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/dokumenty/skolni/PRA1/1-Cavendish/cavendish.pdf
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/dokumenty/skolni/PRA1/1-Cavendish/cavendish.tex
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\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{Cavendishův experiment}
\author{Jakub Kákona, kaklik@mlab.cz}
\date{9.11.2009}
\maketitle
\thispagestyle{empty}
\begin{abstract}
\end{abstract}
\section{Úvod}
\begin{enumerate}
\item Odvoďte vztah pro výpočet chyby měření.
\item Zkontrolujte měřící aparaturu.
\item Dynamickou metodou změřte časový průběh torzních kmitů kyvadla v obou možných pozicích olověných koulí.
\item Naměřenou závislost nafitujte funkcí a zjistěte její fyzikální parametry.
\item Z takto získaných údajů dopočítejte gravitační konstantu a její chybu.
\item Výsledek srovnejte s tabulkovou hodnotou gravitační konstanty.
\end{enumerate}
\section{Postup měření}
Měření silového momentu, kterým působí dvě olověné koule na torzní kyvadlo jsme provedli tak, že koule byly nejprve umístěny křížem v blízkosti hmotností na koncích torzního kyvadla tak, aby na kyvadlo působili maximálním silovým momentem způsobeným gravitačním přitahováním. Následně jsme změřili střední polohu kyvadla dynamickou metodou. A koule prohodili tak, aby nyní působily svým silovým momentem na opačnou stranu. Po opětovném změření střední polohy jsme nyní dokázali určit silový moment, kterým koule působí na kyvadlo. Obrázky dobře popisující tento postup, jsou ve zdroji \cite{Cavendish}. Při měření bylo také důležité odstranit z aparatury elektrický náboj, který by značně ovlivňoval měření, neboť elektrická síla je nesrovnatelně větší než síla gravitační. Tento problém jsme ale vyřešili uzemněním celého přístroje k vodovodnímu potrubí.
\begin{figure}
\includegraphics[width=150mm]{./poloha1.pdf}
\caption{Časový průběh výchylky torzního kyvadla v 1. pozici koulí.}
\end{figure}
\begin{figure}
\includegraphics[width=150mm]{./poloha2.pdf}
\caption{Časový průběh výchylky torzního kyvadla v 2. pozici koulí.}
\end{figure}
Fitem naměřených dat funkcí
\begin{equation}
x=A \exp (- \delta t) \sin(2 \cdot \pi /T + \varphi)
\end{equation}
Jsme dostali žádané fyzikální parametry potřebné pro výpočet gravitační konstanty.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline
A [mm] & 429.751 \pm 2.133 \\ \hline
\delta & 0.000470283 \pm 7.339e-06 \\ \hline
T [s] & 497.817 \pm 0.3167 \\ \hline
\varphi [rad] & -0.634433 \pm 0.005027 \\ \hline
s [mm] & 1062.89 \pm 0.6162 \\ \hline
\end{tabular}
\caption{Vypočtené hodnoty pro první pozici koulí.}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline
A [mm] & 254.373 \pm 0.4389 \\ \hline
\delta & 0.000457669 \pm 2.783e-06 \\ \hline
T [s]& 498.212 \pm 0.1441 \\ \hline
\varphi [rad] & -2.72838 \pm -2.72838 \\ \hline
s [mm]& 961.297 \pm 0.1507 \\ \hline
\end{tabular}
\caption{Vypočtené hodnoty pro druhou pozici koulí.}
\end{center}
\end{table}
K výpočtu je ale nutné znát ještě i některé parametry aparatury. Jako hmotnosti závaží 1.25kg , délku ramena laserového paprsku 6m poloměr kuliček na torzním kyvadle 9.55mm poloměr vnějších kouli působících na kyvadlo 50mm a délky ramena torzního kyvadla 46.5mm.
Všechny tyto hodnot jsme dosadili do odvozeného vzorce
\begin{equation}
G=\frac{\pi^2 b^2 s}{T^2 m_2 L} \cdot \frac{d^2 + \frac{5}{2} r^2}{d (1-\beta)} \qquad \beta = \frac{b^3}{(b^2 + 4d^2)^\frac{3}{2}}
\end{equation}
Konečným cílem pak bylo pomocí této gravitační konstanty spočítat hmotnost Země. Za tímto účelem jsme Zemi aproximovali dokonalou koulí. O poloměru r=6372,796km s gravitačním zrychlením u povrchu g=9,81m/s. Výpočtem nám vyšla hmotnost Země m_z=(6.21 \pm 0.06 \cdot 10^{24} kg)
\section{Diskuse}
Při porovnání našich výsledků s tabulkovou hodnotou $G = (6.67428 \pm 0.00067) * 10^-11 m^2 / kg s^2 $ se ukázalo, že námi naměřená hodnota $G = (6.41 \pm 0.04) * 10^-11 m^2 / kg s^2 $ se liší o 3.93\%. Což je pro mne osobně docela velkým překvapením, protože jsem netušil, že lze gravitační konstantu těmito prostředky vůbec změřit a natož s takovou přesností. Fakt, že se tabulková hodnota nevejde do našeho výsledku včetně chyby bych přisuzoval tomu, že naše měření mohlo být zatíženo nějakou systematickou chybou, které jsme nevěnovali dostatečnou pozornost. Například by to mohla být příliš velká prodleva při přehazování koulí z jedné do druhé polohy. Nějaká teplotní změna a podobně.
\section{Závěr}
Pomocí torzních vah jsme úspěšně určili gravitační konstantu $\kappa = (6.41 \pm 0.04) \cdot 10^{-11} m^3 / kg s^2 $ s chybou 4\% oproti tabulkové hodnotě.
\begin{thebibliography}{99}
\bibitem{Cavendish}{Zadání úlohy 1 - Cavendishův experiment}\\.\href{http://rumcajs.fjfi.cvut.cz/fyzport/FundKonst/Cavendish/cav.pdf}{http://rumcajs.fjfi.cvut.cz/fyzport/FundKonst/Cavendish/cav.pdf}
\end{thebibliography}
\end{document}

/dokumenty/skolni/PRA1/1-Cavendish/cavendish.gp
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set terminal postscript
set samples 2000
set key off
set xlabel "Time [s]"
set ylabel "Amplitude [mm]"
f(x)=a*exp(-d*x)*sin((2*pi/T)*x+fi)+s
s=1063
d=0.0002
a=430
T=498
fi=-0.1
set output "poloha2.ps"
fit f(x) "poloha2A.txt" using 1:2:3 via a,d,T,fi,s
plot "poloha2A.txt" using 1:2:3 with errorbars, f(x)
s=961
a=254
fi=-0.728
set output "poloha1.ps"
fit f(x) "poloha1B.txt" using 1:2:3 via a,d,T,fi,s
plot "poloha1B.txt" using 1:2:3 with errorbars, f(x)
#fit f(x) "poloha1A.txt" using 1:2:3 via a,d,T,fi,s
#plot "poloha1A.txt" using 1:2:3 with errorbars, "poloha2A.txt" using 1:2:3 with errorbars, "poloha1B.txt" using 1:2:3 with errorbars, f(x)
!ps2pdf poloha1.ps
!ps2pdf poloha2.ps

/dokumenty/skolni/PRA1/1-Cavendish/fit.log
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*******************************************************************************
Tue Nov 17 22:02:24 2009
FIT: data read from "poloha2A.txt" using 1:2:3
#datapoints = 64
function used for fitting: f(x)
fitted parameters initialized with current variable values
Iteration 0
WSSR : 84429 delta(WSSR)/WSSR : 0
delta(WSSR) : 0 limit for stopping : 1e-05
lambda : 25719.1
initial set of free parameter values
a = 430
d = 0.0002
T = 498
fi = -0.1
s = 1063
After 7 iterations the fit converged.
final sum of squares of residuals : 119.509
rel. change during last iteration : -3.41463e-06
degrees of freedom (FIT_NDF) : 59
rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 1.42323
variance of residuals (reduced chisquare) = WSSR/ndf : 2.02558
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = 429.751 +/- 2.116 (0.4923%)
d = 0.000470283 +/- 7.339e-06 (1.561%)
T = 497.817 +/- 0.3167 (0.06361%)
fi = -0.634433 +/- 0.005027 (0.7923%)
s = 1062.89 +/- 0.6162 (0.05797%)
correlation matrix of the fit parameters:
a d T fi s
a 1.000
d 0.862 1.000
T 0.035 0.044 1.000
fi 0.026 0.025 0.813 1.000
s -0.030 0.035 -0.037 -0.062 1.000
*******************************************************************************
Tue Nov 17 22:02:24 2009
FIT: data read from "poloha1B.txt" using 1:2:3
#datapoints = 64
function used for fitting: f(x)
fitted parameters initialized with current variable values
Iteration 0
WSSR : 354368 delta(WSSR)/WSSR : 0
delta(WSSR) : 0 limit for stopping : 1e-05
lambda : 10536.2
initial set of free parameter values
a = 254
d = 0.000470283
T = 497.817
fi = -0.728
s = 961
After 10 iterations the fit converged.
final sum of squares of residuals : 7.14462
rel. change during last iteration : -3.92415e-09
degrees of freedom (FIT_NDF) : 59
rms of residuals (FIT_STDFIT) = sqrt(WSSR/ndf) : 0.347987
variance of residuals (reduced chisquare) = WSSR/ndf : 0.121095
Final set of parameters Asymptotic Standard Error
======================= ==========================
a = 254.373 +/- 0.4091 (0.1608%)
d = 0.000457669 +/- 2.783e-06 (0.6082%)
T = 498.212 +/- 0.1441 (0.02893%)
fi = -2.72838 +/- 0.002387 (0.08749%)
s = 961.297 +/- 0.1507 (0.01568%)
correlation matrix of the fit parameters:
a d T fi s
a 1.000
d 0.815 1.000
T -0.191 -0.148 1.000
fi -0.222 -0.205 0.837 1.000
s 0.060 -0.073 0.036 0.073 1.000

/dokumenty/skolni/PRA1/1-Cavendish/poloha1.pdf
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/languagelevel where
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%
%
%End of PostScript Level 2 code
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%
% Substitute for Level 2 pattern fill codes with
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%
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/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
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end
%%EndProlog
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stroke 2355 4671 M
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stroke 3488 1042 M
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stroke 3772 1630 M
2 10 V
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stroke 4055 2893 M
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stroke 4339 3999 M
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stroke 4905 3569 M
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stroke 5472 1535 M
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3 4 V
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2 3 V
stroke 5755 1481 M
3 4 V
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2 5 V
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3 5 V
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stroke 6039 2233 M
2 9 V
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1A prvni mereni.. prvni pozice kouli
2A druhe mereni jina pozice kouli
1B treti mereni prvni pozice kouli..

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