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cleartomark
%%EndFont
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 48 /prime put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 20 /kappa put
dup 73 /I put
dup 80 /P put
dup 86 /V put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
currentdict end
currentfile eexec
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986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE
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5D0740C16ADF0AFD47C4938F373575C6CA91E46D88DE24E682DEC44B57EA8AF8
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53F994EC95CA3D23F1909BE3B7CF6357C31EC4790B7B85DF7D942056C0E809EA
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cleartomark
%%EndFont
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 20 /kappa put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 67 /C put
dup 84 /T put
dup 86 /V put
dup 98 /b put
dup 103 /g put
dup 104 /h put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
currentdict end
currentfile eexec
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5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC
4391C9DF440285B8FC159D0E98D4258FC57892DCC57F7903449E07914FBE9E67
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cleartomark
%%EndFont
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 16 /dotlessi put
dup 19 /acute put
dup 20 /caron put
dup 46 /period put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 68 /D put
dup 77 /M put
dup 80 /P put
dup 82 /R put
dup 85 /U put
dup 90 /Z put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 122 /z put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /fi put
dup 16 /dotlessi put
dup 19 /acute put
dup 20 /caron put
dup 23 /ring put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 47 /slash put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 61 /equal put
dup 64 /at put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 72 /H put
dup 74 /J put
dup 75 /K put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 90 /Z put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-34 -251 988 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 65 /A put
dup 97 /a put
dup 98 /b put
dup 107 /k put
dup 114 /r put
dup 115 /s put
dup 116 /t put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
currentdict end
currentfile eexec
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/dokumenty/PRA1/4-Poisson/poisson.tex
0,0 → 1,124
\documentclass[12pt,a4paper,oneside]{article}
\usepackage[colorlinks=true]{hyperref}
\usepackage[utf8]{inputenc}
\usepackage[czech]{babel}
\usepackage{graphicx}
\textwidth 16cm \textheight 24.6cm
\topmargin -1.3cm
\oddsidemargin 0cm
\pagestyle{empty}
\begin{document}
\title{Určení Poissonovy konstanty vzduchu}
\author{Jakub Kákona, kaklik@mlab.cz}
\date{30.11.2009}
\maketitle
\thispagestyle{empty}
\begin{abstract}
 
\end{abstract}
 
\section{Úvod}
\begin{enumerate}
\item Změřte kompresí plynu objem baňky systému s kmitajícím pístkem.
\item Změřte Poissonovu konstantu metodou adiabatické expanze a současně metodou kmitajícího pístku.
\item Oba výsledky porovnejte. Výsledek metody kmitajícího pístku považujte za tabulkovou hodnotu Poissonovy konstanty.
\end{enumerate}
 
\section{Úvod}
\subsection{Modul pružnosti v tahu}
Poissonova konstanta $\kappa $ je poměr měrného tepla $C_{P}$ při stálém tlaku ke měrnému teplu $C_{V}$ při stálém objemu
\begin{displaymath} \kappa = \frac{C_P }{C_V }. \end{displaymath}
 
$\kappa $ má ve všech soustavách stejnou číselnou hodnotu. Pro všechny plyny je poměr specifických tepel $\kappa $ větší než 1 a závisí na počtu atomů v molekule plynu. Hodnotu $\kappa $ můžeme určit ze změny tlaku při adiabatickém ději, který je popsán Poissonovou rovnicí
 
\begin{displaymath} pV^\kappa = konst. \end{displaymath}
 
kde $p$, $V$ jsou tlak a objem plynu.
 
Pro měření Poissonovy konstanty vzduchu jsme mimo jiné použili Clémentovu-Désormesovu metodu. Jejíž princip spočívá ve stlačení vzduchu ve velké báni, aby měl proti vnějšímu barometrickému tlaku $b$ přetlak $h$. Ten odečteme jako výškový rozdíl hladin otevřeného manometru, který je k báni připojen. Tlak vzduchu v báni je pak
\begin{displaymath} p_1 = b + h. \end{displaymath}
 
Příslušný objem vzduchu před adiabatickou expanzí je $V_{1}$, po adiabatické expanzi $V_{2}$. Teplotu plynu, která by měla být shodná s teplotou okolí báně rozumíme $T_{1}$. Vnitřek báně je od vnějšího vzduchu oddělen membránovým závěrem. Otevřeme-li jej dostatečně ale na velmi krátkou dobu, vyrovnají se tlaky uvnitř a vně na hodnotu barometrického tlaku $b$.
 
\begin{displaymath} p_2 = b. \end{displaymath}
 
Proběhne tedy adiabatická expanze vzduchu v báni z počátečních podmínek $V_{1}$, $T_{1}$, $p_{1}$ do stavu určeného veličinami $V_{2}$, $T_{2} \quad < T_{1}$, $p_{2}$ (kde $V_{2}$ je objem báně). Po dosti dlouhé době se teplota vzduchu v báni vyrovná na vnější teplotu $T_{1}$ a tlak přitom stoupne o přírůstek $ h'$, který změříme. Tato změna je izochorická. Celkový tlak je
\begin{displaymath} p_3 = b + h^I. \end{displaymath}
 
Pro první (adiabatickou) změnu stavu vzduchu dostaneme z Poissonovy rovnice za předpokladu, že vzduch můžeme považovat za ideální plyn,
\begin{displaymath} \frac{p_1 }{p_2 } = \left( {\frac{V_2 }{V_1 }} \right)^\kappa . \end{displaymath}
 
Změna ze stavu $V_{1}$, $T_{1}$, $p_{1}$ do stavu $V_{3}= V_{2}$, $T_{3} = T_{1}$, $p_{3}$ je izotermická a platí pro ni Boyle-Mariotteův zákon
\begin{displaymath} \frac{p_1 }{p_3 } = \frac{V_2 }{V_1 }. \end{displaymath}
 
Sloučíme-li předchozí rovnice dostaneme
\begin{displaymath} \kappa=\frac{log(b+h)-logb}{log(b+h)-log(b+h')} \end{displaymath}
 
Po odlogaritmování pak dostaneme výsledný vztah
 
\begin{displaymath} \kappa=\frac{h}{h-h'} \end{displaymath}
 
Který ovšem neuvažuje chybu způsobenou nenulovou dobou otevření ventilu. Z toho důvodu je přesnější vytvořit aparaturu pracující na analogickém principu a ale s definovanou dobou otevření ventilu.
Velice efektivním řešením tohoto problému je využít vlastností harmonických kmitů a uzávěr (píst) uvést do dynamické rovnováhy s protékajícím vzduchem.
Pro takto zkonstruovanou aparaturu založenou na kmitajícím pístu lze odvodit vztah pro Poissonovu konstantu jako
\begin{displaymath} \kappa=\frac{4mV}{T^2 pr^4} \end{displaymath}
 
\section{Postup měření}
 
Během měření Poissonovy konstanty Clement-Desormesovo metodou jsme se snažili o maximální zkrácení času otevření ventilu, po krátkém tréninku bylo jasné, že nemá smysl dobu otevření snižovat pod mez zhruba 70ms neboť se nestačí dostatečně vyrovnat tlak v aparatuře s atmosférickým tlakem.
 
Uvedené výsledky jsou proto nad touto hranicí. Průměr z naměřených hodnot je $1,40 \pm 0,02$
 
\begin{table}[htbp]
\caption{Naměřené hodnoty Poissonovy konstanty Clement-Desormesovo metodou}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
h1 [mm] & t[ms] & h2 [mm] & t[ms] & $ \kappa $ \\ \hline
31,5 & 88 & 9,10 & 88 & 1,41 \\ \hline
31,2 & 85 & 9,00 & 85 & 1,41 \\ \hline
31,4 & 103 & 9,20 & 103 & 1,41 \\ \hline
31,2 & 130 & 8,70 & 130 & 1,39 \\ \hline
31,4 & 138 & 8,50 & 138 & 1,37 \\ \hline
\end{tabular}
\end{center}
\label{Clement-Desormes}
\end{table}
 
V celém průběhu měření jsme paralelně měřili Poissonovu konstantu i pomocí kmitajícího pístu. A tabulka \ref{Kmit} udává naměřená i vypočtená data, jako objem baňky jsme zvolili asistentem doporučený objem $0,001067 m^3$ jelikož ho nebylo možné změřit žádným z v návodu \cite{objemy} popsaných postupů. Průměrnou změřenou hodnotou je $1,29 \pm 0,03 $. Ostatní konstanty potřebné pro výpočet jsou převzaté z návodu k úloze.
 
\begin{table}[htbp]
\caption{Pocty period pistu v petiminutovem mericim intervalu a vypoctena hodnota Poissovnovy konstanty}
\begin{center}
\begin{tabular}{|c|c|}
\hline
period & $\kappa$ \\ \hline
859 & 1,27 \\ \hline
862 & 1,28 \\ \hline
865 & 1,29 \\ \hline
863 & 1,29 \\ \hline
864 & 1,29 \\ \hline
865 & 1,29 \\ \hline
866 & 1,29 \\ \hline
867 & 1,30 \\ \hline
858 & 1,27 \\ \hline
862 & 1,28 \\ \hline
866 & 1,29 \\ \hline
870 & 1,31 \\ \hline
872 & 1,31 \\ \hline
\end{tabular}
\end{center}
\label{Kmit}
\end{table}
\section{Diskuse}
Z výsledků naměřených v chronologickém pořadí lze vypozorovat mírnou stabilní variaci v průběhu měření. Tento jev by mohl být způsoben buď změnou vlastností aparatury, nebo skutečně změnou složení vzduchu. Nejpravděpodobnější asi bude kombinace obou těchto jevů. Kdy při měření metodou kmitajícího pístu může docházet k mírným změnám v průtoku vzduchu a navíc i doby po které čítač počítá kmity válečku nemusí být úplně časově ekvidistantní. Zřejmé také je, že v místnosti, která není příliš větraná se pohybuje značné množství lidí kteří souhrnně obohacují vzduch o víceatomové molekuly..
 
\section{Závěr}
Závěrem lze říci že hodnota Poissonovy konstanty vzduchu za běžných podmínek leží někde okolo hodnoty 1,35. Což je v souladu s předpokladem, že vzduch obsahuje převážně dvou nebo více atomové plyny.
 
\begin{thebibliography}{99}
\bibitem{objemy}{Měření dutých objemů vážením a kompresí plynu} \href{http://praktika.fjfi.cvut.cz/objemy/}{http://praktika.fjfi.cvut.cz/objemy/}
\end{thebibliography}
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