From ff6bf08cae10109c2b63d5f172960186652ffe3d Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 20 Jan 2022 23:34:02 +0100 Subject: [PATCH] fisica: aggiunge la visione vettoriale del moto circolare --- Fisica/fisica.pdf | Bin 168930 -> 179076 bytes Fisica/fisica.tex | 93 ++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 93 insertions(+) diff --git a/Fisica/fisica.pdf b/Fisica/fisica.pdf index 2c5a1d4597604a341365154879447c041695e0b5..5c94023e10112c4b9927775049297fa81b2c5e75 100644 GIT binary patch delta 100572 zcmce;cUV-*(l1QTIcJ6(1!jO@h76J=OAeBA&Y2-+Nkaxn0wMw;AQ?%5Btb=T7LlAq zka!2(`)toS@AudH-RtwvYfY{0>Z)4ZtAAb9<(FqTQD-TbHMc|>B)MWE&2>m4)0LKzV=uQGwcqI-I+yoyBPJ$5j6Hd(U-iQVH1pkyu=uY!H4e!_r z^ZA+x2naab6<&9R&s|}0S2*7lza9kM-Tf*o;2+Q_kbU8XSS;wj>cGuVsqk#L{?=IN z&X^Pg#zIhl06-hSui)Tq3V^>Pri8!e#Y6>)fU=KFQMh2y<6^kI0%Yhbon2p&w^DN! z#OIR8oqM1yara*nqNGzizFq#@0)@~+7EXHB1yEyaS74q838sm{aS0_Ka2C)ih2SJI z4t zG$Ev=hQcFElD(2`Qg9_OaNdK2E{IX7Qw;{Eq-`_=;FXN$b7#iQuWU3#@+|Q@6Q8)# 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zs&X!tVU)cA!7@6r!xXO;(}9sxrqj}l*qY16L2z=ixg!1?1!u{~2SP8TRn`Nbamus| z5v!vWro)(mWTD^(O`XySF^O4*S1t^dS5?a?63eBAz6TmJMv!Cf zS~EHvM{ys4uh5*1?Zr&53zsk?f?J) diff --git a/Fisica/fisica.tex b/Fisica/fisica.tex index 1dada4e..264cd20 100644 --- a/Fisica/fisica.tex +++ b/Fisica/fisica.tex @@ -7,10 +7,19 @@ \usepackage[italian]{babel} \usepackage[utf8]{inputenc} \usepackage[parfill]{parskip} +\usepackage{wrapfig} + +\usepackage{pgfplots} +\pgfplotsset{compat=1.15} +\usepackage{mathrsfs} +\usetikzlibrary{arrows,angles,quotes} + \newcommand{\gfrac}[2]{\displaystyle \frac{#1}{#2}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\norm}[1]{\lVert \vec{#1} \rVert} +\newcommand{\nnorm}[1]{\lVert #1 \rVert} + \begin{document} @@ -191,6 +200,7 @@ ed è calcolata mediante le seguente equazione: \begin{equation} a=\frac{v^2}{r} + \label{eq:acc_c} \end{equation} \begin{proof} @@ -224,4 +234,87 @@ il risultato desiderato: \end{proof} +\newpage + +\subsection{Il moto circolare visualizzato vettorialmente} + +\vskip 0.1in + +\begin{wrapfigure}{l}{0.45\textwidth} + \begin{tikzpicture} + \coordinate (a) at (2.236, 0); + \coordinate (b) at (0.89, 0.92); + \coordinate (o) at (0, 0); + + \draw [rotate around={0:(0,0)}] (0,0) ellipse (2.236 and 1); + \draw [->,thick] (0,0) -- (0,2.5) node[midway, right] {$\vec{\omega}$}; + \draw [->,thick] (0,0) -- (a) node[right] {$\vec{r}(t)$}; + \draw [->,thick] (0,0) -- (b) node[above right] {$\vec{r}(t+dt)$}; + + \draw [->] (a) -- (2.6, 0.7) node[right] {$\vec{v}(t)$}; + + \draw pic["$d\theta$", draw=black, angle eccentricity=1.5, angle radius=0.5cm] {angle=a--o--b}; + + \end{tikzpicture} + \caption{Il moto circolare nel piano $O_{xy}$} \label{fig:moto_circolare} +\end{wrapfigure} + +Per visualizzare in modo più intuitivo, ma anche più formale, il +moto circolare, è possibile costruire un sistema di riferimento +basandosi su alcune assunzioni. + +Basandosi sul figura \ref{fig:moto_circolare}, assumiamo +$\vec{d\theta} = d\theta \cdot \hat{z}$, attraverso cui +possiamo concludere che $\vec{\omega}=\frac{\vec{d\theta}}{dt}$ è anch'esso +parallelo a $\hat{z}$. + +Inoltre, $d\vec{r}$ deve essere perpendicolare a $\vec{r}$, e, poiché +appartiene al piano $O_{xy}$, $d\vec{r}$ deve essere perpendicolare anche +a $\vec{\omega}$. + +Perciò è possibile riscrivere $d\vec{r}$ nella seguente forma: + +\begin{equation*} + d\vec{r}=\frac{d\theta \cdot \norm{r}}{\lVert \vec{w} \times + \vec{r} \rVert} \vec{w} \times \vec{r} = + \frac{d\theta}{\norm{\omega}} \vec{w} \times \vec{r} +\end{equation*} + +Poiché la velocità $\vec{v}$ è pari a $\frac{d\vec{r}}{dt}$, si ottiene, +conoscendo $d\vec{r}$, la seguente relazione: + +\begin{equation} + \vec{v}=\vec{w}\times\vec{r} +\end{equation} + +Dalla quale si ricava che $\vec{v}$ è perpendicolare sia a $\vec{r}$ che +a $\vec{\omega}$. + +Analogamente, è possibile ricavare l'accelerazione: + +\begin{equation} + \vec{a}=\frac{d\vec{v}}{dt}=\frac{d(\vec{\omega} + \times \vec{r})}{dt}=\vec{\alpha} \times \vec{r} + + \vec{\omega} \times \vec{v} +\end{equation} + +È interessante notare che $\vec{\alpha} \times \vec{r}$ +è perpendicolare a $\vec{\omega} \times \vec{v}$, permettendoci +di calcolare facilmente il modulo dell'accelerazione: + +\begin{equation} + \norm{a} = \sqrt{\nnorm{\vec{\alpha} \times \vec{r}}^2 + \nnorm{\vec{\omega} \times \vec{v}}^2} +\end{equation} + +Non solo: $\vec{\alpha} \times \vec{r}$ è perpendicolare a $\vec{r}$ e +$\vec{\omega} \times \vec{v}$ gli è parallelo, ma possiede un verso opposto. +Per questa serie di motivi, $\vec{\alpha} \times \vec{r}$ viene chiamata +\textbf{accelerazione tangenziale} ($\vec{a_t}$), mentre +$\vec{\omega} \times \vec{v}$ viene chiamata \textbf{accelerazione centripeta} +($\vec{a_c}$). + +Nel moto circolare uniforme, ove $\vec{\alpha}=0$, infatti l'accelerazione +centripeta è costante (vd. eq. \ref{eq:acc_c}) e l'accelerazione tangenziale +è nulla (quindi $\vec{a}=\vec{a_c}$). + \end{document} \ No newline at end of file