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z0WvrU=RUx|caH5-**6XhM&sTuAOe&%?l?3IhI>8;5`??|zz`$^cg|oa8kiu?`JiYh z_K64k<^n^4an}V5Xhbm1T7ls(Bot>|02vyIvxk7Ygufi4#t%sp}=!oaK<5_ z5IjAEgh6mI1L%7&?mh)%zyji)4}pT?>{BojiNrq^g~V1=@0&9cjsLx&KnPqcfKgxs z2zL#DJ%qcjz$hpj7jIw`4A_)7*9E8oSC;@f8wJAMLnz=raOaEy^dIgX0%X`$r(gHx z>}X^OJl+Oqa$*-$wRAVdaKEEk|4xI@rzKn1Qddngh@$>vj6uXdhCF4 v_{cr}$, la legge di Stokes non è più valida. + \end{remark} + + \begin{example} + Un esempio di forza viscosa è la resistenza aerodinamica + al moto del proiettile, spesso trascurata. + \end{example} + + \begin{remark} + La costante $\beta$ della legge di Stokes dipende dalla + viscosità del mezzo e dalle dimensioni e dalla forma del + corpo. + \end{remark} + + \begin{example} (senza alcuna forza) + Si pongano le condizioni $t_0 = 0$ e $\Vec{v_0} = \Vec{v}(t_0) \neq 0$. Se non agiscono altre forze sul corpo, si starà + allora trattando un moto unidimensionale. Si considera + allora il seguente sistema di equazioni: + + \[ \begin{cases} F = ma, \\ F = -\beta v, \end{cases} \] + + da cui si ricava che: + + \[ ma=-\beta v \implies \dv=-\frac{\beta}{m} v. \] + + Si definisce la costante $\tau = \frac{m}{\beta}$, + la cui unità di misura è il secondo. + + L'eq.~differenziale si riscrive allora come: + + \[ \dv = -\frac{1}{\tau} v. \] + + Risolvendo quest'eq.~differenziale, si ottiene allora + dunque che: + + \[ v(t) = c e^{-\frac{t}{\tau}}. \] + + Poiché $c = v(t_0) = v_0$, si conclude dunque che: + + \[ \system{v(t) = v_0 e^{-\frac{t}{\tau}}, \\ a(t) = -\frac{1}{\tau} v(t).} \] + + \vskip 0.1in + + In particolare, integrando la velocità, si ottiene lo + spostamento: + + \[ x(t) = \int_{t_0}^t v(t) dt = x_0 + v_0 \tau (1- e^{-\frac{t}{\tau}}). \] + + Quindi, la distanza percorsa all'infinito\footnote{Ossia, con + buona approssimazione, dopo alcuni periodi di $\tau$.} è + data da $x_\infty - x_0 = v_0 \tau$, dove $x_\infty = \lim_{t \to \infty} x(t) = x_0 + v_0 \tau$. + + \end{example} + + \begin{remark} + Si osserva che la velocità inizia a diventare + trascurabile dopo + \end{remark} + + \begin{example} (con forza di gravità\footnote{In generale, + con qualsiasi forza costante.}) Si supponga che $\Vec{v_0}$ + ed $\Vec{F} = \Vec{F_0}$ siano paralleli, e che dunque il + moto sia ancora completamente unidimensionale. + Si deve ora considerare il seguente sistema di forze: + + \[ \system{\Vec{F_v} = -\beta \Vec{v}, \\ \Vec{F} = \Vec{F_0} = m\vec{g}}, \] + + ossia, passando alle coordinate unidimensionali: + + \[ \system{F_v = -\beta v, \\ F = mg.} \] + + Da questo sistema si ottiene l'eq.~del sistema: + + \vskip 0.1in + + \[ F = mg - \beta v \implies m \dv = mg - \beta v \implies \dv = g - \frac{1}{\tau} v, \] + + ossia un'eq.~differenziale la cui associata omogenea è + esattamente quella analizzata nello scorso esempio. Allora + la soluzione generale è data dalla somma della soluzione + omogenea a quella particolare $v = \tau g$, detta + \textit{velocità limite} $v_{lim}$: + + \[ v(t) = c e^{-\frac{t}{\tau}} + \tau g. \] + + Ponendo allora $v(0) = v_0$, si ricava che $v_0 = c - \tau g \implies c = v_0 - \tau g$. Quindi si conclude che: + + \[ v(t) = (v_0 - v_{lim}) e^{-\frac{t}{\tau}} + v_{lim}, \] + + da cui chiaramente si osserva che $v(t) \tendstot v_{lim}$. + + \end{example} + + \begin{example} (approssimazione al moto uniformemente accelerato) + Si assumano $t << \tau$ e $v_0 << v_{lim}$. Allora + $\frac{t}{\tau} << 1$. Pertanto si può approssimare + $e^{-\frac{t}{\tau}}$ con $1 - \frac{t}{\tau}$. + In questo modo si ricava che: + + \[ v(t) = (v_0 - v_{lim})(1 - \frac{t}{\tau}) + v_{lim} = + v_0 - v_0 \frac{t}{\tau} + \frac{v_{lim}}{\tau} t + \approx v_0 + \frac{v_{lim}}{\tau} t = v_0 + gt,\] + + ossia che il moto, per queste assunzioni, è un moto + uniformemente accelerato. + \end{example} + + \begin{center} + \Large \textbf{Lavoro ed energia} + \end{center} + + Supponiamo che su un corpo di massa $m$ agisca una sola forza + costante $\vec{F}$ (e quindi che ci si stia riferendo + ad un caso unidimensionale). Supponiamo ancora che + in questa semplificazione il corpo si sia spostato + di una lunghezza $\Delta x$ dal punto $A$ al + punto $B$. In questo caso si chiamerà + lavoro svolto dalla forza $\vec{F}$ sul corpo la quantità + scalare: + + \[ L_{AB} = F \Delta x. \] + + In generale, dato il vettore spostamento + $\Delta \Vec{r}$, se $\Vec{F}$ non è l'unica forza + che agisce sul corpo, si ricava che il lavoro è il seguente: + + \[ L_{AB} = \vec{F} \cdot \Delta \Vec{r}. \] + + \begin{remark} Si osservano le seguenti proprietà. \\ + + \li Se la proiezione di $\vec{F}$ sul vettore spostamento ha + direzione opposta a $\Delta \vec{r}$ (ossia se l'angolo + compreso tra i due vettori è maggiore a $\frac{\pi}{2}$), + il lavoro è negativo. + + \li Il lavoro è additivo: $L_{AC} = L_{AB} + L_{BC}$. + + \li Il lavoro da $A$ a $B$, se $\Vec{F}$ non è costante, + può essere ricavato come una somma degli infinitesimi lavori + compiuti dalla forza, ossia: + + \[ dL_{AB} = \Vec{F}(\Vec{r}) \cdot d\Vec{r}, \] + + da cui si ricava la fondamentale identità che coinvolge + un integrale di linea: + + \[ L_{AB} = \int_{\gamma(A, B)} \vec{F}(\Vec{r}) \cdot d \vec{r}, \] + + dove $\gamma(A, B)$ è la traiettoria percorsa dal corpo + negli estremi $A$ e $B$. + \end{remark} + + +\end{document} diff --git a/Geometria I/PDF/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.pdf b/Geometria I/PDF/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.pdf index 0730c7820f7c810177b6e471b0758723c144f6cf..2d67be4ecd0d1f6b533ec7180b244c475bb11427 100644 GIT binary patch delta 133 zcmaEGiu=JS?uHh|ElfY}uo@Ye8<i>0fniG`u5nYoFhiJgKCK_#&Yc6MCFC5c5P6-B9OT!t3r=H@0`s;aL3Zd?G< C?ji?OMjrK!1@sfDSb zlZm6FfwQTZfrXQ!v4x?rk)47KK_#&Yc6MCFC5c5P6-B9OT!t3r=H@0`s;aL3Zd?G> C=p`8d diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index 67089f5..440222b 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -17,6 +17,7 @@ \newcommand{\system}[1]{\begin{cases} #1 \end{cases}} % Modalità matematica/fisica +\let\oldvec\vec \renewcommand{\vec}[1]{\underline{#1}} \newcommand{\lemmaref}[1]{\textit{Lemma \ref{#1}}} @@ -231,4 +232,10 @@ \let\oldlor\lor \let\lor\undefined -\DeclareMathOperator{\lor}{\oldlor} \ No newline at end of file +\DeclareMathOperator{\lor}{\oldlor} + +\DeclareOption{physics}{ + \let\vec\oldvec +} + +\ProcessOptions\relax \ No newline at end of file