From e3193459b1b958227d9cfb4575d3ba549b1e602b Mon Sep 17 00:00:00 2001 From: JBR Date: Fri, 20 Mar 2026 20:59:49 +0100 Subject: [PATCH] Aggiunto a onde la sezione energetica, aggiunte alla lib le derivate con x e x^2 --- jacbr.sty | 2 ++ onde/onde.pdf | Bin 185941 -> 199601 bytes onde/onde.tex | 94 ++++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 96 insertions(+) diff --git a/jacbr.sty b/jacbr.sty index 758ce5b..e74ce9e 100644 --- a/jacbr.sty +++ b/jacbr.sty @@ -116,6 +116,8 @@ %derivata \newcommand{\dert}[1]{\frac{d #1}{dt}} \newcommand{\dertt}[1]{\frac{d^2 #1}{dt^2}} +\newcommand{\derx}[1]{\frac{d #1}{dx}} +\newcommand{\derxx}[1]{\frac{d^2 #1}{dx^2}} \newcommand{\der}[2]{\frac{d #1}{d #2}} %per le parentesi \newcommand{\8}{\left(} diff --git a/onde/onde.pdf b/onde/onde.pdf index c1f1c9918503cc81ca36bea50951d4be548ffa16..dfa646b18156c325ac2be2dbf254a8460ab1e063 100644 GIT binary patch delta 116997 zcmZU&V{D*pw6_5%P4DRZ{cc5%*Mu*gaY;xm`;{+SYkpByLqSe9ZZa{ 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0 +\end{align*} + +Ciò è intuitivo se pensiamo che per avere un oscillatore bisogna essere in una buca di potenziale. +Matematicamente significa che sono in un minimo e che la funzione lì ha concavità verso l'alto. +Il valore del potenziale in se è arbitrario in quanto io posso settare lo 0 ovunque voglia. + +Cerchiamo ora di capirci qualcosa in più. +\[ + F = - \derx{U} +\] + +Il potenziale nella nostra molla sappiamo essere: + +\[ + U(x) = \frac 1 2 K x^2 +\] + +Ne consegue che la forza sia: + +\[ + F = - \frac 1 2 2 K x = - K x +\] + +Studiamo l'energia meccanica totale e chiamiamo il paramentro $\alpha$ per parlare di un caso generale dove la forza abbia la stessa forma: + +\begin{align*} + E_m = K + U\\ + K = \frac 1 2 m \dert{x}\\ + U = \frac 1 2 \alpha x^2 +\end{align*} + +Cosa capiamo dall'equazione del moto? + +\begin{align*} + \dertt{x} + \omega_p^2 x = 0\\ + \omega_p = \sqrt{\frac \alpha m}\\ +\end{align*} +Riportiamo l'equazione del moto distribuendo la massa +\begin{align*} + m \dertt{x} + \alpha x = 0 \\ +\end{align*} + +Noi sappiamo, dall'analisi dimensionale qualcosa di molto interessante. +\begin{align*} + [F*x] = \si{\newton\meter} = \si{\joule}\\ + [F*\dert{x}] = \si{\newton\meter\per\second} = \si{\watt} +\end{align*} + +Moltiplichiamo l'equazione del moto per $\dert{x}$. +\[ + m \dertt{x} \dert{x} + \alpha x \dert{x} = 0 +\] + +Questo tipo di equazione lo troviamo quando deriviamo un quadrato, infatti: +\[ + \dert{f^2(t)} = 2 f(t) * \dert{f(t)}\\ +\] + +Sostituendo questo risultato dentro l'equazione del moto abbiamo: +\begin{align*} + m \frac 1 2 \frac{d}{dt} \8 \dert{x} \9^2 + \alpha \frac 1 2 \dert{x^2} = 0\\ + \frac{d}{dt} \8 \frac 1 2 m \8\dert{x}\9^2 + \frac 1 2 \alpha x^2 \9 = 0 \\ + \imp \frac 1 2 m \8 \dert{x} \9^2 + \frac 1 2 \alpha x^2 = const. \\ + K + U = const. +\end{align*} + +Abbiamo ritrovato dall'equazione del moto la conservazione dell'energia meccanica. +Possiamo quindi provare a vedere cosa risulta inserendo la soluzione dell'equazione. + +\begin{align*} + x(t) = A \cos(\omega_p t + \varphi)\\ + \dert{x} = - A \omega_p \sin(\omega_p t + \varphi)\\ + \text{Sostituendo}\\ + E_m = \frac m 2 A^2 \omega_p^2 \sin^2(\omega_p t + \varphi) + \frac \alpha 2 A^2 \cos^2(\omega_p t + \varphi)\\ + \text{Ricordando } \omega_p = \sqrt{\frac \alpha m}\\ + E_m = \frac \alpha 2 A^2 \8 \sin^2(\omega_p t + \varphi) + \cos^2(\omega_p t + \varphi) \9\\ + E_m = \frac \alpha 2 A^2 +\end{align*} + +Anche in questo caso si nota come $E_m$ è una costante. + +Un altra osservazione molto interessante giunge dallo studio di $x$ e $\dert{x}$ in quanto queste 2 funzioni sono sfasate di $\frac \pi 2$, si dice che sono in quadratura di fase. + + \end{document}
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