From c2baa2628ac7d616308d4792d827d5fff1e772c8 Mon Sep 17 00:00:00 2001 From: JBR Date: Fri, 20 Mar 2026 12:12:51 +0100 Subject: [PATCH] Aggiunto onde, fatta la trattazione dell'oscillatore armonico semplice, modifiche minime alla libreria base --- jacbr.sty | 7 +- loghi/onde.svg | 8 ++ onde/favicon.png | Bin 0 -> 3021 bytes onde/onde.pdf | Bin 0 -> 185941 bytes onde/onde.tex | 202 +++++++++++++++++++++++++++++++++++++++++++++++ 5 files changed, 215 insertions(+), 2 deletions(-) create mode 100644 loghi/onde.svg create mode 100644 onde/favicon.png create mode 100644 onde/onde.pdf create mode 100644 onde/onde.tex diff --git a/jacbr.sty b/jacbr.sty index ab4984d..758ce5b 100644 --- a/jacbr.sty +++ b/jacbr.sty @@ -16,7 +16,7 @@ \usepackage{booktabs} \usepackage{fancyhdr} \usepackage{svg} -\usepackage{siunitx-v2} %unità di misura +\usepackage{siunitx} %unità di misura \usepackage{listings} % Rende il titolo e l'autore delle variabili @@ -123,9 +123,12 @@ % Massimo Comune Divisore 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sTu*LBspfWrl>7i~i+{g{ot%vvo!ua@1eqNO;9zG%e*0ETK^*!215mwloB#j- literal 0 HcmV?d00001 diff --git a/onde/onde.tex b/onde/onde.tex new file mode 100644 index 0000000..b4dfb84 --- /dev/null +++ b/onde/onde.tex @@ -0,0 +1,202 @@ +\documentclass[a4paper,12pt]{article} +\title{Onde} +\author{Jacopo Basso Ricci} +\date{\today} +\usepackage{./../jacbr} + +\begin{document} + \maketitle + \tableofcontents + \pagebreak + +\section{Notazione} +In questo documento userò sempre la scrittura + +\[ + \dert{x} \text{ e } \dertt{x} +\] + +rispettivamente per derivate temporali di primo e secondo ordine. + + + +\section{L'oscillatore armonico} +L'oscillatore è un modello idealizzato che noi studieremo per avere una base solida da cui partire. + +\begin{defJ}{Oscillatore Armonico}{def: oscilArm} + Immaginiamo di avere un piano orizzontale senza alcun attrito sulla quale posizioniamo una massa vincolata ad un punto fisso tramite una molla ideale. +\end{defJ} + +NOTA: La molla è definita ideale in quanto non ha massa e rispetta perfettamente la legge di Hooke. + +Definito questo modello di base iniziamone lo studio. +Immaginiamo di spostare la massa dalla posizione di equlibrio della molla. +In questa condizione, per ora statica, facciamo l'analisi delle forze. + +\[ + \sum_i \vec F_i = m \dertt{\vec x} +\] + +Poichè il moto avviene soltanto lungo un asse, ha un solo grado di libertà, non è necessaria la notazione vettoriale. + +\begin{align*} + F = -k x = m \dertt{x} \\ + \dertt{x} + \frac k m x = 0 +\end{align*} + +L'equazione differenziale che abbiamo trovato, chiamta anche equazione del moto, ha come soluzione una famiglia di funzioni. +In particolare questa è un equazione differenziale lineare (compaiono solo x e le sue derivate), omogenea (a "destra" c'è 0), del secondo ordine (le derivate compaiono al massimo all'ordine 2). + +Per semplificare la trattazione matematica: + +\begin{defJ}{Pulsazione}{def:omp} + \[ + \omega_p^2 = \frac k m + \] +\end{defJ} + +In questo modo l'equazione differenziale diventa: + +\[ + \dertt{x} = - \omega_p^2 x +\] + +Se noi guardiamo questa ODE ci rendiamo conto che dobbiamo trovare una funzione che differenziata 2 volte è la stessa funzione a meno di una costante. +Ci ricordiamo che questa è una caratteristica delle funzioni trigonometriche $\cos(x)$ e $\sin(x)$. +Proviamo a vedere cosa succede sostituendo queste funzioni a x. + +\begin{align*} + x(t) = \cos(t)\\ + \dert{x} = - \sin(t)\\ + \dertt{x} = - \cos(t)\\ +\end{align*} + +Siamo sulla buona strada, ora dobbiamo introdurre $ \omega_p $. + +\begin{align*} + x(t) = \cos(\omega_p t)\\ + \dert{x} = - \omega_p \sin(\omega_p t)\\ + \dertt{x} = - \omega_p^2 \cos(\omega_p t) = - \omega_p^2 x(t) +\end{align*} + +Abbiamo una soluzione valida, ma non è l'unica che ci viene in mente, come detto prima possiamo fare lo stesso giochetto anche con $\sin(x)$. + +\begin{align*} + x(t) = \sin(\omega_p t)\\ + \dert{x} = \omega_p \cos(\omega_p t)\\ + \dertt{x} = - \omega_p^2 \sin(\omega_p t) = - \omega_p^2 x(t) +\end{align*} + +Abbiamo quindi 2 soluzioni particolari indipendenti (l'una non è il prodotto dell'altra). +La soluzione generale dell'oscillatore armonico risulta essere la combinazione lineare delle 2 soluzioni particolari. + +\[ + x(t) = A \cos(\omega_p t) + B \sin(\omega_p t) +\] + +Un altro modo per scrivere la stessa cosa è questo: + +\[ + x(t) = R \cos(\omega_p t + \varphi) +\] + +\begin{dimJ}{Trasformazione dell'equazione}{dim:mate} + Partiamo dalla soluzione: + \[ + A\cos(\omega t) + B\sin(\omega t). + \] + Vogliamo riscriverla come + \[ + R\cos(\omega t + \varphi). + \] + Usiamo la formula di addizione: + \[ + \cos(\omega t + \varphi) = \cos(\omega t)\cos\varphi - \sin(\omega t)\sin\varphi + \] + Dunque: + \[ + R\cos(\omega t + \varphi) = R\cos\varphi\,\cos(\omega t) - R\sin\varphi\ \sin(\omega t). + \] + Uguagliando i coefficienti otteniamo il sistema: + \[ + \begin{cases} + A = R\cos\varphi,\\[6pt] + B = -R\sin\varphi. + \end{cases} + \] + Da cui: + \[ + R = \sqrt{A^2 + B^2}, \qquad \tan\varphi = -\,\frac{B}{A} + \] + + Pertanto: + \[ + A\cos(\omega t) + B\sin(\omega t) = \sqrt{A^2 + B^2}\,\cos\8\omega t + \varphi\9 + \] +\end{dimJ} + + + +\subsection{Condizioni iniziali} +Per ora abbiamo solo guardato ad una soluzione generale, come possiamo calarla in un caso specifico? +Se osserviamo il nostro sistema ci viene in mente che noi possiamo farlo partire in 2 modi: spostando la massa dal punto di riposo della molla, dando una leggera velocità iniziale alla molla o con una combinazione di queste 2 situazioni. +Partiamo dall'equazione generale: + +\[ + x(t) = A \cos(\omega_p t) + B \sin(\omega_p t) +\] + +Poichè noi vogliamo definire questa equazione a partire dalle condizioni iniziali capiamo cosa succede in 0. + +\begin{align*} + x(t) |_{t=0} = x(0) = A \\ + \dert{x} = - A \omega_p \sin(\omega_p t) + B \omega_p \cos(\omega_p t) \\ + \dert{x(0)} = B \omega_p = v(0)\\ + B = \frac{v(0)}{\omega_p} +\end{align*} + +In questo modo abbiamo fissato i valori di A, B in modo tale che siano definite a partire dai valori fisici del sistema. +Con le formule ottenute prima possiamo anche muoverci tra le 2 soluzioni. + +\begin{align*} + x(t) = R \cos(\omega_p t + \varphi)\\ + R = \sqrt{A^2 + B^2}\\ + \tan \varphi = - \frac B A +\end{align*} + + + +\subsection{Analisi Dimensionale e implicazioni} +Tutto quello che abbiamo trovato fin ora deve avere unità di misura coerenti. +Partiamo dall'analisi di $\omega_p$. + +\[ + [\omega_p] = \left[\sqrt{\frac{K}{m}}\right] = \sqrt{\frac{\si{\newton\per\meter}}{\si{\kilogram}}} = \sqrt{\frac{\si{\kilogram\per\second\squared}}{\si{\kilogram}}} = \sqrt{\si{\per\second\squared}} = \si{\per\second} +\] + +Noi sappiamo che i $\si{\rad}$ sono un unità di misura che è in realtà adimensionale, ma per maggiore chiarezza preferiamo scrivere: +\[ + [\omega_p] = \si{\rad\per\second} +\] + +Ragionando sulle caratteristiche delle funzioni trigonometriche risulta chiaro che questa $\omega_p$ è legata al periodo $T$. In particolare: + +\begin{defJ}{Periodo}{def:periodo} + \begin{align*} + T = \frac{2\pi}{\omega_p}\\ + [T] = \frac{1}{\si{\per\second}} = \si{\second} + \end{align*} +\end{defJ} + +Come si nota anche le unità di misura sono coerenti. + +Esiste anche un'altra grandezza ricavata da $\omega_p$, che è strettamente legata al periodo. + +\begin{defJ}{Frequenza}{def:freq} + \begin{align*} + f = \frac 1 T \imp& \omega_p = 2 \pi f\\ + [f] =& \si{\per\second} + \end{align*} +\end{defJ} + +\end{document}