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z|NYmjAQ1N-4hL)T@H=Vz!!beZuyFhs#=*+*C~yHm?Cg(&53EZG>wn_{N{4ER{`yW}6l^y1qk0-Np{ApGW&>vBhm4o9?Yk^^DA4c!D{r_uzSSy=9B;@4a zcw`wMD;MZbf?Qm%o~VzsgKf`8F#wiOIoSV4;eYKvb`aMe8neUJ?y)c&>@fR3Vgk!A z+>d<$7QBD=ocV{TVR`b8-NeZT_ z*#3C_1GBR|_H#f%U)7f3WNX`Xg_1!X6(VIUFou zvHgCF`u+Kj&@gv<9D-r-gX52-<75N>etrAz+@~<}bFy&(xuNNG03664ykNWF@ltcL zvpwncMTNOJxx~4J#K7EQ;^IR5r2l^u9a30F dC_8%tTYG0)BV!a;dV=jJ6sjjwVshdr{|_-W^s4{> diff --git a/Fisica/fisica.tex b/Fisica/fisica.tex index 264cd20..a323ef3 100644 --- a/Fisica/fisica.tex +++ b/Fisica/fisica.tex @@ -152,15 +152,15 @@ dimostrare le seguenti equazioni: \end{dcases} \end{equation} -\section{Il moto circolare uniforme} +\section{Il moto circolare} Definendo alcune grandezze fisiche in modo analogo a come vengono proposte nel m.u.a., è possibile riproporre le equazioni \ref{eq:mua} mediante l'impiego di grandezze esclusivamente angolari. -\subsection{Le equazioni del moto circolare uniforme} +\subsection{Le equazioni del moto circolare} -Si definiscono dunque le seguenti grandezze: +Si definiscano dunque le seguenti grandezze: \begin{itemize} \item $\theta$ in funzione del tempo @@ -192,49 +192,30 @@ le analoghe seguenti: Oltre all'accelerazione tangenziale, direttamente proporzionale a quella lineare, è possibile definire anche un altro tipo di -accelerazione: \textbf{l'accelerazione centripeta} ($a_c$), diretta dal +accelerazione: l'\textbf{accelerazione centripeta} ($a_c$), diretta dal corpo verso il centro della circonferenza sulla quale questo muove. Questo tipo di accelerazione è costante nel moto circolare uniforme -ed è calcolata mediante le seguente equazione: +($\alpha=0$) ed è calcolata mediante le seguente equazione: \begin{equation} - a=\frac{v^2}{r} + a_c=\frac{v^2}{r} \label{eq:acc_c} \end{equation} -\begin{proof} -La velocità può essere espressa vettorialmente nella seguente -forma $\vec{v}(-v\sin(\theta),v\cos(\theta))$, mentre -le funzioni trigonometriche possono essere sostituite utilizzando -le coordinate del punto ed il raggio della circonferenza su cui -si muove il corpo secondo le seguenti relazioni: - -\begin{equation*} - \begin{dcases} - \cos(\theta)=\frac{x}{r} \\ - \sin(\theta)=\frac{y}{r} - \end{dcases} -\end{equation*} - -Perciò, derivando la velocità, si ottiene l'accelerazione -nella seguente forma: - -\begin{equation*} - \vec{a}=(-\frac{v}{r}\cdot v_y, \frac{v}{r}\cdot v_x) -\end{equation*} +Qualora non ci si riferisse ad un moto circolare uniforme, +l'accelerazione centripeta non sarà costante, ma variabile in +funzione della velocità con la quale si muove il corpo. -Computando il modulo dell'accelerazione e tenendo -conto che $v=\sqrt{v_x^2+v_y^2}$, si ottiene -il risultato desiderato: +Inoltre, vale la seguente relazione: -\begin{equation*} - a=\frac{v^2}r -\end{equation*} - -\end{proof} +\begin{equation} + a=\sqrt{a_t^2 + a_c^2} + \label{eq:acc_moto_cir} +\end{equation} -\newpage +Nel caso del moto circolare uniforme, l'unica +accelerazione agente sul corpo è quindi quella centripeta. \subsection{Il moto circolare visualizzato vettorialmente} @@ -263,16 +244,17 @@ 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 +Basandosi sulla 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 $\vec{\omega}$. +Inoltre, $d\vec{r}$ deve essere perpendicolare sia a $\vec{r}$ che +a $\vec{\omega}$, poiché appartiene al piano $O_{xy}$. \\ \\ \\ -Perciò è possibile riscrivere $d\vec{r}$ nella seguente forma: +Perciò è possibile riscrivere $d\vec{r}$ nella seguente forma, tenendo +conto che il suo modulo è pari a $d\theta \cdot \norm{r}$ (ovvero +l'arco di circonferenza percorso per $d\theta$): \begin{equation*} d\vec{r}=\frac{d\theta \cdot \norm{r}}{\lVert \vec{w} \times @@ -317,4 +299,8 @@ 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 $\vec{a}=\vec{a_c}$). +Attraverso questa visualizzazione del moto, è possibile ricavare tutte +le formule proposte all'inizio della sezione (ed è soprattutto +possibile giustificare l'equazione \ref{eq:acc_moto_cir}). + \end{document} \ No newline at end of file