MLABlibrarysvnkaklikMLAB_E8magsvn

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/dokumenty/PRA1/11-Rotace/Obrazky/Steiner.jpg
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/dokumenty/PRA1/11-Rotace/Rotace.pdf
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/dokumenty/PRA1/11-Rotace/Rotace.tex
18,24 → 18,56
\end{abstract}
\section{Úvod}
\begin{enumerate}
.
\item V domácí přípravě odvoďte vzorec pro výpočet momentu setrvačnosti válce a dutého válce.
\item Změřte momenty setrvačnosti přiložených rotačních objektů experimentálně a porovnejte je s hodnotami z teoretických vzorců. Měření proveďte alespoň pětkrát. Použijte disk, disk + prstenec a pomocí nich stanovte moment setrvačnosti samotného prstence.
\item Změřte moment setrvačnosti disku, umístěného na dráze mimo osu rotace a pomocí výsledků z předchozího úkolu ověřte platnost Steinerovy věty.
\item Ověřte zákon zachování momentu hybnosti. Do protokolu přiložte graf závislosti úhlové rychlosti rotace na čase.
\item Změřte rychlost precese gyroskopu jak přímo senzorem, tak i nepřímo z měření rychlosti rotace disku. Měření proveďte alespoň pětkrát. Obě hodnoty porovnejte.
\end{enumerate}
\section{Postup měření}
\subsection{Měření momentu setrvačnosti}
Moment hybnosti jsme měřili roztáčením tělesa přes kladku pomocí definovaného závaží, Ze záznamu časového průběhu rychlosti lze proložením přímkou určit moment setrvačnosti tělesa. Změřené výsledky jsou v následujících grafech.
Moment hybnosti jsme měřili roztáčením tělesa přes kladku pomocí definovaného závaží, Ze záznamu časového průběhu rychlosti lze proložením přímkou určit moment setrvačnosti tělesa. Změřená data jsou vidět v následujících grafech.
\begin{figure}
\includegraphics[width=150mm]{./data/rotace1.pdf}
\caption{Data z mereni momentu setrvacnosti disku}
\end{figure}
\begin{figure}
\includegraphics[width=150mm]{./data/rotace_prstenec1.pdf}
\caption{Data z mereni momentu setrvacnosti disku a prstence}
\end{figure}
\subsection{Zachování momentu hybnosti}
Při měření jsme postupovali prakticky totožně, jako při měření momentu setrvačnosti, ale hmotnost byla rozložena ve dvou závažích, které jsme během otáčení zatažením za šňůrku sesunuli k sobě.
\includegraphics[width=150mm]{./data/rotace_zmena.pdf}
\subsection{Precese gyroskopu}
Precesi gyroskopu jsme měřili tak že v jsme jej v klidu vyvážili a na straně s gyroskopem zatížili definovaným závažím o hmotnosti 17,9 g. Které působí na gyroskop tíhovou silou. Která způsobuje precesi gyroskopu kolem svislé osy.
Precesi gyroskopu jsme měřili tak že v jsme jej v klidu vyvážili a na straně s gyroskopem zatížili definovaným závažím o hmotnosti 17,9 g. Které působí na gyroskop tíhovou silou. Která způsobuje precesi gyroskopu kolem svislé osy.
\begin{table}[htbp]
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|l|}{$\omega_{P} [rad/s]$} & \multicolumn{1}{l|}{$\delta_{\omega P} [\%]$} & \multicolumn{1}{l|}{$\omega_S [rad/s]$} & \multicolumn{1}{l|}{$\Omega_S [rad/s]$} & \multicolumn{1}{l|}{$\Omega_M [rad/s]$} & \multicolumn{1}{l|}{$\delta_{\omega M} [rad/s]$} & \multicolumn{1}{l|}{$\Delta_{\Omega} [-]$} & \multicolumn{1}{l|}{$\delta_{\Omega} [\%]$} \\ \hline
77.3891 & 0.030 & 157.2051 & 0.0181 & 0.0107 & 59.280 & 0.0074 & 41.04 \\ \hline
89.5982 & 0.015 & 182.0062 & 0.0157 & 0.0154 & 66.040 & 0.0002 & 1.42 \\ \hline
97.5382 & 0.022 & 198.1352 & 0.0144 & 0.0086 & 199.800 & 0.0058 & 40.13 \\ \hline
71.8631 & 0.032 & 145.9798 & 0.0195 & 0.0309 & 10.810 & -0.0114 & 58.32 \\ \hline
61.4821 & 0.030 & 124.8923 & 0.0228 & 0.0199 & 28.180 & 0.0029 & 12.81 \\ \hline
\end{tabular}
\caption{Změřené a vypočtené hodnoty precese}
\label{Precese}
\end{table}
\section{Diskuse}
Největším problémem bylo měření zachování momentu hybnosti, kdy aparatura přecházela při změně konfirugace závaží do neopakovatelně definovaných stavů, což způsobilo značnou chybu.
Největším problémem bylo měření zachování momentu hybnosti, kdy aparatura přecházela při změně konfirugace závaží do neopakovatelně definovaných stavů, což pravděpodobně způsobylo značnou chybu.
\section{Závěr}
Z naměřených dat lze potvrdit že momenty setvačností objektů rotujících na téže ose se sčítají.
\begin{thebibliography}{99}
\bibitem{oscilace}{Zadání úlohy 11 - Dynamika rotačního pohybu}. \href{http://rumcajs.fjfi.cvut.cz/fyzport/Mechanika/HarmOscilator/osc.pdf}{http://rumcajs.fjfi.cvut.cz/fyzport/Mechanika/HarmOscilator/osc.pdf}.
\bibitem{pohl_kyv}{Pohlovo torzni kyvadlo}. \href{http://praktika.fjfi.cvut.cz/PohlKyv}{http://praktika.fjfi.cvut.cz/PohlKyv}
\bibitem{oscilace}{Zadání úlohy 11 - Dynamika rotačního pohybu}. \href{http://praktika.fjfi.cvut.cz/RotacniPohyb/}{http://praktika.fjfi.cvut.cz/RotacniPohyb/}.
\end{thebibliography}
\end{document}

/dokumenty/PRA1/11-Rotace/data/rotace.gp
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set terminal postscript
set output "rotace1.ps"
set key off
set xlabel "Time [s]"
set ylabel "Velocity [rad/s]"
#set data style lines
plot "rotace1.txt" using 1:2
!ps2pdf rotace1.ps
set output "rotace_prstenec1.ps"
plot "rotace_prstenec1.txt" using 1:2
!ps2pdf rotace_prstenec1.ps

/dokumenty/PRA1/11-Rotace/data/rotace1.pdf
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/dokumenty/PRA1/11-Rotace/data/rotace1.ps
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/dokumenty/PRA1/11-Rotace/data/rotace1.txt
1,79 → 1,77
Velocity, Run #16
Time ( s ) Velocity ( rad/s )
0,357259 2,99
0,551776 3,51
0,720205 3,98
0,870275 4,41
1,006880 4,80
1,133120 5,16
1,251110 5,50
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3,664122 12,24
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4,919902 15,78
4,959563 15,90
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5,076865 16,26
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4.196271 13.73
4.241727 13.92
4.286793 13.97
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4.463375 14.53
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4.549293 14.76
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5.038049 16.12
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5.115435 16.32
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/dokumenty/PRA1/11-Rotace/data/rotace_prstenec.pdf
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/dokumenty/PRA1/11-Rotace/data/rotace_prstenec1.txt
1,80 → 1,78
Velocity, Run #16
Time ( s ) Velocity ( rad/s )
0,329202 2,57
0,556018 3,00
0,753661 3,38
0,931450 3,71
1,094524 4,01
1,246117 4,29
1,388272 4,56
1,522239 4,83
1,649424 5,06
1,770811 5,30
1,887089 5,51
1,998908 5,73
2,106752 5,92
2,211195 6,11
2,312423 6,30
2,410724 6,48
2,506318 6,67
2,599159 6,87
2,689624 7,02
2,777927 7,21
2,864171 7,36
2,948568 7,53
3,031233 7,67
3,112378 7,82
3,192012 7,96
3,270247 8,10
3,347118 8,25
3,422459 8,43
3,496486 8,54
3,569362 8,70
3,641086 8,82
3,711707 8,98
3,781304 9,08
3,850019 9,21
3,917810 9,33
3,984763 9,44
4,050875 9,57
4,115994 9,73
4,180198 9,84
4,243640 9,97
4,306377 10,07
4,368353 10,21
4,429629 10,30
4,490307 10,41
4,550337 10,52
4,609811 10,61
4,668710 10,73
4,726894 10,87
4,784365 10,99
4,841297 11,08
4,897767 11,17
4,953660 11,31
5,009031 11,39
5,063953 11,49
5,118367 11,60
5,172397 11,66
5,226027 11,77
5,279089 11,91
5,331587 12,03
5,383678 12,10
5,435398 12,20
5,486691 12,30
5,537597 12,39
5,588124 12,48
5,638292 12,57
5,688173 12,63
5,737722 12,73
5,786812 12,86
5,835427 12,98
5,883716 13,04
5,931762 13,12
5,979435 13,24
6,026768 13,31
6,073832 13,40
6,120590 13,48
6,167126 13,52
6,213409 13,63
6,259310 13,75
0.329202 2.57
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0.753661 3.38
0.931450 3.71
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1.522239 4.83
1.649424 5.06
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1.887089 5.51
1.998908 5.73
2.106752 5.92
2.211195 6.11
2.312423 6.30
2.410724 6.48
2.506318 6.67
2.599159 6.87
2.689624 7.02
2.777927 7.21
2.864171 7.36
2.948568 7.53
3.031233 7.67
3.112378 7.82
3.192012 7.96
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4.668710 10.73
4.726894 10.87
4.784365 10.99
4.841297 11.08
4.897767 11.17
4.953660 11.31
5.009031 11.39
5.063953 11.49
5.118367 11.60
5.172397 11.66
5.226027 11.77
5.279089 11.91
5.331587 12.03
5.383678 12.10
5.435398 12.20
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5.537597 12.39
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5.638292 12.57
5.688173 12.63
5.737722 12.73
5.786812 12.86
5.835427 12.98
5.883716 13.04
5.931762 13.12
5.979435 13.24
6.026768 13.31
6.073832 13.40
6.120590 13.48
6.167126 13.52
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6.259310 13.75

/dokumenty/PRA1/11-Rotace/data/rotace_zmena.gp
0,0 → 1,8
set terminal postscript
set output "rotace_zmena.ps"
set key off
set xlabel "Time [s]"
set ylabel "Velocity [rad/s]"
#set data style lines
plot "rotace_zmena1.txt" using 1:2
!ps2pdf rotace_zmena.ps

/dokumenty/PRA1/11-Rotace/data/rotace_zmena.pdf
Cannot display: file marked as a binary type.
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Property changes:
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0,0 → 1,681
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/Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto
neg 0 rlineto closepath} bind def
/Square {dup Rec} bind def
/Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def
/S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def
/S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def
/S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def
/S3 {BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare} bind def
/S4 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def
/S5 {BL [] 0 setdash 2 copy 2 copy vpt Square fill
exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def
/S6 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare} bind def
/S7 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill
2 copy vpt Square fill Bsquare} bind def
/S8 {BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare} bind def
/S9 {BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare} bind def
/S10 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill
Bsquare} bind def
/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill
Bsquare} bind def
/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def
/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
2 copy vpt Square fill Bsquare} bind def
/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
2 copy exch vpt sub exch vpt Square fill Bsquare} bind def
/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def
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/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def
/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def
/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def
/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def
/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def
/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def
/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def
/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def
/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def
/D12 {gsave translate 45 rotate 0 0 S12 stroke grestore} bind def
/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def
/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def
/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def
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hpt neg vpt 1.62 mul V
hpt 2 mul 0 V
hpt neg vpt -1.62 mul V closepath stroke} def
/PentE {stroke [] 0 setdash gsave
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
closepath stroke grestore} def
/CircE {stroke [] 0 setdash
hpt 0 360 arc stroke} def
/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def
/DiaW {stroke [] 0 setdash vpt add M
hpt neg vpt neg V hpt vpt neg V
hpt vpt V hpt neg vpt V Opaque stroke} def
/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M
0 vpt2 neg V hpt2 0 V 0 vpt2 V
hpt2 neg 0 V Opaque stroke} def
/TriUW {stroke [] 0 setdash vpt 1.12 mul add M
hpt neg vpt -1.62 mul V
hpt 2 mul 0 V
hpt neg vpt 1.62 mul V Opaque stroke} def
/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M
hpt neg vpt 1.62 mul V
hpt 2 mul 0 V
hpt neg vpt -1.62 mul V Opaque stroke} def
/PentW {stroke [] 0 setdash gsave
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
Opaque stroke grestore} def
/CircW {stroke [] 0 setdash
hpt 0 360 arc Opaque stroke} def
/BoxFill {gsave Rec 1 setgray fill grestore} def
/Density {
/Fillden exch def
currentrgbcolor
/ColB exch def /ColG exch def /ColR exch def
/ColR ColR Fillden mul Fillden sub 1 add def
/ColG ColG Fillden mul Fillden sub 1 add def
/ColB ColB Fillden mul Fillden sub 1 add def
ColR ColG ColB setrgbcolor} def
/BoxColFill {gsave Rec PolyFill} def
/PolyFill {gsave Density fill grestore grestore} def
/h {rlineto rlineto rlineto gsave closepath fill grestore} bind def
%
% PostScript Level 1 Pattern Fill routine for rectangles
% Usage: x y w h s a XX PatternFill
% x,y = lower left corner of box to be filled
% w,h = width and height of box
% a = angle in degrees between lines and x-axis
% XX = 0/1 for no/yes cross-hatch
%
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PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate
PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec
gsave 1 setgray fill grestore clip
currentlinewidth 0.5 mul setlinewidth
/PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def
0 0 M PFa 5 get rotate PFs -2 div dup translate
0 1 PFs PFa 4 get div 1 add floor cvi
{PFa 4 get mul 0 M 0 PFs V} for
0 PFa 6 get ne {
0 1 PFs PFa 4 get div 1 add floor cvi
{PFa 4 get mul 0 2 1 roll M PFs 0 V} for
} if
stroke grestore} def
%
/languagelevel where
{pop languagelevel} {1} ifelse
2 lt
{/InterpretLevel1 true def}
{/InterpretLevel1 Level1 def}
ifelse
%
% PostScript level 2 pattern fill definitions
%
/Level2PatternFill {
/Tile8x8 {/PaintType 2 /PatternType 1 /TilingType 1 /BBox [0 0 8 8] /XStep 8 /YStep 8}
bind def
/KeepColor {currentrgbcolor [/Pattern /DeviceRGB] setcolorspace} bind def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke}
>> matrix makepattern
/Pat1 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke
0 4 M 4 8 L 8 4 L 4 0 L 0 4 L stroke}
>> matrix makepattern
/Pat2 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop 0 0 M 0 8 L
8 8 L 8 0 L 0 0 L fill}
>> matrix makepattern
/Pat3 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop -4 8 M 8 -4 L
0 12 M 12 0 L stroke}
>> matrix makepattern
/Pat4 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop -4 0 M 8 12 L
0 -4 M 12 8 L stroke}
>> matrix makepattern
/Pat5 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop -2 8 M 4 -4 L
0 12 M 8 -4 L 4 12 M 10 0 L stroke}
>> matrix makepattern
/Pat6 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L
0 -4 M 8 12 L 4 -4 M 10 8 L stroke}
>> matrix makepattern
/Pat7 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L
12 0 M -4 8 L 12 4 M 0 10 L stroke}
>> matrix makepattern
/Pat8 exch def
<< Tile8x8
/PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L
-4 0 M 12 8 L -4 4 M 8 10 L stroke}
>> matrix makepattern
/Pat9 exch def
/Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def
/Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def
/Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def
/Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def
/Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def
/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def
/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def
} def
%
%
%End of PostScript Level 2 code
%
/PatternBgnd {
TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse
} def
%
% Substitute for Level 2 pattern fill codes with
% grayscale if Level 2 support is not selected.
%
/Level1PatternFill {
/Pattern1 {0.250 Density} bind def
/Pattern2 {0.500 Density} bind def
/Pattern3 {0.750 Density} bind def
/Pattern4 {0.125 Density} bind def
/Pattern5 {0.375 Density} bind def
/Pattern6 {0.625 Density} bind def
/Pattern7 {0.875 Density} bind def
} def
%
% Now test for support of Level 2 code
%
Level1 {Level1PatternFill} {Level2PatternFill} ifelse
%
/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall
currentdict end definefont pop
end
%%EndProlog
%%Page: 1 1
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%%Trailer
%%DocumentFonts: Helvetica
%%Pages: 1

/dokumenty/PRA1/11-Rotace/data/rotace_zmena1.txt
1,51 → 1,48
Velocity, Run #16
Time ( s ) Velocity ( rad/s )
0,337444 2,69
0,571983 2,67
0,808548 2,64
1,047003 2,63
1,286544 2,62
1,526028 2,63
1,742193 3,25
1,885989 6,65
1,969245 8,72
2,041020 8,79
2,112857 8,70
2,185412 8,62
2,258495 8,58
2,331903 8,54
2,405546 8,52
2,479318 8,51
2,553192 8,50
2,627072 8,51
2,701073 8,47
2,775253 8,47
2,849642 8,43
2,924468 8,37
2,999692 8,34
3,075223 8,30
3,150961 8,29
3,226766 8,29
3,302598 8,29
3,378400 8,29
3,454329 8,26
3,530475 8,25
3,606897 8,20
3,683821 8,14
3,761199 8,10
3,838931 8,06
3,916885 8,06
3,994897 8,05
4,072940 8,05
4,150938 8,06
4,229046 8,03
4,307354 8,02
4,385874 7,98
4,464818 7,93
4,544137 7,91
4,623762 7,87
4,703618 7,86
4,783556 7,86
4,863533 7,85
4,943482 7,86
5,023550 7,83
0.571983 2.67
0.808548 2.64
1.047003 2.63
1.286544 2.62
1.526028 2.63
1.742193 3.25
1.885989 6.65
1.969245 8.72
2.041020 8.79
2.112857 8.70
2.185412 8.62
2.258495 8.58
2.331903 8.54
2.405546 8.52
2.479318 8.51
2.553192 8.50
2.627072 8.51
2.701073 8.47
2.775253 8.47
2.849642 8.43
2.924468 8.37
2.999692 8.34
3.075223 8.30
3.150961 8.29
3.226766 8.29
3.302598 8.29
3.378400 8.29
3.454329 8.26
3.530475 8.25
3.606897 8.20
3.683821 8.14
3.761199 8.10
3.838931 8.06
3.916885 8.06
3.994897 8.05
4.072940 8.05
4.150938 8.06
4.229046 8.03
4.307354 8.02
4.385874 7.98
4.464818 7.93
4.544137 7.91
4.623762 7.87
4.703618 7.86
4.783556 7.86
4.863533 7.85
4.943482 7.86
5.023550 7.83