\[ + \dertt{x} + \omega_p^2 x = 0 +\] + +\subsection{Moto 2d} +Iniziamo ora a considerare un moto nel piano. +\begin{dimJ}{Moto 2d}{dim:mt2} + \[ + \vec r = x \un{x} + y \un{y} + \] + \[ + \vec v = \dert{\vec r } = \dert{x} \un{x} + \dert{y} \un{y} + \] + Ora consideriamo un nuovo versore $\un{r}$ che è direzionato da $O$ al punto materiale. + \[ + \vec v = \dert{r \un{r}} = \dert{r} \un{r} + r \dert{\un{r}} + \] + A qusto punto siamo costretti a derivare il versore $\un{r}$ + \[ + \vec v = \dert{r} \un{r} + r \dert{\theta} \un{t} + \] + \begin{defJ}{Velocità radiale}{def:vrad} + \[ + \vec v_r = \dert{r} \un{r} + \] + \end{defJ} + \begin{defJ}{Velocità trasversale}{def:vtra} + \[ + \vec v_t = r \dert{\theta} \un{t} + \] + \end{defJ} + Ora valutiamo l'accelerazione ricordandoci che la velocità è sempre tangente alla curva.\\ + Dichiaro che $T = $tg, $N = $norm. + \[ + \vec a = \dert{\vec v} = \dert{v \un{T}} = \dert{v} \un{t} + v \dert{\un{T}} = \dert{v}\un{T} + v \dert{\phi}\un{N} + \] + Poichè in ogni punto la traiettoria può essere localmente approssimata ad un arco di circonferenza. + \[ + ds = R d\phi \imp \dert{\phi} = \frac 1 R \dert{s} = \frac v R + \] + \[ + \vec a = \dert{v} \un{T} + \frac{v^2}{R} \un{N} + \] + \begin{defJ}{Accelerazione Tangenziale}{def:at} + \[ + \vec a_T = \dert{v} \un{T} + \] + \end{defJ} + \begin{defJ}{Accelerazione Normale o centripeta}{def:an} + \[ + \vec a_N = \frac{v^2}{R} \un{N} + \] + \end{defJ} +\end{dimJ} + +\subsubsection{Moto Circolare} +La traiettoria in questo caso è una circonferenza che per semplicità è posta con cetro in $ O $. +Coordinate cartesiane: +\[ + x(t) = R \cos \theta(t) +\] +\[ + y(t) = R \sin \theta(t) +\] +Coordinate polari: +\[ + r = R \text{, } \theta = \theta(t) +\] +Coordinate curvilinee: Definisco un "punto di partenza" e identificano la posizione come distranza da esso. +\[ + s(t) = R \theta(t) +\] +\begin{dimJ}{Moto Circolare}{dim:mc} + La velocità + \[ + \vec v = \dert{R} \un{r} + R \dert{\theta} \un{t} = R \dert{\theta} \un{t} + \] + \begin{defJ}{Velocità ancolare}{def:va} + \[ + \omega = \dert{\theta} + \] + \end{defJ} + Piccola osservazione: + \[ + \omega = \frac 1 R \dert{s} \imp R \omega = \dert{s} = v + \] + Ne risulta quindi: + \[ + \vec v = R \omega \un{t} + \] + Valutiamo l'accelerazione + \[ + \vec a = \dert{v} \un{T} + \frac{v^2}{R} \un{N} = R \dert{\omega} \un{T} + \omega^2 R \un{N} + \] + \begin{defJ}{Accelerazione ancolare}{def:aa} + \[ + \alpha = \dert{\omega} + \] + \end{defJ} + Anche in questo caso una piccola osservazione. + \[ + \alpha = \frac 1 R \dert{v} \imp a_T = \frac{a_T}{R} + \] + Ne risulta + \[ + \vec a = R \alpha \un{T} + R \omega^2 \un{N} + \] +\end{dimJ} + +\subsubsection{Moto Circolare Uniforme} +La traiettoria è una circonferenza e la velocità ha modulo costante. +\begin{dimJ}{Moto Circolare Uniforme}{dim:mcu} + Osserviamo prima il moto in un sistema di riferimento polare che ha per $r = R$. + \[ + \omega = const. + \] + \[ + \theta = \theta_0 + \int_{t_0}^{t_f} \omega = \theta_0 + \omega \Delta t + \] + Ora valutiamo la velocità in modulo sfruttando le coordinate curvilinee: + \[ + s = R \theta = R (\theta_0 + \omega \Delta t) + \] + \[ + v = \dert{s}= R \omega = const. + \] + Il modulo della velocità è costante, ma l'accelerazione non è $0$. + \[ + \vec a = \dert{v} \un{T} + \frac{v^2}{R} \un{N} = \frac{v^2}{R} \un{N} = \omega^2 R \un{N} + \] +\end{dimJ} +Osservazione: il moto è periodico, questo non implica che sia armonico. +\[ + T = \frac{2\pi}{\omega} = \frac{2\pi R}{v} +\] + +\subsubsection{Moto Circolare Uniformemente accelerato} +La traiettoria è una circonferenza e il modulo della velocità aumenta linearmente. +\begin{dimJ}{Moto Circolare Uniformemente accelerato}{dim:mua} + Valutiamo prima il moto utilizzando un sistema di riferimento polare che ha per $r = R$. + \[ + \alpha = const. + \] + \[ + \omega = \omega_0 + \int_{t_0}^{t_f} \alpha dt = \omega_0 + \alpha \Delta t + \] + \[ + \theta = \theta_0 + \int_{t_0}^{t_f} \omega dt = \theta_0 + \omega_0 \Delta t + \frac 1 2 \alpha \left( \Delta t \right)^2 + \] + Ora, come prima, valutiamo il moto con un sistema di riferimento in coordinate curvilinee. + \[ + s = R \theta + \] + \[ + v = \dert{s} = R \omega + \] + Valutiamo l'accelerazione + \[ + \vec a = \dert{v} \un{T} + \frac{v^2}{R} \un{N} = \alpha R \un{T} + \left( \omega_0 + \alpha \Delta t \right)^2 R \un{N} + \] +\end{dimJ} + +\subsubsection{Il vettore velocità angolare} +Per ora abbiamo definito solamente il modulo di tale oggetto ignorandone completamente l'orientamento. +\begin{defJ}{Velocità angolare}{def:va} + In modulo + \[ + \omega = \dert{\theta} + \] + La direzione è ortogonale al piano su cui stà avvenendo la rotazione.\\ + Il verso è tale che dalla "punta" il moto risulti sempre antiorario.\\ + In questo modo otteniamo: + \[ + \vec v = \vec \omega \x \vec R + \] +\end{defJ} + +\section{Dinamica} +Lo studio del motivo dei moti a partire dalle cause necessita delle leggi di Newton. +\begin{defJ}{I legge di Newton}{def:I} + Un corpo permane nel suo stato di quite o moto rettilineo uniforme a meno che non intervenga una forza esterna. +\end{defJ} +\begin{defJ}{II legge di Newton}{def:II} + Una forza produrrà una accelerazione di modulo inversamente proporzionale alla massa, direttamente proporzionale all'intesità della forza e con stessa direzione e verso. + \[ + \vec F_{TOT} = m \vec a + \] +\end{defJ} +\begin{defJ}{II legge di Newton}{def:III} + Dati 2 corpi A e B. Se il corpo A esrcita una forza sul corpo B allora il corpo B esercita una forza uguale e contraria sul corpo A. + \[ + \vec F_{AB} = -\vec F_{BA} + \] +\end{defJ} + +\subsection{Nota importante} +\[ + \vec a = 0 \nimp \text{Nessuna forza agisce sul corpo} +\] +\[ + \vec a = 0 \imp \vec F_{TOT} = 0 +\] + +\section{Quantità di moto} +\begin{defJ}{Quantità di moto}{def:qdm} + \[ + \vec \rho = m \vec v + \] +\end{defJ} +Possiamo quindi riscrivere la seconda legge di Newton. +\[ + \vec F = m \vec a = m \dert{\vec v } = \dert{ m \vec v } = \dert{\vec \rho} +\] +In questo modo ci accorgiamo che: +\[ + \vec F dt = d \vec \rho +\] +\begin{defJ}{Impulso}{def:imp} + \[ + \vec J = \int_{t_0}^{t_f} \vec F dt + \] +\end{defJ} +Da qui otteniamo l'enunciato del teorema dell'Impulso. +\begin{teoJ}{Impulso}{teo:imp} + \[ + \vec J = \Delta \rho = \int_{t_0}^{t_f} \vec F dt + \] +\end{teoJ} + +\section{Definiamo ora un po' di forze} +\begin{defJ}{Forza peso}{def:fp} + Sappiamo che su ogni corpo in prossimità della superficie terrestre agisce un accelerazione di gravità. + \[ + \vec a = -g \un{z} + \] + \[ + \vec P = m \vec a = - m g \un{z} + \] +\end{defJ} +\begin{defJ}{Reazione vincolare}{def:rv} + Chiamata anche Normale essa è la forza che permette ad un oggetto di non sprofondare in ogni piano. Questa non verrà discussa oltre qui, ma è semplicemente la forza esercita da un vincolo.\\ + Nel caso speficico di un corpo su un piano. + \[ + \vec F_T = \vec P + \vec N = 0 + \] +\end{defJ} +\begin{defJ}{Attrito Radente}{def:ar} + Se applichiamo una forza orizzontale al piano ci accorgiamo che il corpo non si muove fino al superamento di un valore critico. + \[ + \vert\vec{F}\vert > \mu_s N + \] + Ci accorgiamo che il corpo non si muove, quindi esiste $\vec{f}_s$ che \'e adattiva rispetto alla forza applicata. + \[ + \vec{F} + \vec{f}_s = 0 \imp F = f_s \text{ finch\'e } F \leq \mu_sN + \] + Si osserva che esiste una forza di attrito anche durante il moto che si oppone allo stesso. Sperimentalmente: + \[ + \vec{f}_d = - \mu_d \abs{\vec N} \un{x} + \] + \[ + \mu_d < \mu_s + \] +\end{defJ} +\begin{defJ}{Attrito Viscoso}{def:av} + Resistenza che un fluido (liquido o gas) oppone al moto. Per piccole velocità e per un corpo soggetto alla gravità vale: + \[ + \vec f_v = -b \vec v + \] + Per comprendere come questo ha effetti su un moto che subisce solo l'attrito viscoso e la forza peso che sia verticale facciamo uno studio del moto. + \[ + \vec F = \vec P + \vec f_v = m \vec a + \] + Se togliamo la notazione vettoriale, poichè il moto avviene su una retta, prestando attenzione ai segni + \[ + -mg +bv = -ma + \] + \[ + - \frac b m v + g = \dert{v} + \] + \[ + dt = \frac{dv}{g - \frac b m v} + \] + Integrando da tutte e 2 le parti considerando $t_0 = 0$ e $v_0 = 0$ otteniamo che: + \[ + v = m \frac g b \left( 1 - e^{-\frac b m t} \right) + \] +\end{defJ} +\begin{defJ}{Forza elastica}{def:fe} + Una forza elastica è di tipo centrale. + \[ + \vec F_e = -k \Delta \vec x = -k (\vec x - \vec x_0) + \] + Con $\vec x_0$ che è la posizione di riposo della molla. + L'equazione del moto di un oggetto soggetto alla sola forza elastica risulta: + \[ + \vec F_e = m \vec a + \] + Anche in questo caso il moto avviene su una retta quindi possiamo evitare la notazione vettoriale + \[ + -kx = ma + \] + \[ + a + \frac k m x = \dertt{x} + \frac k m x = 0 + \] + Questa è l'equazione differenziale di un moto armonico con + \[ + \omega_p = \sqrt{\frac k m} + \] +\end{defJ} + +\section{Studiamo 2 moti che sono casi patologici} +\begin{dimJ}{Pendolo semplice}{dim:pensem} + Una massa puntiforme vincolata ad un filo ideale soggetta alla forza peso costituisce il sistema studiato. Poichè il moto è circolare conviene usare le coordinate con assi legati al moto del corpo.\\ + \[ + F_c : T - mg\cos \theta = m a_N \imp a_N = \frac{T - mg\cos \theta}{m} + \] + \[ + F_t : -mg\sin\theta = m a_T \imp a_T = -g\sin\theta + \] + Cosideriamo per ora la seconda equazione: + \[ + a_t + g \sin\theta = 0 \imp L \dertt{\theta} + g \sin\theta = 0 + \] + Questa equazione differenziale ricorda quella del moto armonico soltato se applichiamo approssimazioni per piccoli angoli: $ \sin \theta = \theta$ + \[ + \dertt{\theta} + \frac g L \theta = 0 + \] + Così facendo ora conosciamo tutto di questo moto. +\end{dimJ} +\begin{dimJ}{Pendolo conico}{dim:pencon} + Un pendolo viene mosso in moto circolare uniforme con un angolo rispetto la verticale $ \theta $. + \[ + \vec F = \vec T + \vec P = m \vec a + \] + Scomponiamo questa equazione vettoriale nelle sue 3 componenti. + \[ + F_z = T\cos \theta -mg = 0 + \] + \[ + F_n = T \sin \theta = m a_n = m \frac{v^2}{R} + \] + \[ + F_t = 0 + \] + Le 2 prime equazioni ci permettono di osservare che: + \[ + T = \frac{mg}{\cos\theta} = m \frac{v^2}{R \sin\theta} + \] + \[ + v^2 = gR \tan\theta + \] + Otteniamo quindi + \[ + v = \pm \sqrt{gR\tan\theta} = \pm \sqrt{gL\sin\theta\tan\theta} + \] + Sfruttando le approssimazioni per angoli piccoli ricaviamo + \[ + v = \theta \sqrt{gL} + \] + \[ + \omega = \theta \frac{\sqrt{gL}}{L\sin\theta} + \] +\end{dimJ} + +\section{Energia e Lavoro} +\begin{defJ}{Lavoro}{def:W} + \[ + W = \vec F \cdot \Delta \vec s + \] + Si parla di: + \begin{itemize} + \item Lavoro motore: $W > 0 \implies \theta < \frac{\pi}{2}$ + \item Lavoro nullo: $W = 0 \implies \theta = \frac{\pi}{2}$ o $\vec{F} = 0$ o $\Delta\vec{s} = 0$ + \item Lavoro resistente: $W < 0 \implies \theta > \frac{\pi}{2}$ + \end{itemize} + Se generalizziamo a curve qualunque e forze qualunque possiamo dire: + \[ + W \approx \sum_{i=1}^n \vec{F}_i \cdot \Delta \vec{s}_i + \] + Al limite questo diventa: + \[ + W = \int_{A}^{B} \vec{F} \cdot d\vec{s} + \] + Notiamo che + \[ + \vec{F} = \sum_{i = 1}^n \vec{F}_i + \] + \[ + W = \int_{A}^{B} \vec{F} \cdot d\vec{s} = \int_{A}^{B} \sum_{i = 1}^n \vec{F}_i \cdot d\vec{s} = \sum_{i = 1}^n \int_{A}^{B} \vec{F}_i \cdot d\vec{s} = \sum_{i = 1}^n W_i + \] +\end{defJ} +\begin{defJ}{Potenza}{def:potenza} + \[ + P = \dert{W} + \] +\end{defJ} +Introduciamo ora un teorema molto importante. +\begin{teoJ}{Energia cinetica}{teo:K} + \[ + W = \Delta K + \] +\end{teoJ} +\begin{dimJ}{Energia cinetica}{dim:K} + Introduciamo la seconda legge della dinamica nella definizione di lavoro + \[ + dW = \vec F \cdot d\vec s = F_t ds = m a_t ds = m \dert{v} ds = m dv \dert{s} = m v dv + \] + Possiamo quindi integrare + \[ + W_{AB} = \int_{v_0}^{v_f} m v dv = \frac 1 2 m \Delta v^2 = \Delta K + \] +\end{dimJ} +\begin{dimJ}{Lavoro della forza peso}{dim:fpeso} + \[ + W = \int_A^B \vec F \cdot d\vec s = \int_A^B m \vec g \cdot d \vec s = m \vec g \cdot \int_A^B d \vec s = -mg \un{z} \cdot \vec s_{AB} + \] + \[ + W = -mg(z_B -z_A) = -mg \Delta z + \] + \begin{defJ*}{Energia potenziale gravitazionale} + \[ + U_p = mgz + \] + \end{defJ*} + \[ + W = - \Delta U_p + \] +\end{dimJ} +\begin{dimJ}{Lavoro della forza elastica}{dim:felastica} + \[ + W = \int_A^B \vec F \cdot d \vec s = \int_A^B -k\vec s \cdot d \vec s = \int_A^B -k s ds = - \frac 1 2 k \Delta s^2 + \] + \begin{defJ*}{Energia potenziale elastica} + \[ + U_e = k s^2 + \] + \end{defJ*} + \[ + W = - \Delta U_e + \] +\end{dimJ} +\begin{dimJ}{Lavoro della forza d'attrito}{dim:fattrito} + \[ + W = \int_A^B \vec F \cdot d \vec s = \int_A^B - \mu N \un{s} \cdot d \vec s = - \mu N \int_A^B ds = - \mu N \Delta s + \] + In questo caso non esiste un modo per definire un energia potenziale. +\end{dimJ} + +\subsection{Forze conserivative} +Osserviamo che tutte le forze viste fin ora ammettono l'esistenza di una Energia potenziale che è determinata univocamente dalla posizione, ma la forza di attrito non ammette la stessa funzione. Questo perché il lavoro compiuto dal punto A al punto B non dipendono solamente da essi, ma anche dal percorso compiuto. +\begin{defJ}{Forza conservativa}{def:fconservativa} + Una forza è conservativa se il lavoro di tale forza NON dipende dal percorso, ma solo dalla posizione iniziale e finale. + \[ + \oint \vec{F} \cdot d\vec{s} = 0 + \] + Il lavoro in un qualunque circuito è 0 +\end{defJ} +Lavorare con forze conservative ci permette di: +\begin{align*} + W = \Delta K = -\Delta U \\ + K_b - K_a = U_a - U_b \\ + K_a + U_a = K_b + U_b \\ + E_{Ma} = E_{Mb} +\end{align*} +Abbiamo quindi identificato che se nel nostro sistema se intervengono solo forze conservative l'energia si conserva. \\ +Consideriamo il lavoro di una forza conservativa +\[ + dW = \vec F \cdot d \vec s +\] +Nel caso monodimensionale +\[ + dW = Fdx = -dU \imp F = \frac{-dU}{dx} +\] +Questo può essere generalizzato nelle 3 dimensioni +\[ + \vec F = - \nabla U +\] +Cosa succede nel caso generale in cui sono presenti anche forze non conservative? +\begin{align*} + W &= W_{cons} + W_{non-cons}\\ + W_{non-cons} &= W - W_{cons}\\ + W_{non-cons} &= K_b - K_a + U_b - U_a\\ + W_{non-cons} &= E_{Mb} - E_{Mb} = \Delta E_{M} +\end{align*} + +\section{Momenti} +\begin{defJ}{Momento angolare}{def:mangolare} + \[ + \vec L = \vec r \x \vec \rho + \] +\end{defJ} +\begin{defJ}{Momento di una forza}{def:mforza} + \[ + \vec M = \vec r \x \vec F + \] +\end{defJ} +Studiamo ora la variazione del momento angolare nel tempo. Questo studio lo introduciamo perchè studiando le 2 grandezze ci rendiamo conto che: +\[ + [ \vec L ] = Kg \frac{m^2}{s} +\] +\[ + [ \vec M ] = Kg \frac{m^2}{s^2} +\] +Queste 2 grandezze differiscono solo di un $\frac 1 s$ in unità fondamentali quindi ci deve essere un legame tra $ \dert{ \vec L} $ e $ \vec M $ +\begin{teoJ}{Momento angolare}{teo:mang} + \[ + \dert{\vec L} = \vec M + \] +\end{teoJ} +\begin{dimJ}{Teorema del momento angolare}{dim:angolare} + \[ + \dert{\vec L} = \8\dert{\vec r} \x m \vec v\9 + \8\vec r \x m \dert{\vec v}\9 = \8\vec v \x m \vec v\9 + \8\vec r \x m \vec a\9 = 0 + \vec r \x \vec F = \vec M + \] +\end{dimJ} +\begin{teoJ}{Momento dell'impulso}{teo:momimp} + \[ + \Delta \vec L \simeq \vec r \x \vec J + \] +\end{teoJ} +\begin{dimJ}{Teorema del momento dell'impulso}{dim:momimp} + \[ + d \vec L = \vec M dt + \] + Integriamo a destra e sinistra + \[ + \int_{t_0}^{t_f} \vec M dt = \Delta \vec L + \] + Scomponiamo il momento + \[ + \int_{t_0}^{t_f} \vec M dt = \int_{t_0}^{t_f} \8\vec r \x \vec F \9 dt \simeq \vec r \x \int_{t_0}^{t_f} \vec F dt = \vec r \x \vec J + \] +\end{dimJ} + +\section{Moti relativi} +Questo capitolo si concentra sulla domanda: "Le leggi della Fisica dipendono dal sistema di riferimento scelto?".\\ +Da questo momento in poi utilizzeremo 2 sistemi di riferimento il primo "fisso" che ha come centro $ O $ e assi $x$, $y$, $z$; il secondo in moto rispetto a $ O' $, $x'$, $y'$, $z'$. +\begin{teoJ}{Posizione relativa}{dim:pr} + \[ + \vec r = \vec r_{O'} + \vec r' + \] + \begin{itemize} + \item $\vec r $ posizione del punto P nel sistema di riferimento O + \item $\vec r_{O'}$ posizione del punto O' nel sistema di rifermento O + \item $\vec r '$ posizione del punto P nel nel sistema di riferimento O' + \end{itemize} +\end{teoJ} +Ora valutiamo la velocità rispetto che sappiamo essere $\dert{\vec r}$ +\begin{teoJ}{Velocità relativa}{teo:vr} + \[ + \vec v = \dert{\vec r} = \dert{\vec r_{O'}} + \dert{\vec r'} + \] + Ora farò la scomposizione solo per l'asse x, ma lo stesso procedimento deve essere firatto per ogni asse. + \[ + = \dert{x_{O'}} \un{x} + \8 \dert{x'}\un{x'} + x'\dert{\un{x'}} \9 \dots = \vec v_{O'} +\vec v' + \8 x' \dert{\un{x'}} \dots \9 = \vec v_{O'} +\vec v' \8 \vec \omega \x \un{x'} \dots \9 + \] + \[ + \vec v = \vec v_{O'} + \vec v' + \vec \omega \x \vec r' + \] +\end{teoJ} +Anche in questo caso è necessaria qualche spiegazione +\begin{itemize} + \item $\vec v $ velocità di P da O + \item $\vec v_{O'}$ velocità di O' da O + \item $\vec v '$ velocità di P da O' + \item $\vec \omega $ vettore rotazione di O' da O + \item $\vec r' $ posizione di P da O' +\end{itemize} +\begin{teoJ}{Accelerazione relativa}{teo:are} + La dimostrazione in questo caso è molto caotica. + \[ + \vec a = \dert{\vec v} = \dert{\vec v_{O'}} + \dert{\vec v'} +\dert{\vec \omega} \x \vec r' + \vec \omega \x \dert{\vec r'} + \] + Scomponiamo le operazioni e consideriamo come prima solo l'asse x, basta duplicare per gli altri assi. + \[ + \dert{\vec v_{O'} } = \vec a_{O'} + \] + \[ + \dert{\vec v'} = \dert{v_{x'} \un{x'} + \dots } = \dert{v_{x'}} \un{x'} + v_{x'} \dert{\un{x'}} +\dots = \vec a' + v_{x'} \8 \vec \omega \x \un{x'}\9 + \dots = \vec a' + \vec \omega \x \vec v' + \] + \[ + \dert{\vec \omega} \x \vec r' = \alpha \x \vec r' + \] + \[ + \vec \omega \x \dert{\vec r'} = \vec \omega \x \8\vec v' + \vec \omega \x \vec r'\9 = \vec \omega \x \vec v' + \vec \omega \x\8\vec \omega \x \vec r' \9 + \] + Ricomponiamo tutti i pezzi della dimostrazione + \[ + \vec a = \vec a' + \vec a_{O'} + \vec \omega \x \8\vec \omega \x \vec r'\9 + \vec \alpha \x \vec r' + 2 \vec\omega \x \vec v' + \] +\end{teoJ} +Piccola spiegazione +\begin{itemize} + \item $\vec a'$ accelerazione di P da O' + \item $\vec a_{O'}$ accelerazione di O' da O + \item $\vec a_{O'} + \vec \omega \x \8 \vec \omega \x \vec r'\9 + \vec \alpha \x \vec r' $ Accelerazione di trascinamento + \item $ 2 \vec \omega \x \vec v'$ Accelerazione di Coriolis +\end{itemize} +Definiti tutti questi nomi ricordo che nello studio di un moto in un sistema di riferimento non inerziale bisogna considerare $ \vec F_t $ e $ \vec F_c$ +\begin{defJ}{sistema di rifermento inerziale}{def:sri} + Un sistema di rifermento è inerziale se vale la 2' legge della dinamica: + \[ + \vec F = m \vec a + \] +\end{defJ} +Definito un primo sistema di riferimento inerziale se $ \vec a_{O'} = 0$ e $ \vec \omega = 0$ allora anche il secondo sistema di rifermento è inerziale poichè vale $ \vec a = \vec a'$ quindi la prima legge della dinamica è vera. + +\section{Dinamica dei sistemi di punti materiali} +\begin{defJ}{Forze interne}{def:fi} + Sono le forze che interagiscono tra punti materiali del sistema +\end{defJ} +\begin{defJ}{Forze esterne}{def:fe} + Solo le forze che interagiscono tra punti materiali del sistema e oggetti che ne sono al di fuori. +\end{defJ} +Dalle definizioni è chiaro che una forza possa solo essere interna o esterna, ne consegue che: +\[ + \vec F_i = \vec F_i^E + \vec F_i^E +\] +\begin{teoJ}{Forze interne}{teo:fi} + Per la 3' legge della dinamica sappiamo che + \[ + \vec F_{AB} = - \vec F_{BA} + \] + Ne consegue che: + \[ + \vec F^I = \sum \vec F_{AB} + \vec F_{BA} = 0 + \] + La somma di tutte le forze interne è 0 +\end{teoJ} +\begin{defJ}{Centro di massa}{def:CM} + \[ + \vec R = \frac{\sum m_i \vec r_i}{ \sum m_i} = \frac 1 M \sum m_i \vec r_i + \] +\end{defJ} +In generale il centro di massa si sposta +\begin{dimJ}{velocità del centro di massa}{dim:vcm} + \[ + \vec V = \dert{\vec R} = \dert{\frac 1 M \sum m_i \vec r_i} = \frac 1 M \sum m_i \dert{\vec r_i} = \frac 1 M \sum m_i v_i = \frac{\vec \rho}{M} + \] +\end{dimJ} +\begin{dimJ}{accelerazione del centro di massa}{dim:acm} + \[ + \vec A = \dert{\vec V} = \frac 1 M \sum m_i \dert{\vec v_i} = \frac 1 M \sum m_i \vec a_i = \frac 1 M \8\sum F^E_i + \sum F_i^I \9 = \frac{\vec F^E}{M} + \] +\end{dimJ} +\begin{teoJ}{I equazione cardinale della dinamica dei sistemi}{teo:Ieqc} + Abbiamo visto come + \[ + M \vec A = \vec F^E + \] + Questo può essere riscitto come + \[ + M \dert{\vec V} = \dert{M \vec V} = \dert{\vec \rho} = \vec F^E + \] +\end{teoJ} +Nel caso di $\dert{\vec \rho} = 0$ si parla di sistema isolato. In altre parole la somma delle forze esterne sul sistema è nulla. Un sitema isolato si muove in moto rettilineo uniforme (se la sua massa non varia) +\begin{teoJ}{II equazione cardinale della dinamica dei sistemi}{teo:mangol} + \[ + \vec L = \sum \vec L_i = \sum \8 \vec r_i \x m_i \vec v_i\9 + \] + Ora deriviamo il momento angolare rispetto al tempo per vedere come cambia il teorema definito. + \[ + \dert{\vec L} = \sum \dert{\vec r_i \x m_i \vec v_i} = \sum \8\dert{r_i} \x \8m_i \vec v_i\9 + \vec r_i \x \dert{m_i \vec v_i}\9 + \] + $ \vec r_i$ è il vettore posizione rispetto al ceontro di massa + \[ + \dert{\vec L} = \sum \8\vec v_i -\vec v_O\9 \x m_i \vec v_i + \sum \8 \vec r_i \x \8\vec F_i^E + \vec F_i^I\9\9 + \] + \[ + \dert{\vec L} = \sum \vec v_i \x m_i \vec v_i - \sum \vec v_O \x m_i \vec v_i + \sum \vec r_i \x \vec F_i^E + \sum \vec r_i \x \vec F_i^I = 0 - \8\vec v_O \x \vec \rho\9 + \vec M^E + \vec M^I + \] + Ora dimostriamo che $\vec M^I$ è sempre uguale a 0. + \[ + \vec M^I = \vec r_A \x \vec F_{AB} + \vec r_B \x \vec F_{BA} = \vec r_A \x \vec F_{AB} - \vec r_B \x \vec F_{AB} = \8 \vec r_A - \vec r_B \9 \x \vec F_{AB} = \vec r_{BA} \x \vec F_{AB} + \] + Questi 2 vettori sono paralleli quindi il prodotto vettorial è 0. + \[ + \vec M^I = 0 + \] + Quindi risulta che: + \[ + \dert{\vec L} = \vec \rho \x \vec v_O + \vec M^E = M \vec V_{CM} \x \vec V_O + \vec M^E + \] +\end{teoJ} +\begin{itemize} + \item $\vec v_{CM}$ la velocità del centro di massa + \item $\vec v_O$ la velocità del polo O + \item $\vec M^E$ momenti delle forze esterne +\end{itemize} +Ora Scomponiamo il momento angolare e l'energia cinetica in un contributo "interno" ed uno del "CM". +\begin{teoJ}{I teorema di Köning}{teo:IKöning} + Polo = origine sistema inerziale + \[ + \vec L = \sum \vec r_i \x m_i \vec v_i + \] + \[ + \vec r_i = \vec r_i' + \vec r_{CM} + \] + \[ + \vec v_i = \vec v_i' + \vec v_{CM} + \] + \[ + \vec L = \sum \8\vec r_i' + \vec r_{CM}\9 \x m_i \8\vec v_i' + \vec v_{CM}\9 + \] + \[ + \vec L = \sum \vec r_i' \x m_i \vec v_i' + \sum \vec r_i' \x m_i \vec v_{CM} + \sum \vec r_{CM} \x m_i \vec v_i' + \sum \vec r_{CM} \x m_i \vec v_{CM} = \vec L' + 0 + \vec r_{CM} \x M \vec v_{CM} + \] + \[ + \sum m_i \vec r_i' = 0 \text{ Nel sistema di riferimento del CM il CM è nell'O} + \] + \[ + \sum m_i \vec v_i' = \vec \rho' = 0 \text{ Nel sistema di riferimento del CM la somma delle velocità del CM è 0} + \] + Riassumendolo e semplificando + \[ + \vec L = \vec L' + \vec L_{CM} + \] +\end{teoJ} +\begin{itemize} + \item $\vec L'$ momento angolare nel sistema di riferimento del CM + \item $\vec L_{CM}$ momento angolare del CM nel sistema di riferimento inerziale +\end{itemize} +\begin{teoJ}{II teorema di Köning}{teo:IIKöning} + Nel sistema di riferimento inerziale. + \[ + K = \sum \frac 1 2 m_i v_i^2 = \sum \frac 1 2 m_i \8\vec v_i' + \vec v_{CM}\9^2 = \sum \frac 1 2 m_i v_i'^2 + \sum \frac 1 2 m_i v_{CM}^2 + \sum m_i \vec v_i' \cdot \vec v_{CM} + \] + \[ + K = \frac 1 2 \sum m_i v_i'^2 + \frac 1 2 \sum m_i v_{CM}^2 = K' + K_{CM} + \] +\end{teoJ} +\begin{itemize} + \item $K'$ energia cinetica "interna" rispetto al CM. 0 nel caso di corpo rigido + \item $K_{CM}$ energia cinetica dovuta alla velocità del CM rispetto al sistema inerziale +\end{itemize} + +\section{Gli urti} +Durante l'urto, la forza tra le due particelle è trascurabile rispetto alle forze esterne. Il tempo $ \tau $ è trascurabile rispetto al "tempo di osservazione".\\ +Facciamo degli esempi: una pallina da tennis che urta la racchetta ha un $ \tau \simeq 0.01 s $ e un tempo di osservazione di circa 1 secondo. 2 galassie urtano in milioni di anni, ma han un tempo di osservazione di centinaia di milioni. + +Poichè durante l'urto le forze esterne son trascurabili rispetto alle forze interne possiamo affermare che: +\[ + \vec F^E = 0 \imp \vec \rho = const. \imp M\vec v_{CM} = const. +\] +In altre parole, se la massa non varia, la velocità del CM resta costante. +\begin{dimJ}{Conservazione di $ \rho $ per le singole particelle}{dim:cqmsp} + Per le singole particelle vale $ \Delta \vec \rho = \vec J$, inoltre $ \vec F_{AB} = - \vec F _{BA} $. + \[ + \Delta \vec \rho_A = m_A \vec v_{Af}-m_A \vec v_{Ai} = \vec J_{BA} \simeq \tau \vec F_{BA} = - \tau \vec F_{AB} + \] + \[ + \Delta \vec \rho_B = m_B \vec v_{Bf} -m_B \vec v_{Bi} = \vec J_{AB} \simeq \tau \vec F_{AB} + \] + Ne consegue che + \[ + \Delta \rho_A = - \Delta \rho_B + \] +\end{dimJ} + +\subsection{Conservazione dell'eneriga in un urto} +Abbiamo osservato come la quantità di moto si conservi, vale lo stesso per l'energia?\\ +Sappiamo che +\[ + \Delta E_M = \Delta K + \Delta U +\] +Poiche durante un urto la posizione non varia sappiamo che non c'è variazione nell'energia potenziale $ \Delta U = 0$.\\ +Per il II teorema di Köning +\[ + K = K' + K_{CM} +\] +L'ultimo termine è costante: +\[ + K_{CM} = \frac 1 2 \8m_A + m_B\9 v^2_{CM} +\] +Invece +\[ + K' = \frac 1 2 m_A v'^2_A + \frac 1 2 m_B v'^2_B +\] +Chiameremo quindi: +\begin{itemize} + \item Urto elastico: un urto in cui interagiscono solo forze conservative $K' = const$. + \item Urto anaelastico: urto in cui non si conserva l'energia cinetica $K'_f < K'_i $. +\end{itemize} + +\subsection{Urti nel sitema di riferimento} +Ricordo che +\[ + \vec v = \vec v' + \vec v_{CM} +\] +Da questo è facile osservare come nel CM +\[ + \vec \rho' = \sum m_i \vec v'_i = 0 +\] +La quantità di moto nel CM risulta sempre essere 0. Questo non implica che ogni urto sia elastico, ma solo che per per il CM tutti gli urti avvengono su una retta e che $ \vec \rho'_A = - \vec \rho'_B $ è una relazione valida in qualunque istante. + +\subsection{Urti speciali} +\begin{dimJ}{Urto completamente anaelastico}{dim:uca} + Un urto è completamente anaelastico quando dopo l'urto i 2 corpi sono un unico punto di massa.\\ + La quantità di moto. + \[ + m_A \vec v_A + m_B \vec v_B = \8 m_A + m_B \9 \vec v_f + \] + \[ + \vec v_f = \frac{m_a \vec v_A + m_B \vec v_B}{m_A + m_B} = \vec v_{CM} + \] + Per quel che riguarda l'energia cinetica prima e dopo l'urto + \[ + K = K' + K_{CM} = \frac 1 2 m_A v'_A^2 + \frac 1 2 m_B v'_B^2 + \frac 1 2 \8 m_A + m_B\9 v_{CM}^2 + \] + Dopo l'urto + \[ + K = K' + K_{CM} = K_{CM} = \frac 1 2 \8 m_A + m_B\9 v_{CM}^2 + \] + In un urto completamente anaelastico la velocità relativa rispetto al CM delle particelle è nulla quindi scompare il termine $ K' $ +\end{dimJ} +\begin{dimJ}{Urto elastico monodimensionale}{dim:uemd} + \[ + \vec \rho_f = \vec \rho_i \text{, } K_i = K_f + \] + Nel caso monodimensionale queste 2 equazioni sono riscrivibili + \[ + m_A v_{Ai} + m_B v_{Bi} = m_A v_{Af} + m_B v_{Bf} + \] + \[ + \frac 1 2 m_A v_{Ai}^2 + \frac 1 2 m_B v_{Bi}^2 = \frac 1 2 m_A v_{Af}^2 + \frac 1 2 m_B v_{Bi}^2 + \] + Ne risultano 2 equazioni di cui ne scriverò una sola, l'altra ho solo gli indici A e B scambaiti. + \[ + v_{Af} = \frac{\8 m_A - m_B \9 v_{Ai} + 2m_B v_{Bi}}{m_A + m_B} + \] +\end{dimJ} +\begin{dimJ}{Urto anaelastico}{dim:urtoanaelastico} + Studieremo il tutto nel sistema del CM.\\ + Poiché non tutta l'enrgia cinetica viene restituita definiamo il coefficiente di restituzione. + \[ + e = \frac{\abs{\vec \rho_{Af}}}{\abs{\vec \rho_{Ai}}} \leq 1 + \] + \[ + K_f = \frac 1 2 m_A v'_{Af}^2 + \frac 1 2 m_A v'_{Af}^2 = e^2 \8 \frac 1 2 m_A v'_{Ai}^2 + \frac 1 2 m_A v'_{Ai}^2 \9 + \] + La frazione di energia cinetica assorbita sarebbe + \[ + \frac{\Delta K'}{K'_i} = e^2 - 1 + \] + Nel sistema inerziale vale: + \[ + v_{Af} = \frac{\8 m_A - e m_B \9 v_{Ai} + \8 1 + e \9 m_B v_{Bi}}{m_A + m_B} + \] + Nel sistema del CM vale: + \[ + v'_{Af} = e v'_{Ai} + \] +\end{dimJ} + + +\pagebreak +\section{Corpo rigido} +\includegraphics{corpoRigidoGira.png} + +\begin{defJ}{Corpo rigido}{def:corporigido} + Sistema di punti materiali in chi per ogni coppia $P_A$ e $P_B$ vale $\vec r_{AB} = const$. + Questo ci permette di dire: + \[ + W = W^E + W^I = \Delta K + \] + $W^I = 0$ Poichè non vi sono spostamenti interni. + Consideriamo una distribuzione continua della materia fatta da elementi infinitesimi di volume ($dV$) e di massa ($dm$). + \[ + \rho = \frac{dm}{dV} \imp M = \int_V dm = \int_V \rho(\vec r) dV = \int_x \int_y \int_z \rho(x,y,z) dxdydz + \] + Se $\rho = const$ si parla di corpo omegeneo per cui vale una comoda semplificazione $ M = V \rho$. + +\end{defJ} + +\begin{dimJ}{rotazione del corpo rigido}{dim:rcr} + Rotazione intorno ad un asse fisso $ z $ no simmetrie. + \[ + \vec v_i = \vec \omega \x \vec r_i + \] + Calcoliamo il momento angolare di $P_i$ rispetto $O$. + \[ + \vec L_i = \vec r_i \x m_i \vec v_i \imp \abs{\vec L_i} = r_i m_i v_i = r_i m_i \8 \omega R_i \9 + \] + Sull'asse $z$ + \[ + L_{iz} = L_i \cos \8 \frac \pi 2 - \theta_i \9 = L_i \sin \theta_i = r_i m_i R_i \omega \sin \theta_i = R_i^2 m_i \omega + \] + Se ora sommiamo tutte le componenti $L$ sull'asse $z$ + \[ + L_z = \sum L_{iz} = \sum R_i^2 m_i \omega = I_z \omega + \] + + \begin{defJ}{momento d'inerzia}{def:mi} + Il momento di inerzia rispetto un determinato asse $z$ + \[ + I_z = \int_V R^2 dm \simeq \sum_i m_i R_i^2 + \] + Ci accorgiamo che + \[ + I_z = \int_V \8 x^2 + y^2 \9 \rho dV + \] + Poichè non vogliamo calcolare questo integrale troviamo delle belle tabelle per i momenti di inerzia. (Bersanelli 22-9) + \end{defJ} + + Per ora abbiamo valutato solo quello che succede sull'asse $z$, ma + \[ + \vec L = \vec L_z + \vec L_\perp + \] + $\vec L_\perp$ Ruota con il corpo ed in generale $ \not = 0$. \\ + Se un corpo è simmetrico rispetto all'asse, vale che $\forall P_i \exists P_j $ t.c. + \[ + \vec L_i = \vec r_i \x \vec \rho_i \imp \vec L_{i\perp} = - \vec L_{j\perp} \imp \vec L_\perp = 0 + \] + Sempre nel caso simmetrico vale: + \[ + \vec M = \dert{\vec L} = \dert{I_z \vec \omega} = I_z \dert{\vec \omega} = I_z \vec \alpha + \] +\end{dimJ} + +\begin{teoJ}{Huygens-Steiner}{teo:HS} + Dato il momento di inerzia rispetto a un asse passante per il CM vogliamo trovare il momento d'inerzia per un asse parallelo.\\ + Coordinate del punto $P_i$ nei 2 sistemi di riferimento + \[ + x_i = x'_i \text{, } y_i = y'_i + b \text{, } z_i = z'_i + \] + \[ + I_{zi} = m_i \8 x_i^2 + y_i^2 \9 \imp I_z = \sum m_i \8 x_i^2 + y_i^2 \9= \sum m_i x'_i^2 + \sum m_i \8 y_i'^2 + b \9^2= + \] + \[ + = \sum m_i \8 x'_i^2 + y'_i^2 \9 + 2\sum m_i y'_i b + \sum m_i b^2 = I_{CM} + M b^2 + \] + \[ + I_z = I_{CM} + M b^2 + \] +\end{teoJ} + +Dobbiamo fare un osservazione importante sull'eneriga. +\begin{dimJ}{energia cinetica rotazione}{dim:Krot} + Se il corpo sta ruotando intorno al CM vale + \[ + K = \sum \frac 1 2 m_i v_i^2 = \frac 1 2 \sum m_i R_i^2 \omega^2 = \frac 1 2 I_z \omega^2 + \] + Nel caso simmetrico valgono queste 2 formule $ \vec L / / \vec \omega$ + \[ + K = \frac 1 2 I_z \8 \frac{L}{I_z} \9^2 = \frac{L^2}{2I_z} + \] + \[ + dK = d\8 \frac 1 2 I_z \omega ^2\9 = I_z\omega d \omega = I_z \dert{\theta}\alpha dt = I_z \alpha d\theta = M d\theta = dW \imp P = \dert{W} = M \dert{\theta} = M \omega + \] + + Se il corpo ruota intorno ad un qualunque asse $z$ vuol dire che il CM di massa si sta spostando quindi dobbiamo aggiungere la sua energia usando il teo Huygens-Steiner. + \[ + K = \frac 1 2 I_z \omega^2 = \frac 1 2 \8 I_{CM} + M b^2\9\omega^2= \frac 1 2 I_{CM} \omega^2 + \frac 1 2 M b^2 \omega^2 = \frac 1 2 I_{CM} \omega^2 + \frac 1 2 M v_{CM}^2 + \] +\end{dimJ} + +\section{Il pendolo fisico} +Il pendolo fisico è un corpo rigido che viene imperneato in un punto. Non ci sono assunzioni sul corpo particolari, eccetto la realtà fisica che viene modellizzata. +\[ + I_z \alpha = I_z \dertt{\theta} = M = -Mgh \sin\theta +\] +Ci accorgiamo che per piccole oscillazioni per cui $\sin \theta = \theta$ +\[ + \dertt{\theta} + \frac{Mgh}{I_z} \theta = 0 +\] +Questa è l'equazione del moto armonico +\[ + \omega_p = \sqrt{\frac{Mhg}{I_z}} \imp T = \frac{2 \pi}{\omega_p} = 2\pi \sqrt{\frac{I_z}{Mh}g} +\] + +\begin{defJ}{Lunghezza ridotta}{def:lr} + \[ + l_R = \frac{I_z}{Mh} + \] + La lunghezza ridotta risulta utile nelle situazioni sperimentali: + \[ + I_z = I_{CM} + M h^2 = Mh \8\frac{I_{CM}}{Mh} + h\9= Mh \8 h'+ h \9 + \] + \[ + l_R = h' + h + \] + Ora consideriamo $O'$ come centro dell'oscillazione + \[ + l'_R = \frac{I'_z}{Mh'} = \frac{I_{CM} + M h'^2}{Mh'} = h + h' = l_R + \] + $l_R$ non è altro che una distanza tra 2 buchi per cui posso far ruotare il corpo rigido.\\ + Poichè so che $l_R = l'_R \imp T_O = T_{O'}$ mi basta che un corpo abbia 2 punti per cui il periodo di oscillazione sia uguale e vale + \[ + l'_R = l_R = L \imp T = 2\pi \sqrt{\frac L g} \imp g = \frac{4 \pi^2}{T^2} L + \] + Così facendo possiamo misurare $g$ con precisione elevata $\delta g / g \sim 10^{-6}$ come fece Kater con il suo pendolo reversibile 1817 +\end{defJ} + +\section{Rotolamento} +Un corpo rigido è in puro rotolamento quando il punto di contatto C è fermo e nel tempo infinitesimo $dt$ il moto è una rotazione intorno a C con velocità angolare $\omega$. + +\subsection{Forza costante orizzontale applicata all'asse passante CM} +"Ruota imperneata ad un asse passante per il suo centro di un carretto tirato da dei cavalli" +\[ + \vec F_T = m \vec a_{CM} \imp \text{asse x } F - f = m a +\] +\[ + \vec M = \vec r \x \vec f = I \vec \alpha \imp rf = I \frac{a}{r} +\] +Ricaviamo ora f per poter risolvere il sistema e conoscere l'accelerazione del CM +\[ + f = \frac{I}{r^2}a +\] +\[ + a = \frac F m \frac{1}{\8 1+\frac{I}{mr^2}\9} = \frac F m \frac{1}{ \8 1+k \9} \leq \frac F m +\] +$k$ lo usiamo per semplificare poiché dipende solo dalla forma del corpo.\\ +Quello che è molto interessante è notare che rotolare riduce l'accelerazione rispetto a scivolare, che può risultare antintuitivo. \\ +Ritorniamo a valutare la forza d'attrito ora che conosciamo l'accelerazione. +\[ + f = \frac I r^2 \frac F m \frac{1}{\8 1 + k \9} = F \frac{k}{1+k} +\] +Sappiamo che la forza di attrito statico è adattiva, ma sappiamo quale è il suo valore massimo +\[ + f \leq \mu_s N = \mu_s mg \imp F \leq \mu_s m g \frac{k +1}{k} = \mu_s m g \8 \frac{mr^2}{I} + 1\9 +\] + +\subsection{Momento costante applicato all'asse} +"Ruota di un automobile in accelerazione" +\[ + \vec F_{T} = m \vec a_{CM} \imp \text{asse x } f = m a +\] +\[ + \vec M = \vec M_o + \vec r \x \vec f = I \vec \alpha \imp M_o - rf = I \frac{a}{r} +\] +Anche in questo caso risolviamo il sistema per identificare $a$. +\[ + f = \frac 1 r \8 M_o - I \frac a r \9 +\] +Applicando la stessa procedura di prima. +\[ + a = \frac{M_o}{mr}\frac{1}{\8 1+k \9} +\] +Anche in questo caso identifichiamo il massimo momento applicabile +\[ + f \leq \mu_s N = \mu_s mg \imp M_o \leq \mu_s m g r \8 1 + k \9 +\] + +\subsection{Energia di un corpo in moto di puro rotolamento} +\[ + a_{CM} = \frac{F_o}{m} \frac{1}{\8 1 + k\9} = \frac{M_o}{m r} frac{1}{\8 1 + k\9} +\] +\[ + \alpha = \frac{a_{CM}}{r} \text{, } \omega = \omega_0 + \alpha t +\] +\[ + K = \frac 1 2 I_C \omega^2 = \frac 1 2 \8 kmr^2 + mr^2\9\omega^2 = \frac 1 2 m r^2 \8 k + 1 \9\omega^2 +\] + +\subsection{Attrito volvente} +Sperimentalmente ci accorgiamo che una biglia non rotola su ogni superficie piana, inoltre dopo un po' si ferma dopo averla lanciata. +\[ + M_v = h_v N = h_v m g \imp F_v = \frac{h_v m g}{r} +\] +La forza motrice minima è dipendente dal raggio del corpo che ruota. + +\section{Precessione} +Studiamo cosa succede se $\vec L_\perp \not = 0$. Sappiamo che la componente $\vec L_\perp$ ruota intorno all'asse.\\ +Assumiamo $\vec \omega = const.$ +\[ + \vec M = \dert{\vec L} = \dert{} \sum \vec r_i \x m_i \vec v_i = \sum \dert{\vec r_i} \x m_i \vec v_i + \sum \vec r_i \x m_i \dert{\vec v_i} = \sum \vec r_i \x m_i \8 \dert{\vec \omega \x \vec r_i}\9 = +\] +\[ + = \sum \vec r_i \x m_i \8 \dert{\vec \omega} \x \vec r_i + \vec \omega \x \dert{\vec r_i}\9 = \sum \vec r_i \x m_i \8 \vec \omega \x \vec v_i \9 = - \8 \sum \vec r_i \x m_i \vec v_i \9 \x \vec \omega = \vec \omega \x \vec L +\] +Semplifichiamo i conti +\[ + \dert{\vec L} = \vec M = \vec \omega \x \vec L \imp d\vec L = \vec M dt +\] +In modulo +\[ + dL = L_\perp d \phi \imp M = \dert{L} = L_\perp \dert{\phi} = L_\perp \omega +\] + +\section{Forze centrali} +Una forza centrale ha origine in un punto O, centro della forza.\\ +In qualsiasi punto P dello spazio la forza è nella direzione OP.\\ +Il modulo della forza P è funzione solo di $r$. +\[ + \vec F = \pm F(r)\un{r} +\] + +\subsection{Proprietà notevoli delle forze centrali} +\begin{teoJ}{Momento angolare}{teo:mafc} + \[ + \dert{\vec L} = \vec r \x \vec F = \pm \vec r \x F(r)\un{r} = 0 + \] + Il vettore momento angolare si conserva. + \[ + \vec L = \vec r \x m \vec v \imp \vec L \perp \8 \vec r , \vec v \9 + \] + Posizione e velocità si mantengono sullo stesso piano. +\end{teoJ} +\begin{teoJ}{"Velocità areale" costante}{teo:vac} + \[ + \vec L = \vec r \x m \vec v = \vec r \x m \8 \vec v_r + \vec v_\theta\9 = \vec r \x m \vec v_\theta + \] + \[ + L = rmv_\theta = m r^2 \dert{\theta} = 2 m \dert{A} + \] + $dA = \frac 1 2 r^2 d\theta$ L'area spazzata è un triangolo che ha per base $r$ e altezza $rd\theta$ + \[ + VA = \dert{A} = \frac{L}{2m} + \] +\end{teoJ} +\begin{teoJ}{Lavoro di una forza centrale}{teo:forcen} + \[ + \vec F = \pm F(r)\un{r} + \] + \[ + W = \int_A^B F(r)\un{r} \cdot d \vec s = \int_A^B F(r)dr = f(\vec r_B) - f(\vec r_A) + \] + Le forze centrali sono tutte conservative. +\end{teoJ} + +\section{Le leggi di Keplero} +\begin{teoJ}{I Keplero}{teo:IK} + Le orbite dei pianeti sono ellittiche +\end{teoJ} +\begin{teoJ}{II Keplero}{teo:IIK} + La velocità areale del raggio che unisce il Sole al pianeta è costante $\dert{A} = const.$ +\end{teoJ} +\begin{teoJ}{III Keplero}{teo:IIIK} + Il quadrato del periodo di rivoluzione è proporzionale al cubo del semiasse maggiore + \[ + T^2 = k_S a^3 + \] + Questa costante $k_S$ è legata al Sole. +\end{teoJ} + +\subsection{Newton} +Assumiamo un orbita circolare e partiamo dalla II legge di Keplero. +\[ + \dert{A} = const. = \frac 1 2 r^2 \dert{\theta} \imp \dert{\theta} = const. +\] +Quindi si ha un moto circolare uniforme che genera una forza centripeta. +\[ + F = m \omega^2 r = m \frac{4 \pi^2}{T^2} r = \frac{4 \pi^2}{k_S} \frac{m}{r^2} +\] +La forza è inversamente proporzionale al quadrato della distanza, ma ancora non sappiamo cosa sia $k_S$.\\ +Se il Sole esercita una forza sulla Terra allora anche la Terra deve esercitare una forza sul Sole. +\[ + \abs{\vec F_{ST}} = \abs{\vec F_{TS}} \imp \frac{4 \pi^2}{k_S} \frac{m_T}{r^2} = \frac{4 \pi^2}{k_S} \frac{m_S}{r^2} \imp \frac{4 \pi^2}{m_S k_S} = \frac{4 \pi^2}{m_T k_T} = G +\] +\[ + \vec F = - G \frac{m_S m_T}{r^2} \un{r} +\] +In questa formula compaiono 2 masse che sono agenti della forza gravitazionale, mentre per ora abbiamo sempre considerato la massa come resistente alla variazione di velocità. +\[ + G \frac{m_T^g m^g}{r^2} = m^i g \imp g = G \frac{m_T}{r^2} +\] +Questa semplificazione è valida solo se $m^g = m^i$ la massa gravitazionale è uguale a quella inerziale. Sperimentalmente questa ugualianza vale fino ad una precisione di $10^{-12} m/s^2$ + +\begin{defJ}{Campo Gravitazionale}{def:campg} + \[ + \vec F = - G \frac{m_1 m_2}{r^2} \un{r} = \8 -G \frac{m_1}{r^2} \un{r}\9 m_2 = \vec \eta m_2 + \] + Il vettore $\vec \eta$ lo chiamiamo vettore campo gravitazionale, e $m_2$ massa campione.\\ + Il Campo totale generato in P da n masse: + \[ + \vec \eta(P) =\sum_{i = 1}^n \vec \eta_i(P) = -G \sum_{i = 1}^n \frac{m_i}{r_i^2}\un{r_i} + \] +\end{defJ} + +\begin{teoJ}{di Gauss}{teo:Gauss} + Chiamata $S$ una superficie chiusa che racchiude la massa puntiforme $m$. + \[ + d \vec S= dS\un{n} + \] + Il vettore superficie infinitesima ha come modulo l'area della superficie infinitesima, per direzione la normale alla superficie e verso uscente.\\ + Il flusso infinitesimale del campo $\vec \eta$: + \[ + d\Phi = \vec \eta \cdot d\vec S + \] + Il flusso attraverso l'intera superficie S: + \[ + \Phi = \int_S \vec \eta \cdot d\vec S = - \int_S \eta \un{r} \cdot d\vec S = - \int_S \frac{Gm}{r^2} dS \cos\theta = -Gm \int_S \frac{dS\cos\theta}{r^2} = + \] + Riconosciamo che $dS\cos\theta$ è l'angolo solido che al massimo e $4 \pi$. + \[ + \Phi = -Gm \int_S \frac{dS_n}{r^2} = - Gm \int_S d\Omega = -4\pi Gm + \] + m è una qualunque massa dentro la superficie. +\end{teoJ} + +Valutiamo il flusso totale del campo $\vec \eta = - \eta(r)\un{r}$ attraverso la sfera S. +\[ + \Phi = \int_S \vec \eta \cdot d\vec S = - \int_S \eta(r) \un{r} \cdot d\vec S = - \eta(r) \int_S dS = -4\pi r^2 \eta(r) +\] +Ricordo che $\un{r} // d\vec S$ per questo l'angolo compreso è $0$.\\ +Uguagliando con il teorema di Gauss. +\[ + \Phi = -4\pi r^2 \eta(r) = -4\pi G m \imp \eta(r) = \frac{Gm}{r^2} +\] + +\begin{dimJ}{Guscio sferico}{dim:guscio} + Prendiamo un guscio sferico con distribuzione di massa simmetrica.\\ + Valutiamo il campo di pun qualunque punto P appartenente all'interno del guscio. + \[ + \eta_1 = -G \frac{dm_1}{r_1^2} = -G \frac{\8 \rho dS_1 dr\9}{r_1^2} = -G \frac{\rho dr}{r_1^2} \pi \8 r_1 d\alpha\9^2 = -G \rho dr \pi \8 d\alpha\9^2 = - d \eta_2 + \] + Questa lunga ugualianza ci dice che preso uno "spicchio" infinitesimale di gusio il suo campo viene annullato dallo "spicchio" opposto. + Sappiamo quindi che il campo gravitazionale dentro un guscio sferico generato da se stesso è nullo. +\end{dimJ} + +\begin{dimJ}{Campo gravitazionale interno ad una sfera omogenea}{dim:cgis} + Immaginiamo un pianeta omogeneo. + La massa contenuta nel raggio r è data da $m(r) = \frac 4 3 \pi r^3 \rho$. + \[ + \frac{m(r)}{M} = \frac{r^3}{R^3} \imp \Phi = -4 \pi r^2 \eta(r) = -4\pi G m(r) = -4 \pi G M \frac{r^3}{R^3} + \] + Ricaviamo il campo. + \[ + \eta(r) = \frac{GM}{R^3} r = \frac g R r + \] +\end{dimJ} + +Facciamo un piccolo giochino: immaginiamo un buco che attraversi un pianeta. +\[ + F = -\eta(r) m = -\frac{GM}{R^3} r m = -k r +\] +In altre parole la forza gravitazionale si comporta come una molla centrata nel centro del pianeta, nel caso della terra partendo dalla superficie e risolvendo si ottiene $T \sim 5*10^3 s \sim 84 \min$. \\ + +\begin{dimJ}{Energia potenziale gravitazionale}{dim:epgra} + Se il comportamento del corpo è del tutto simile alle leggi per una molla, che è una forza centrale sappiamo che $E = U + K = const. $ Vogliamo ottenere una legge per l'eneriga potenziale. + \[ + dW = \vec F \cdot d\vec s = -G \frac{m_1 m_2}{r^2} \un{r} \cdot d\vec s = -G \frac{m_1 m_2}{r^2} dr + \] + Integriamo + \[ + W = -Gm_1m_2\int_A^B \frac{1}{r^2} dr = -Gm_1m_2 \8 - \frac{1}{r_B} + \frac{1}{r_A} \9 = - \Delta E_P + \] +\end{dimJ} + +Osserviamo che se un corpo ad una data distanza da un altro non viaggia abbastanza rapidamente resta legato ad esso. Per conservazione dell'energia diciamo a quale velocità deve viaggiare un corpo per far sì che si fermi solo all'infinito? +\[ + \frac 1 2 m v_f^2 -G \frac{m_T m }{r_T} = 0 \imp v_f = \sqrt{\frac{2 G m_T}{r_T}} = \sqrt{2 g r_T} \simeq 11.2 Km / s +\] +Cosa succede al limite: $v_f \to c$ poichè niente si muove più veloce della luce stiamo ipotizzando un corpo che non fa scappare niente da se stesso. +\[ + \frac 1 2 m c^2 - G \frac{Mm}{r} = 0 \imp r_S = 2 \frac{GM}{c^2} +\] +Questo si chiama raggio di Schwarzschild e un corpo così massivo Buco Nero. + +\subsection{Potere è volere} +No dimostrazione +\[ + \vec \eta = - \nabla V = - \nabla \8 -G \frac{m_1}{r}\9 +\] +\end{document}