Attachment 4220 Details for Bug 22523 – TeX-файл на страничку

TeX-файл на страничку

задача_7-26.tex (text/plain), 1.98 KB, created by Alexei V. Mezin on 2010-01-21 22:43:21 MSK

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Creator: Alexei V. Mezin

Created: 2010-01-21 22:43:21 MSK

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obsolete

>\documentclass[a4paper,12pt]{report}
>\usepackage[cp1251]{inputenc}
>\usepackage[russian]{babel}
>%\usepackage{epsfig}
>\usepackage{amssymb,amsmath}
>%\usepackage{floatflt}
>%\usepackage{setspace}
>%\usepackage{titpg}
>\setlength{\textwidth}{16.cm} 
>\setlength{\textheight}{24.0cm}
>\setlength{\oddsidemargin}{1.0cm}
>\setlength{\evensidemargin}{0.0in} 
>\setlength{\topmargin}{-0.0in}
>%opening
>\title{Çàäà÷êè ïî ôèçèêå. Ðåøåíèÿ.}
>\author{Alexei V. Mezin}
>
>\begin{document}
>%\maketitle
>
>\textbf{Çàäà÷à:}
>×åðåç íåïîäâèæíûé áëîê ñ ìîìåíòîì èíåðöèè $I$ è ðàäèóñîì $r$ ïåðåêèíóòà íèòü, ê
>îäíîìó êîíöó êîòîðîé ïîäâåøåí ãðóç ìàññû $m$. Äðóãîé êîíåö íèòè ïðèâÿçàí ê
>ïðóæèíå ñ çàêðåïë¸ííûìè íèæíèì êîíöîì. Âû÷èñëèòü ïåðèîä êîëåáàíèé ãðóçà, åñëè
>êîýôôèöèåíò óïðóãîñòè ïðóæèíû ðàâåí $k$, à íèòü íå ìîæåò ñêîëüçèòü ïî
>ïîâåðõíîñòè áëîêà.
>
>~
>
>\textbf{Ðåøåíèå:}
>×åñòíî ãîâîðÿ, ëåíèâî ðèñîâòü êàðòèíêó è íàíîñèòü íà íåå ñèëû è èõ íàïðàâëåíèÿ.
>Òàê ÷òî ñàì ïîäóìàé, êàê òàì ÷òî íàïðàâëåíî. À ÿ ïðîñòî ðåøåíèå íàïèøó, çíàêè
>ïåðåä ñèëàìè ``îò áàëäû'', íî ÷òîá îòâåò ïîëó÷èëñÿ.
>
>Ñèëîé òàæåñòè ïðèíåáðåãàåì, òàê êàê îíà íå âëèÿåò íà ïåðèîä, à òîëüêî äàåò
>ñìåùåíèå ïîëîæåíèÿ ðàâíîâåñèÿ.
>
>Çàïèøåì óðàâíåíèå ìîìåíòîâ äëÿ áëîêà:
>\begin{equation}
> I\frac{d\omega}{dt} = r(F_2 - F_1)
>\end{equation}
>ãäå $F_2$ è $F_1$ ñèëû, äåéñòâóþùèå íà áëîê ñî ñòîðîíû ïðóæèíêè è ãðóçà.
>\begin{equation}
>	\begin{array}{rcl}
>		 F_1 = m\ddot{x}\\
>		 F_2 = - kx
>	\end{array}
>\end{equation}
>
>Òàê êàê íèòü íåðàñòÿæèìàÿ è íå ïðîñêàëüçûâàåò, òî ìîæíî çàïèñàòü óñêîðåíèå
>òî÷êè íà êðàþ áëîêà
>\begin{equation}
> r\frac{d\omega}{dt} = a = \ddot x
>\end{equation}
>
>Ñîáåðåì âñå âûðàæåíèÿ â îäíîì ìåñòå
>\begin{equation}
> I\frac{\ddot x}{r} = r(-kx-m\ddot x)
>\end{equation}
>îòêóäà íåñëîæíî ïîëó÷èòü êëàññè÷åñêîå óðàâíåíèå ãàðìîíè÷åñêèõ êîëåáàíèé
>\begin{equation}
> \ddot x(I+mr^2)+kr^2x=0 \qquad \Longrightarrow \qquad \ddot x +
>\frac{kr^2}{I+mr^2}x=0
>\end{equation}
>
>Îòñþäà ñðàçó ìîæíî çàïèñàòü \textbf{îòâåò}:
>\begin{equation}
> T=2\pi\sqrt{\frac{\frac{I}{r^2}+m}{k}}
>\end{equation}
>
>
>
>\end{document}

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