From 72cf5b79a1d2271464b848ddd400a970a19da50b Mon Sep 17 00:00:00 2001 From: nihonium Date: Thu, 23 Mar 2023 23:02:03 +0300 Subject: [PATCH] nya --- 2.1.1/2.1.1(q1).txt | 4 + 2.1.1/2.1.1(q2).txt | 4 + 2.1.1/2.1.1.graph | 33 + 2.1.1/2.1.1.png | 2184 +++++++++++++++++ 2.1.1/2.1.1.ps | 2184 +++++++++++++++++ 2.1.1/2.1.3(22.8).data | 5 + 2.1.1/data.txt | 68 +- 2.1.1/data_fucked_up.txt | 68 + 2.1.1/images/1.png | Bin 78522 -> 19788 bytes 2.1.1/images/2.1.1.ps | 2184 +++++++++++++++++ 2.1.1/images/2.png | Bin 0 -> 15960 bytes 2.1.1/images/3.png | Bin 0 -> 78522 bytes .../lab-gnuplottex-fig1-eps-converted-to.pdf | Bin 0 -> 4586 bytes 2.1.1/lab-gnuplottex-fig1.eps | 1730 +++++++++++++ 2.1.1/lab-gnuplottex-fig1.gnuplot | 22 + 2.1.1/lab-gnuplottex-fig1.tex | 215 ++ 2.1.1/lab.aux | 26 + 2.1.1/lab.gnuploterrors | 1 + 2.1.1/lab.log | 1575 ++++++++++++ 2.1.1/lab.pdf | Bin 231185 -> 314091 bytes 2.1.1/lab.tex | 620 ++--- 2.1.1/lab.tex1 | 275 +++ 2.1.1/texput.log | 21 + 2.1.2/1.jpg | Bin 0 -> 9596 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Malinovskii_V2.5.1/plot2.png create mode 100644 Malinovskii_V2.5.1/plot3 create mode 100644 Malinovskii_V2.5.1/plot3.png create mode 100644 Malinovskii_V2.5.1/texput.log create mode 100644 Senokosov_A/2.5.1.aux create mode 100644 Senokosov_A/2.5.1.log create mode 100644 Senokosov_A/2.5.1.out create mode 100644 Senokosov_A/2.5.1.pdf create mode 100644 Senokosov_A/2.5.1.tex create mode 100644 Senokosov_A/2.5.1.xlsx create mode 100644 Senokosov_A/251 (2).docx create mode 100644 Senokosov_A/igla.jpg create mode 100644 Senokosov_A/ust.jpg create mode 100644 mls.py diff --git a/2.1.1/2.1.1(q1).txt b/2.1.1/2.1.1(q1).txt new file mode 100644 index 0000000..fffe80e --- /dev/null +++ b/2.1.1/2.1.1(q1).txt @@ -0,0 +1,4 @@ +1.0367 3.78 1 +0.8644 3.12 1 +0.7919 2.95 1 +0.6618 2.51 1 diff --git a/2.1.1/2.1.1(q2).txt b/2.1.1/2.1.1(q2).txt new file mode 100644 index 0000000..9b1aa36 --- /dev/null +++ b/2.1.1/2.1.1(q2).txt @@ -0,0 +1,4 @@ +0.984 4.57 1 +0.7049 3.22 1 +0.5844 2.65 1 +0.5005 2.29 1 diff --git a/2.1.1/2.1.1.graph b/2.1.1/2.1.1.graph new file mode 100644 index 0000000..2ae28b4 --- /dev/null +++ b/2.1.1/2.1.1.graph @@ -0,0 +1,33 @@ +#!/usr/bin/gnuplot -persist +set terminal postscript solid color eps enhanced +#set terminal png enhanced +set output "2.1.1.ps" +set encoding koi8r + +#set label 11 center at graph 0.5,char 0.5 "Рис.2 Результаты измерений напряжения V_{B} в зависимости от тока I_{A} для проволок разной длины l и их линейная аппроксимация y = kx" font "Arial,10" +set grid +set xlabel "N, Вт" +set ylabel '/Symbol D T, ^\circ C' + +set multiplot +set yrange [0:8] +set xrange [0:1.5] +set key spacing 2 +set key bottom right +set key off + +set style line 1 lt 1 pt 7 ps 0.5 lc rgb "red" +set style line 2 lt 1 pt 7 ps 0.5 lc rgb "blue" +#set style line 3 lt 1 pt 7 ps 0.5 lc rgb "green" +set style line 4 lc black + +set key box linestyle 4 +set key opaque Left + +plot "2.1.1(q1).txt" using 1:2:3 notitle with yerrorbars linestyle 1 +plot "2.1.1(q2).txt" using 1:2:3 notitle with yerrorbars linestyle 2 +#plot "1.1.1(20).txt" using 2:1:3 notitle with yerrorbars linestyle 3 + +plot 3.68*x smooth csplines title "q_{1}" linestyle 1, \ + 4.6*x smooth csplines title "q_{2}" linestyle 2 +# 2.006*x smooth csplines title "l = 20 см\nk=2.006^+/-0.023 Ом" linestyle 3 diff --git a/2.1.1/2.1.1.png b/2.1.1/2.1.1.png new file mode 100644 index 0000000..0f5ed87 --- /dev/null +++ b/2.1.1/2.1.1.png @@ -0,0 +1,2184 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: 2.1.1.png +%%Creator: gnuplot 5.4 patchlevel 3 +%%CreationDate: Sun Mar 13 16:23:26 2022 +%%DocumentFonts: (atend) +%%BoundingBox: 50 50 410 302 +%%EndComments +%%BeginProlog +/gnudict 256 dict def +gnudict begin +% +% The following true/false flags may be edited by hand if desired. +% The unit line width and grayscale image gamma correction may also be changed. +% +/Color true def +/Blacktext false def +/Solid false def +/Dashlength 1 def +/Landscape false def +/Level1 false def +/Level3 false def +/Rounded false def 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180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C3 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C4 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C5 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc + 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc} bind def +/C6 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C7 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C8 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C9 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 450 arc closepath fill + vpt 0 360 arc closepath} bind def +/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill + 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C11 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C12 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C13 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C14 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 360 arc closepath fill + vpt 0 360 arc} bind def +/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/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 +/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def +/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def +/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def +/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 +/DiaE {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke} def +/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke} def +/TriUE {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 closepath stroke} def +/TriDE {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 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 closepath gsave fill grestore stroke} 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 +% +/PatternFill {gsave /PFa [ 9 2 roll ] def + 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 + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse + 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 +dup 2 lt + {/InterpretLevel1 true def + /InterpretLevel3 false def} + {/InterpretLevel1 Level1 def + 2 gt + {/InterpretLevel3 Level3 def} + {/InterpretLevel3 false def} + ifelse } + 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 +% +% +% encoding for code page koi8r (8-bit Russian) +% version 1.0: - Mainly letters are mapped. The following positions +% (JFi) are left blank (undefined): +% -- first 32 positions, +% -- frame characters, +% -- a few special characters. + +/reencodeKOI8R { +dup dup findfont dup length dict begin +{ 1 index /FID ne { def }{ pop pop } ifelse } forall +currentdict /CharStrings known { + CharStrings /Idieresis known { + /Encoding KOI8REncoding def } if +} if +currentdict end definefont +} def +/KOI8REncoding [ +/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef +/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef +/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef +/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef +/space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quoteright +/parenleft/parenright/asterisk/plus/comma/minus/period/slash +/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon +/less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N +/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/backslash/bracketright +/asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m +/n/o/p/q/r/s/t/u/v/w/x/y/z/braceleft/bar/braceright/asciitilde/.notdef +/SF100000/SF110000/SF010000/SF030000/SF020000/SF040000/SF080000/SF090000 +/SF060000/SF070000/SF050000/SF600000/SF570000/SF610000/SF580000/SF590000 +/SF140000/SF150000/SF160000/SS260000/filledbox/bullet/radical/approxequal +/lessequal/greaterequal/space/SS270000/degree/twosuperior/periodcentered/divide +/SF430000/SF240000/SF510000/afii10071/SF520000/SF390000/SF220000/SF210000 +/SF250000/SF500000/SF490000/SF380000/SF280000/SF270000/SF260000/SF360000 +/SF370000/SF420000/SF190000/afii10023/SF200000/SF230000/SF470000/SF480000 +/SF410000/SF450000/SF460000/SF400000/SF540000/SF530000/SF440000/copyright +/afii10096/afii10065/afii10066/afii10088/afii10069/afii10070/afii10086/afii10068 +/afii10087/afii10074/afii10075/afii10076/afii10077/afii10078/afii10079/afii10080 +/afii10081/afii10097/afii10082/afii10083/afii10084/afii10085/afii10072/afii10067 +/afii10094/afii10093/afii10073/afii10090/afii10095/afii10091/afii10089/afii10092 +/afii10048/afii10017/afii10018/afii10040/afii10021/afii10022/afii10038/afii10020 +/afii10039/afii10026/afii10027/afii10028/afii10029/afii10030/afii10031/afii10032 +/afii10033/afii10049/afii10034/afii10035/afii10036/afii10037/afii10024/afii10019 +/afii10046/afii10045/afii10025/afii10042/afii10047/afii10043/afii10041/afii10044 +] def +/Metrics {ExtendTextBox Gswidth} def +/Lwidth {currentpoint stroke M 0 vshift R Metrics} def +/Rwidth {currentpoint stroke M dup stringwidth pop neg vshift R Metrics} def +/Cwidth {currentpoint stroke M dup stringwidth pop -2 div vshift R Metrics} def +/GLwidth {currentpoint stroke M 0 vshift R {ExtendTextBox} forall} def +/GRwidth {currentpoint stroke M dup Gwidth vshift R {ExtendTextBox} forall} def +/GCwidth {currentpoint stroke M dup Gwidth 2 div vshift R {ExtendTextBox} forall} def +/GLwidth2 {0 Gwidth AddGlyphWidth} def +/GRwidth2 {Gwidth -1 mul 0 AddGlyphWidth} def +/GCwidth2 {Gwidth 2 div dup -1 mul AddGlyphWidth} def +/AddGlyphWidth { dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse + dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse } def +/MFshow { + { dup 5 get 3 ge + { 5 get 3 eq {gsave} {grestore} ifelse } + {dup dup 0 get findfont exch 1 get scalefont setfont + [ currentpoint ] exch dup 2 get 0 exch R dup 5 get 2 ne {dup dup 6 + get exch 4 get {textshow} {Metrics pop 0 R} ifelse }if dup 5 get 0 eq + {dup 3 get {2 get neg 0 exch R pop} {pop aload pop M} ifelse} {dup 5 + get 1 eq {dup 2 get exch dup 3 get exch 6 get Gswidth pop -2 div + dup 0 R} {dup 6 get Gswidth pop -2 div 0 R 6 get + textshow 2 index {aload pop M neg 3 -1 roll neg R pop pop} {pop pop pop + pop aload pop M} ifelse }ifelse }ifelse } + ifelse } + forall} def +/Gswidth {dup type /stringtype eq {stringwidth} {pop (n) stringwidth} ifelse} def +/MFwidth {0 exch { dup 5 get 3 ge { 5 get 3 eq { 0 } { pop } ifelse } + {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont + 6 get Gswidth pop add} {pop} ifelse} ifelse} forall} def +/MLshow { currentpoint stroke M + 0 exch R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/MRshow { currentpoint stroke M + exch dup MFwidth neg 3 -1 roll R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/MCshow { currentpoint stroke M + exch dup MFwidth -2 div 3 -1 roll R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/XYsave { [( ) 1 2 true false 3 ()] } bind def +/XYrestore { [( ) 1 2 true false 4 ()] } bind def +/Helvetica reencodeKOI8R def +Level1 SuppressPDFMark or +{} { +/SDict 10 dict def +systemdict /pdfmark known not { + userdict /pdfmark systemdict /cleartomark get put +} if +SDict begin [ + /Title (2.1.1.png) + /Subject (gnuplot plot) + /Creator (gnuplot 5.4 patchlevel 3) +% /Producer (gnuplot) +% /Keywords () + /CreationDate (Sun Mar 13 16:23:26 2022) + /DOCINFO pdfmark +end +} ifelse +% +% Support for boxed text - Ethan A Merritt Sep 2016 +% +/InitTextBox { userdict /TBy2 3 -1 roll put userdict /TBx2 3 -1 roll put + userdict /TBy1 3 -1 roll put userdict /TBx1 3 -1 roll put + /Boxing true def } def +/ExtendTextBox { dup type /stringtype eq + { Boxing { gsave dup false charpath pathbbox + dup TBy2 gt {userdict /TBy2 3 -1 roll put} {pop} ifelse + dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse + dup TBy1 lt {userdict /TBy1 3 -1 roll put} {pop} ifelse + dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse + grestore } if } + {} ifelse} def +/PopTextBox { newpath TBx1 TBxmargin sub TBy1 TBymargin sub M + TBx1 TBxmargin sub TBy2 TBymargin add L + TBx2 TBxmargin add TBy2 TBymargin add L + TBx2 TBxmargin add TBy1 TBymargin sub L closepath } def +/DrawTextBox { PL PopTextBox stroke /Boxing false def} def +/FillTextBox { gsave PopTextBox fill grestore /Boxing false def} def +0 0 0 0 InitTextBox +/TBxmargin 20 def +/TBymargin 20 def +/Boxing false def +/textshow { ExtendTextBox Gshow } def +% +end +%%EndProlog +%%Page: 1 1 +gnudict begin +gsave +doclip +50 50 translate +0.050 0.050 scale +0 setgray +newpath +(Helvetica) findfont 140 scalefont setfont +BackgroundColor 0 lt 3 1 roll 0 lt exch 0 lt or or not {BackgroundColor C 1.000 0 0 7200.00 5040.00 BoxColFill} if +1.000 UL +LTb +LCb setrgbcolor +0.500 UL +LTa +LCa setrgbcolor +518 448 M +6429 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +518 448 M +63 0 V +6366 0 R +-63 0 V +stroke +434 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0)] +] -46.7 MRshow +/Helvetica findfont 140 scalefont setfont +/vshift -46 def +1.000 UL +LTb +LCb setrgbcolor +0.500 UL +LTa +LCa setrgbcolor +518 1004 M +6429 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +518 1004 M +63 0 V +6366 0 R +-63 0 V +stroke +434 1004 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 1)] +] -46.7 MRshow +/Helvetica findfont 140 scalefont setfont +1.000 UL +LTb +LCb setrgbcolor +0.500 UL +LTa +LCa setrgbcolor +518 1561 M +6429 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +518 1561 M +63 0 V +6366 0 R +-63 0 V +stroke +434 1561 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 2)] +] -46.7 MRshow +/Helvetica findfont 140 scalefont setfont +1.000 UL +LTb +LCb setrgbcolor +0.500 UL +LTa +LCa setrgbcolor +518 2117 M +6429 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +518 2117 M +63 0 V +6366 0 R +-63 0 V +stroke +434 2117 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 3)] +] -46.7 MRshow +/Helvetica findfont 140 scalefont setfont +1.000 UL +LTb +LCb setrgbcolor +0.500 UL +LTa +LCa setrgbcolor +518 2674 M +6429 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +518 2674 M +63 0 V +6366 0 R +-63 0 V +stroke +434 2674 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 4)] +] -46.7 MRshow +/Helvetica findfont 140 scalefont setfont +1.000 UL +LTb +LCb setrgbcolor +0.500 UL +LTa +LCa setrgbcolor +518 3230 M +6429 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +518 3230 M +63 0 V +6366 0 R +-63 0 V +stroke +434 3230 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 5)] +] -46.7 MRshow +/Helvetica findfont 140 scalefont setfont +1.000 UL +LTb +LCb setrgbcolor +0.500 UL +LTa +LCa setrgbcolor +518 3786 M +6429 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +518 3786 M +63 0 V +6366 0 R +-63 0 V +stroke +434 3786 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 6)] +] -46.7 MRshow +/Helvetica findfont 140 scalefont setfont +1.000 UL +LTb +LCb setrgbcolor +0.500 UL +LTa +LCa setrgbcolor +518 4343 M +6429 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +518 4343 M +63 0 V +6366 0 R +-63 0 V +stroke +434 4343 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 7)] +] -46.7 MRshow +/Helvetica findfont 140 scalefont setfont +1.000 UL +LTb +LCb setrgbcolor +0.500 UL +LTa +LCa setrgbcolor +518 4899 M +6429 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +518 4899 M +63 0 V +6366 0 R +-63 0 V +stroke +434 4899 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 8)] +] -46.7 MRshow +/Helvetica findfont 140 scalefont setfont +1.000 UL +LTb +LCb setrgbcolor 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+/UP {dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def + /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def} def +/DL {Color {setrgbcolor Solid {pop []} if 0 setdash} + {pop pop pop 0 setgray Solid {pop []} if 0 setdash} ifelse} def +/BL {stroke userlinewidth 2 mul setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +/AL {stroke userlinewidth 2 div setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +/UL {dup gnulinewidth mul /userlinewidth exch def + dup 1 lt {pop 1} if 10 mul /udl exch def} def +/PL {stroke userlinewidth setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +3.8 setmiterlimit +% Classic Line colors (version 5.0) +/LCw {1 1 1} def +/LCb {0 0 0} def +/LCa {0 0 0} def +/LC0 {1 0 0} def +/LC1 {0 1 0} def +/LC2 {0 0 1} def +/LC3 {1 0 1} def +/LC4 {0 1 1} def +/LC5 {1 1 0} def +/LC6 {0 0 0} def +/LC7 {1 0.3 0} def +/LC8 {0.5 0.5 0.5} def +% Default dash patterns (version 5.0) +/LTB {BL [] LCb DL} def +/LTw {PL [] 1 setgray} def +/LTb {PL [] LCb DL} def +/LTa 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add M + hpt2 vpt2 neg V currentpoint stroke M + hpt2 neg 0 R hpt2 vpt2 V stroke} def +/TriU {stroke [] 0 setdash 2 copy 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 closepath stroke + Pnt} def +/Star {2 copy Pls Crs} def +/BoxF {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath fill} def +/TriUF {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 closepath fill} def +/TriD {stroke [] 0 setdash 2 copy 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 closepath stroke + Pnt} def +/TriDF {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 closepath fill} def +/DiaF {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath fill} def +/Pent {stroke [] 0 setdash 2 copy gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore Pnt} def +/PentF {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath fill grestore} def +/Circle {stroke [] 0 setdash 2 copy + hpt 0 360 arc stroke Pnt} def +/CircleF {stroke [] 0 setdash hpt 0 360 arc fill} def +/C0 {BL [] 0 setdash 2 copy moveto vpt 90 450 arc} bind def +/C1 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + vpt 0 360 arc closepath} bind def +/C2 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C3 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C4 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C5 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc + 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc} bind def +/C6 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C7 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C8 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C9 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 450 arc closepath fill + vpt 0 360 arc closepath} bind def +/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill + 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C11 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C12 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C13 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C14 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 360 arc closepath fill + vpt 0 360 arc} bind def +/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/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 +/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def +/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def +/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def +/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 +/DiaE {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke} def +/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke} def +/TriUE {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 closepath stroke} def +/TriDE {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 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 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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 +% +/PatternFill {gsave /PFa [ 9 2 roll ] def + 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 + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse + 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 +dup 2 lt + {/InterpretLevel1 true def + /InterpretLevel3 false def} + {/InterpretLevel1 Level1 def + 2 gt + {/InterpretLevel3 Level3 def} + {/InterpretLevel3 false def} + ifelse } + 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 +% +% +% encoding for code page koi8r (8-bit Russian) +% version 1.0: - Mainly letters are mapped. The following positions +% (JFi) are left blank (undefined): +% -- first 32 positions, +% -- frame characters, +% -- a few special characters. + +/reencodeKOI8R { +dup dup findfont dup length dict begin +{ 1 index /FID ne { def }{ pop pop } ifelse } forall +currentdict /CharStrings known { + CharStrings /Idieresis known { + /Encoding KOI8REncoding def } if +} if +currentdict end definefont +} def +/KOI8REncoding [ +/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef +/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef +/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef +/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef +/space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quoteright +/parenleft/parenright/asterisk/plus/comma/minus/period/slash +/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon +/less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N +/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/backslash/bracketright +/asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m +/n/o/p/q/r/s/t/u/v/w/x/y/z/braceleft/bar/braceright/asciitilde/.notdef +/SF100000/SF110000/SF010000/SF030000/SF020000/SF040000/SF080000/SF090000 +/SF060000/SF070000/SF050000/SF600000/SF570000/SF610000/SF580000/SF590000 +/SF140000/SF150000/SF160000/SS260000/filledbox/bullet/radical/approxequal +/lessequal/greaterequal/space/SS270000/degree/twosuperior/periodcentered/divide +/SF430000/SF240000/SF510000/afii10071/SF520000/SF390000/SF220000/SF210000 +/SF250000/SF500000/SF490000/SF380000/SF280000/SF270000/SF260000/SF360000 +/SF370000/SF420000/SF190000/afii10023/SF200000/SF230000/SF470000/SF480000 +/SF410000/SF450000/SF460000/SF400000/SF540000/SF530000/SF440000/copyright +/afii10096/afii10065/afii10066/afii10088/afii10069/afii10070/afii10086/afii10068 +/afii10087/afii10074/afii10075/afii10076/afii10077/afii10078/afii10079/afii10080 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--git a/2.1.1/data.txt b/2.1.1/data.txt index 5506481..a2715b6 100644 --- a/2.1.1/data.txt +++ b/2.1.1/data.txt @@ -1,68 +1,4 @@ Влажность - 64% -Давление - 745.5 мм. рт. ст. -t = 22.3 C +Давление - 745.5 мм. рт. ст. (99375.15 Па) +t = 22.3 C (295.45 К) p0 = 1.17 кг/м^3 - -______________ -delta T = 1.9656 К - -#1 -I = 0.1 А -U = 2.8 В -t = 70 c -delta V = 5 л -eps = 0.086 мВ - -#2 -I = 0.08 А -U = 2.4 В -t = 110 с -delta V = 5 л -eps = 0.088 мВ - -#3 -I = 0.075 А -U = 2.1 В -t = 95 c -delta V = 3 л -eps = 0.88 - -#4 -I = 0.07 А -U = 2 В -t = 110 с -delta V = 3 л -eps = 0.84 мВ - -___________________ - -delta T = 5.1597 К - -#1 -I = 0.12 А -U = 3.4 В -t = 110 с -delta V = 3 л -eps = 0.21 мВ - -#2 -I = 0.13 А -U = 3.8 В -t = 65 с -delta V = 3 л -eps = 0.217 м - -#3 -I = 0.15 А -U = 4.4 В -t = 64 с -delta V = 5 л -eps = 0.214 м - -#4 -I = 0.17 А -U = 4.8 В -t = 55 с -delta V = 5 л -eps = 0.216 м - diff --git a/2.1.1/data_fucked_up.txt b/2.1.1/data_fucked_up.txt new file mode 100644 index 0000000..5506481 --- /dev/null +++ b/2.1.1/data_fucked_up.txt @@ -0,0 +1,68 @@ +Влажность - 64% +Давление - 745.5 мм. рт. ст. +t = 22.3 C +p0 = 1.17 кг/м^3 + +______________ +delta T = 1.9656 К + +#1 +I = 0.1 А +U = 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grestore} bind def +/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def +/DiaE {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke} def +/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke} def +/TriUE {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 closepath stroke} def +/TriDE {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 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 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fill grestore stroke} 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 +% +/PatternFill {gsave /PFa [ 9 2 roll ] def + 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 + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse + 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 +dup 2 lt + {/InterpretLevel1 true def + /InterpretLevel3 false def} + {/InterpretLevel1 Level1 def + 2 gt + {/InterpretLevel3 Level3 def} + {/InterpretLevel3 false def} + ifelse } + 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 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ps 0.5 lc rgb "red" +set style line 2 lt 1 pt 7 ps 0.5 lc rgb "blue" +set style line 4 lc black + +set key box linestyle 4 +set key opaque Left + +plot "2.1.3(22.8).data" using 1:2 notitle linestyle 1 + diff --git a/2.1.1/lab-gnuplottex-fig1.tex b/2.1.1/lab-gnuplottex-fig1.tex new file mode 100644 index 0000000..1c824cf --- /dev/null +++ b/2.1.1/lab-gnuplottex-fig1.tex @@ -0,0 +1,215 @@ +% GNUPLOT: LaTeX picture with Postscript +\begingroup + \makeatletter + \providecommand\color[2][]{% + \GenericError{(gnuplot) \space\space\space\@spaces}{% + Package color not loaded in conjunction with + terminal option `colourtext'% + }{See the gnuplot documentation for explanation.% + }{Either use 'blacktext' in gnuplot or load the package + color.sty in LaTeX.}% + \renewcommand\color[2][]{}% + }% + \providecommand\includegraphics[2][]{% + \GenericError{(gnuplot) \space\space\space\@spaces}{% + Package graphicx or graphics not loaded% + }{See the gnuplot documentation for explanation.% + }{The gnuplot epslatex terminal needs graphicx.sty or graphics.sty.}% + \renewcommand\includegraphics[2][]{}% + }% + \providecommand\rotatebox[2]{#2}% + \@ifundefined{ifGPcolor}{% + \newif\ifGPcolor + \GPcolorfalse + }{}% + \@ifundefined{ifGPblacktext}{% + \newif\ifGPblacktext + \GPblacktexttrue + }{}% + % define a \g@addto@macro without @ in the name: + \let\gplgaddtomacro\g@addto@macro + % define empty templates for all commands taking text: + \gdef\gplbacktext{}% + \gdef\gplfronttext{}% + \makeatother + \ifGPblacktext + % no textcolor at all + \def\colorrgb#1{}% + \def\colorgray#1{}% + \else + % gray or color? + \ifGPcolor + \def\colorrgb#1{\color[rgb]{#1}}% + \def\colorgray#1{\color[gray]{#1}}% + \expandafter\def\csname LTw\endcsname{\color{white}}% + \expandafter\def\csname LTb\endcsname{\color{black}}% + \expandafter\def\csname LTa\endcsname{\color{black}}% + \expandafter\def\csname LT0\endcsname{\color[rgb]{1,0,0}}% + \expandafter\def\csname LT1\endcsname{\color[rgb]{0,1,0}}% + 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\author{Гришаев Григорий С01-119} \title{Отчёт о выполнении лабораторной работы 2.1.1} \usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} @@ -16,403 +18,255 @@ \begin{document} \maketitle \begin{center} - {\Large Измерение удельной теплоёмкости воздуха при постоянном давленини} + {\Large Измерение удельной теплоёмкости воздуха при постоянном давлении} \end{center} - \paragraph*{Цель работы:} 1) измерение повышения температуры воздуха в результате подвода тепла при стационарном течении через стеклянную трубу; 2) вычисление по результатам измерений теплоёмкости воздуха при постоянном давлении. - \paragraph*{В работе используются:} теплоизолированная трубка; электронагреватель; источник питания постоянного тока; амперметр; вольтметр; термопара; компрессор; газовый счётчик; - секундомер. + \paragraph*{Цель работы:}измерить повышение температуры воздуха в зависимости от мощности подводимого тепла и расхода при стационарном течении через трубу; исключив тепловые потери, по результатам измерений определить теплоёмкость воздуха при постоянном давлении + \paragraph*{В работе используются:}теплоизолированная стеклянная трубка; электронагреватель; источник питания постоянного тока; амперметр, вольтметр (цифровые мультиметры); термопара, подключенная к микровольтметру; компрессор; газовый счётчик; секундомер. + + \section{Теоретические сведения} Определение теплоёмкости обычно производится в калориметрах. При этом регистрируется количество тепла {Q}, полученное телом, и изменение температуры этого тела $ {\Delta T} $. - Теплоёмкость тела определяется как их отношение: $$ C = \frac{Q}{\Delta T} \; (1).$$ - - Необходимо, чтобы количество тепла, затрачиваемое на нагревание исследуемого тела, существенно превосходило тепло, расходуемое на нагревание самого калориметра, а также на потери тепла из установки. - - Для увеличения количества нагреваемого газа при неизменных размерах установки в нашей работе исследуемый газ (воздух) продувается через калориметр, внутри которого установлен нагреватель. При этом - измеряются мощность нагревателя, масса воздуха, протекающего в единицу - времени (расход), и приращение его температуры. - + Теплоёмкость тела определяется как их отношение: + \begin{equation} + \label{eq1} + C = \frac{\delta Q}{dT} \; + \end{equation} + Необходимо, чтобы количество тепла, затрачиваемое на нагревание исследуемого тела, существенно превосходило тепло, расходуемое на нагревание самого калориметра, а также на потери тепла из установки. + Для увеличения количества нагреваемого газа при неизменных размерах установки в нашей работе исследуемый газ (воздух) продувается через калориметр, внутри которого установлен нагреватель. При этом измеряются мощность нагревателя, масса воздуха, протекающего в единицу времени (расход), и приращение его температуры. + Рассмотрим газ, протекающий стационарно слева направо через трубу постоянного сечения, в которой установлен нагревательный элемент (см. рис. 1). Пусть за некоторое время $ dT $ через калориметр прошла малая порция газа массой $ dm = qdt $, где $ q $ [кг/с] — массовый расход газа в трубе. Если мощность нагрева равна $ N $, мощность тепловых потерь на обмен с окружающей средой $ N_{пот} $, то порция получила тепло $ \delta Q = (N - N_{пот})dt $. С другой стороны, по определению теплоёмкости \ref{eq1}, $ \delta Q = c dm \Delta T$, где $ \Delta T = T_{2} - T_{1}$ - приращение температуры газа, и $ c $ — удельная (на единицу массы) теплоёмкость газа в рассматриваемом процессе. При малых расходах газа и достаточно большом диаметре трубы перепад давления на её концах мал, поэтому можно принять, что $ P_{1} \approx P_{2} = P_{0} $, где $ P_{0} $ — атмосферное давление. Следовательно, в условиях опыта измеряется удельная теплоёмкость при постоянном давлении $ c_{p} $. Таким образом, получаем + \begin{equation} + \label{eq2} + c_{p} = \frac{N - N_{пот}}{q \Delta T}; + \end{equation} + \begin{figure}[h!] + \center{\includegraphics[width=0.5\linewidth]{1}} + \caption{Нагрев газа при течении по трубе} + \end{figure} + + \subsection{Экспериментальная установка:} - Схема установки изображена на рис.1. Воздух, нагнетаемый компрессором, прокачивается через калориметр. Калориметр представляет собой стеклянную цилиндрическую трубку с двойными стенками, запаянными с торцов. + Схема установки изображена на рис.2. Воздух, нагнетаемый компрессором, прокачивается через калориметр. Калориметр представляет собой стеклянную цилиндрическую трубку с двойными стенками, запаянными с торцов. \begin{figure}[h] - \center{\includegraphics{1}} + \center{\includegraphics{3}} + \caption{Схема экспериментальной установки} \end{figure} Нагреватель в виде намотанной на пенопласт нихромовой проволоки расположен внутри калориметра непосредственно в воздушном потоке. Нагрев проволоки производится от регулируемого источника постоянного тока (ИП). Напряжение $U$ на нагревателе и ток $I$ через него регистрируются цифровыми мультиметрами. Таким образом, мощность нагрева равна - $$N= UI \; (3).$$ + \begin{equation} + \label{eq3} + N= UI \; + \end{equation} Для измерения разности температур $\Delta T$ служит медно-константановая термопара. Один спай термопары расположен в струе воздуха, входящего в - калориметр, и находится при комнатной температуре, а второй — в струе выходящего нагретого воздуха. Константановая проволока термопары расположена внутри калориметра, а медные проводники подключены к цифровому вольтметру. Возникающая в термопаре ЭДС $\varepsilon$ пропорциональна разности температур $\Delta T$ спаев: $$\varepsilon =\beta \Delta T \; (4),$$ где $\beta = 42.3 \frac{мкВ}{^\circ C}$ — чувствительность медно-константановой термопары в рабочем диапазоне температур (20–30 $^\circ C$ ). ЭДС регистрируется с помощью микровольтметра. - - Вычислим работу, совершаемую при протекании газа через калориметр. Внешняя работа по перемещению моля газа в направлении течения в начале трубки равна $ A_{1} = P_{1}V_{1} $, а в конце трубки давление препятствует движению и внешняя работа над газом отрицательна: $ A_{2}=-P_{2}V_{2}$. Полная работа над газом равна $ A_{1}+A_{2}=P_{1}V_{1}-P_{2}V_{2}$, а работа самого газа равна этой же величине, но с обратным знаком: $$ A =P_{2}V_{2}-P_{1}V_{1}$$ - Внутренняя энергия газа изменяется на величину $ \Delta U = U_{2} - U_{1} $. - Из первого начала термодинамики: $$ Q = U_{2}-U_{1}+P_{2}V_{2}-P_{1}V_{1}=H_{2}-H_{1}$$, - где $H=U+PV$ - энтальпия. - Для идеального газа $ H = C_{p}T$, поэтому $$ Q = C_{p}(T_{2}-T_{1})$$. - Следовательно, в данном эксперименте измеряется теплоемкость при постоянном давлении. - Расчет удельной теплоемкости воздуха: $$ c_{p} = \frac{Q}{m \Delta T}= \frac{IU-N}{m \Delta T}$$, -% Объём воздуха, прошедшего через калориметр, измеряется газовым счётчиком ГС. Для регулировки расхода служит кран К. Время $\Delta t$ прохождения некоторого объема $\Delta V$ воздуха измеряется секундомером. Объёмный расход равен $\frac{\Delta V}{\Delta t} $, массовый расход может быть найден как $$q = \rho_{0} \frac{\Delta V}{\Delta t} \; (5),$$ где $rho_{0}$ — плотность воздуха при комнатной температуре, которая в свою очередь может быть получена из уравнения Менделеева–Клапейрона: $\rho_{0}= \frac{\mu P_{0} }{R T_{0}},$ где $P_{0}$ — атмосферное давление, $T_{0}$ — комнатная температура (в Кельвинах), $\mu = 29,0 {г/моль}$ — средняя молярная масса (сухого) воздуха. - -% Учитывая особенности устройства калориметра, следует ожидать, что мощность нагревателя расходуется не только на нагрев массы прокачиваемого воздуха, но и частично теряется за счет нагрева внутренних стенок термостата и рассеяния тепла через торцы термостата. Можно предположить, что при небольшом нагреве ($\Delta T \ll T_{0}$) мощность потерь тепла $N_{пот}$ прямо пропорциональна разности температур:$$ N_{пот} = \alpha \Delta T \; (6),$$ где $\alpha$ — некоторая константа. При этом условии основное соотношение (2) принимает вид $$N = (c_{P}q +\alpha)\Delta T \;(7)$$ -% Следовательно, при фиксированном расходе воздуха ($q = const$ ) подводимая мощность и разность температур связаны прямой пропорциональностью($\Delta T(N)$ — линейная функция). + калориметр, и находится при комнатной температуре, а второй — в струе выходящего нагретого воздуха. Константановая проволока термопары расположена внутри калориметра, а медные проводники подключены к цифровому вольтметру. Возникающая в термопаре ЭДС $\varepsilon$ пропорциональна разности температур $\Delta T$ спаев: + \begin{equation} + \label{eq4} + \varepsilon =\beta \Delta T \; + \end{equation} + где $\beta = 40.7 \frac{мкВ}{^\circ C}$ — чувствительность медно-константановой термопары в рабочем диапазоне температур (20–30 $^\circ C$ ). ЭДС регистрируется с помощью микровольтметра. + Объём воздуха, прошедшего через калориметр, измеряется газовым счётчиком ГС. Для регулировки расхода служит кран К. Время $ \Delta t $ прохождения некоторого объема $ \Delta V $ воздуха измеряется секундомером. Объёмный расход равен $ \Delta V/\Delta T $, массовый расход может быть найден как + \begin{equation} + \label{eq5} + q = \rho_{o}\frac{\Delta V}{\Delta t} \; + \end{equation} + где $ \rho_{0} $ — плотность воздуха при комнатной температуре, которая в свою очередь может быть получена из уравнения Менделеева–Клапейрона: $ \rho_{0} = \frac{\mu P_{0}}{R T_{0}} $, где $ P_{0} $ — атмосферное давление, $ T_{0} $ — комнатная температура (в Кельвинах), $ \mu $= 29,0 г/моль — средняя молярная масса (сухого) воздуха. + Учитывая особенности устройства калориметра, следует ожидать, что мощность нагревателя расходуется не только на нагрев массы прокачиваемого воздуха, но и частично теряется за счет нагрева внутренних стенок термостата и рассеяния тепла через торцы термостата. Можно предположить, что при небольшом нагреве ($ \Delta T << T_{0} $) мощность потерь тепла $ N_{пот} $ прямо +пропорциональна разности температур: +\begin{equation} + \label{eq6} + N_{пот} = \alpha \Delta T \; +\end{equation} +где $ \alpha $ — некоторая константа. При этом условии основное соотношение \ref{eq2} принимает вид +\begin{equation} + \label{eq7} + N = (c_{p}q + \alpha)\Delta T \; +\end{equation} +Следовательно, при фиксированном расходе воздуха ($ q $ = const) подводимая мощность и разность температур связаны прямой пропорциональностью ($ \Delta T{N} $ — линейная функция). + \subsection{Оборудование и инструментальные погрешности} + Для измерения $ \varepsilon $ использовался вольтметр универсальный В7-78/1, его абсолютная погрешность равна $ \pm (0.005 \frac{\varepsilon}{100} + 3.5) \; мкВ$. + + Для измерения $ U $ использовался цифровой универсальный вольтметр GDM-8145, абсолютная погрешность которого равна $ \pm (0.0003U \pm 4 \; мВ) $. + Его же мы использовали для измерения $ I $, абсолютная погрешность: $ \pm (0.002I+20 \; мкА) $. \subsection{Ход работы} -% \begin{enumerate} -% \item Подготовим к работе газовый счетчик: проверим, что он заполнен водой, установим счетчик по уровню. -% \item Охладим калориметр до комнатной температуры. -% \item Включим вольтметр, предназначенный для измерения ЭДС термопары. -% \item Запишем показания компнатной температуры и давления. $$T_{0} = 297.05 \; ^\circ C, P_{0} = 99325 \pm 13 \; {Па} $$ -% \item С помощью газового счетчика и секундомера измерим максимальный расход воздуха $\frac{\Delta V}{\Delta T}$ (в л/с). Измерения представлены в таблице 1. По найденным значениям определим среднее значение расхода и массовый расход воздуха $q_{max}$ [г/с]. -% -% $$q = \rho_0 \frac{\Delta V}{\Delta t} = \frac{\mu P_0}{RT_0} \frac{\Delta V}{\Delta t}.$$ -% -% Относительная погрешность косвенных измерений может быть найдена по формуле $$\frac{\sigma_{q_{max}кос}}{q_{max}} = \sqrt{(\frac{\sigma_{T_0}}{T_{0}})^2+(\frac{\sigma_{P_0}}{P_{0}})^2+ (\frac{\sigma_t}{t})^2}$$ -% -% -% \begin{table} -% \begin{center} -% \begin{tabular}{|c|c|c|c|c|} -% \hline -% $\Delta V, л$ & $\Delta t, c$ & $\frac{\Delta V}{\Delta t},\frac{л}{с}$ & $q_{max},\frac{г}{c} $ & $\sigma_{q_{max}кос}, \; \frac{г}{с} \cdot 10^{-3}$ \\ -% \hline -% x & 0 & 0 & 0 & 0 \\ -% \hline -% x & 0 & 0 & 0 & 0\\ -% \hline -% x & 0 & 0 & 0 & 0\\ -% \hline -% x & 0 & 0 & 0 &0 \\ -% \hline -% -% \end{tabular} -% \end{center} -% \caption{Измерение расхода воздуха} -% \end{table} -% $$\overline{\frac{\Delta V}{\Delta t}} = 0.1951 \; \frac{л}{с} ,\; \overline{q_{max}} = 0.2276 \; \frac{г}{с} $$ -% -% Случайная погрешность массового расхода может быть найдена по формуле: $$\sigma_{q_{max}сл} = \sqrt{\frac{\sum_{i=1}^{7} (q_{max,i}-\overline{q_{max}})^2}{6}} = 0.0003 \; \frac{г}{c}.$$ -% -% Косвенная погрещность для среднего значения: $q_{max}$ $$\sigma_{\overline{q_{max}}кос} = \sqrt{\frac{\sum_{i=1}^{7} (\sigma_{q_{max}кос})^2}{7^2}} = 0.0003 \; \frac{г}{c}.$$ -% -% Суммарная погрешность: $$\sigma_{\overline{q_{max}}} = \sqrt{(\sigma_{q_{max}сл})+(\sigma_{\overline{q_{max}}кос})^2} = 0.0004 \; \frac{г}{c}.$$ -% -% Окончательное значение: $$q_{max} = 0.2276 \pm 0.0004 \; \frac{г}{с} $$ -% -% \item Оценим величину тока нагревателя $I_{0}$, требуемого для нагрева воздуха на $\delta T = 1 {К}$. -% -% Определим теоретическое значение удельной теплоемкости воздуха при постоянном давлении $C_{теорp} \; \frac{Дж}{г\cdot K}$, считая воздух смесью двухатомных идеальных газов: $Cp = 3.5R\mu \approx 1 \; \frac{Дж}{г\cdot K}.$ -% -% Оценим минимальную мощность $N_0$, необходимую для нагрева газа при максимальном расходе. $N_{0} = c_{p}q\Delta T \approx 0.227 {Вт}.$ -% -% Учитывая, что сопротивление проволоки нагревателя составляет приблизительно $R_{н} \approx 35 {Ом}$ и в процессе опыта практически не меняется, искомое значение тока $I_{0} = q N_{0} R_{н} \approx 0.08 \; {А}.$ -% -% \item Проведем измерение зависимости разности температур от мощности нагрева $\Delta T(N)$ при максимальном расходе воздуха $q_0 = q_{max}.$ -% \begin{table} -% -% \begin{center} -% \begin{tabular}{|c|c|c|c|c|c|c|c|c|} -% \hline -% $I, мA$ & $U, B$ & $N, Вт$ & $R_н, Ом$ & $\varepsilon, \muВ$ & $ \Delta T, K$ -% \\ -% \hline -% 108.18 & 3.19 & 0.345 & 29.49 & 52 & 1.28 -% \\ -% \hline -% 141.62 & 4.18 & 0.592 & 29.52 & 88 & 2.16 -% \\ -% \hline -% 165.70 & 4.89 & 0.810 & 29.51 & 118 & 2.90 -% \\ -% \hline -% 181.06 & 5.34 & 0.967 & 29.49 & 141 & 3.46 -% \\ -% \hline -% 210.7 & 6.21 & 1.308 & 29.48 & 188 & 4.61 -% \\ -% \hline -% \end{tabular} -% \end{center} -% \caption{Измерение $\Delta T (N) \; {при} \; q_{max}$} -% \end{table} -% Следует отметить, что погрешность измерения тока: $\sigma_{I} = 0.01 \; мA$, а напряжения: $\sigma_{U}= 0.01 \; В$, $\sigma_{\varepsilon}= 1 \; \muВ$ -% -% Завершив первую серию измерении, охладим калориметр до комнатнои температуры. -% Для этого отключим источник питания нагревателя, откроем кран К и продуем калориметр при максимальном расходе воздуха до тех пор, пока показания ЭДС не достигнут нуля. -% -% Данные представлены в таблице 2. -% -% \begin{table} -% \begin{center} -% \begin{tabular}{lr} -% \begin{tabular}{|c|c|c|c|c|} -% \hline -% $\Delta V, л$ & $\Delta t, c$ & $\frac{\Delta V}{\Delta t},\frac{л}{с}$ & $q_1, \frac{г}{c}$ & $Z, \; \frac{г}{с} \cdot 10^{-3}$ \\ -% \hline -% 5 & 46.02 & 0.1086 & 0.1266 & 0.550 \\ -% \hline -% 10 & 91.4 & 0.1094 & 0.1275 & 0.279 \\ -% \hline -% 15 & 134.78 & 0.1113 & 0.1297& 0.192 \\ -% \hline -% 20 & 178.96 & 0.1117 & 0.1302 & 0.146\\ -% \hline -% 25 & 221.14 & 0.1130 & 0.1317 & 0.119 \\ -% \hline -% 30 & 266.36 & 0.1126 & 0.1312 & 0.099\\ -% \hline -% 35 & 311.83 & 0.1122 & 0.1308 & 0.084\\ -% \hline -% \end{tabular} -% -% \begin{tabular}{|c|c|c|c|c|} -% \hline -% $\Delta V, л$ & $\Delta t, c$ & $\frac{\Delta V}{\Delta t},\frac{л}{с}$ & $q_2, \frac{г}{c}$ & $Z, \; \frac{г}{с} \cdot 10^{-3}$ \\ -% \hline -% 1 & 13.40 & 0.0747 & 0.0870 & 1.30\\ -% \hline -% 2 & 27.56 & 0.0726 & 0.0846 & 0.61\\ -% \hline -% 3 & 41.31 & 0.0726 & 0.0846 & 0.41\\ -% \hline -% 4 & 54.95 & 0.0733 & 0.0855 & 0.31 \\ -% \hline -% 5 & 69.05 & 0.0724 & 0.0844 & 0.24\\ -% \hline -% 6 & 82.24 & 0.0729 & 0.0850 & 0.21\\ -% \hline -% 7 & 96.18 & 0.0728 & 0.0848 & 0.18 \\ -% \hline -%\end{tabular} -%\end{tabular} -%\end{center} -%\caption{Измерения других расходов; $q_1 = 0.123 \pm 0.002 \; \frac{г}{с}; \; q_2 = 0.085 \pm 0.001 \; \frac{г}{с} $ } -%\end{table} -% Проведем аналогичные измерения для других значений расхода воздуха. Новая температура $T_{0} = 297.4 ^\circ C$. -% Данные представлены в таблице 3 и 4. ($Z \equiv \sigma_{q_{max}кос}$). Погрешности рассчитаны аналогично $q_{max}.$ -% -%\begin{table} -% \begin{center} -% \begin{tabular}{lr} -% \begin{tabular}{|c|c|c|c|c|c|} -% \hline -% $I, A$ & $U, B$ & $N, Вт$ & $R_н, Ом$ & $\varepsilon, \muВ$ & $ \Delta T, K$\\ -% \hline -% 91.34 & 2.69 & 0.246 & 29.45 & 50 & 1.23 \\ -% \hline -% 120.30 & 3.55 & 0.427 & 29.51 & 89 & 2.19 \\ -% \hline -% 145.11 & 4.28 & 0.621 & 29.49 & 128 & 3.14 \\ -% \hline -% 176.45 & 5.20 & 0.918 & 29.47 & 184 & 4.52 \\ -% \hline -% 212.20 & 6.23 & 1.322 & 29.36 & 261 & 6.41 \\ -% \hline -% -% \end{tabular} -% \begin{tabular}{|c|c|c|c|c|c|} -% \hline -% $I, A$ & $U, B$ & $N, Вт$ & $R_н, Ом$ & $\varepsilon, \muВ$ & $ \Delta T, K$\\ -% \hline -% 105.95 & 3.13 & 0.332 & 29.54 & 81 & 1.99 \\ -% \hline -% 125.74 & 3.71 & 0.466 & 29.51 & 128 & 3.14 \\ -% \hline -% 149.46 & 4.41 & 0.659 & 29.50 & 175 & 4.30 \\ -% \hline -% 177.33 & 5.23 & 0.927 & 29.49 & 236 & 5.80 \\ -% \hline -% 211.5 & 6.25 & 1.322 & 29.55 & 310 & 7.62 \\ -% \hline -% \end{tabular} -% \end{tabular} -% \end{center} -% \caption{Измерение $\Delta T (N) \; {при} \; q_{1} \; и \; q_{2}$} -%\end{table} -% -% Погрешности будем считать по слудующим формулам: $$ \sigma_{\Delta T} = \Delta T \frac{\sigma_{\varepsilon}}{\varepsilon}, \sigma_{N}= N\sqrt{( \frac{\sigma_{I}}{I})^2 + (\frac{\sigma_{U}}{U})^2} \approx N \frac{\sigma_{U}}{U}$$ -% -% \begin{table} -% \begin{tabular}{lr} -% \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|} -% \hline -% q, $\frac{г}{с}$ & \multicolumn{4}{|c|}{0.2276} & \multicolumn{4}{|c|}{0.123} -% \\ -% \hline -% k & $\Delta T, \; ^\circ C$& $\sigma_{\Delta T}\; ^\circ C$ & $N$, мВт & $\sigma_{N}$, мВт & $\Delta T, \; ^\circ C$& $\sigma_{\Delta T}\; ^\circ C$ & $N$, мВт & $\sigma_{N}$, мВт -% \\ -% -% \hline -% 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 -% \\ -% \hline -% 1 & 1.28 & 0.02 & 345 & 1.1 & 1.23 & 0.02 & 246 & 0.9 -% \\ -% \hline -% 2 & 2.16 & 0.02 & 592 & 1.4 & 2.19 & 0.02 & 427 & 1.2 -% \\ -% \hline -% 3 & 2.90 & 0.02 & 810 & 1.7 & 3.14 & 0.02 & 621 & 1.5 -% \\ -% \hline -% 4 & 3.46 & 0.02 & 967 & 1.8 & 4.52 & 0.02 & 918 & 1.8 -% \\ -% \hline -% 5 & 4.61 & 0.02 & 1308 & 2.1 & 6.41 & 0.02 & 1322 & 2.1 -% \\ -% \hline -% \end{tabular} -% -% \\ -% -% \begin{tabular} {|l|l|l|l|l|l|l|l|l|l|l|} -% \hline -% q, $\frac{г}{с}$ & \multicolumn{4}{|c|}{0.085} -% \\ -% \hline -% k & $\Delta T, \; ^\circ C$& $\sigma_{\Delta T}\; ^\circ C$ & $N$, мВт & $\sigma_{N}$, мВт -% \\ -% \hline -% 0 & 0 & 0 & 0 & 0 -% \\ -% \hline -% 1 & 1.99 & 0.02 & 332 & 1.1 -% \\ -% \hline -% 2 & 3.14 & 0.02 & 466 & 1.3 -% \\ -% \hline -% 3 & 4.30 & 0.02 & 659 & 1.5 -% \\ -% \hline -% 4 & 5.80 & 0.02 & 927 & 1.8 -% \\ -% \hline -% 5 & 7.62 & 0.02 & 1322 & 2.1 -% \\ -% \hline -% \end{tabular} -% -% -% -% \end{tabular} -% \caption{ погрешности, $\Delta T \; и \; N$} -% \end{table} -% -% -% После завершения опытов выключим источник питания нагревателя и мультиметры. Кран К откроем для максимального продува воздуха через калориметр. -% -%\item Построим на одном графике зависимости $\Delta T (N)$ при разных значениях $q$. -% \begin{center} -% \begin{tikzpicture}[scale = 2] -% \begin{axis}[ -% axis lines = left, -% legend style={at={(1,0.3)}}, -% xlabel = {$N$, {мВт}}, -% ylabel = {$\Delta T, {^\circ C} $ -% }, -% xmin=0, xmax=1350, -% ymin=0, ymax=8, -% ymajorgrids = true, -% xmajorgrids = true, -% minor tick num = 4 -% ] -% \addplot+[only marks ] plot[error bars/.cd, y dir=both, y explicit] -% coordinates { -% (0,0) -% (345,1.28) -% (592, 2.16) -% (810, 2.90) -% (967 , 3.46) -% (1308, 4.61) -% }; -% \addplot+[only marks ] plot[error bars/.cd, y dir=both, y explicit] coordinates { -% (0,0) -% (246,1.23) -% (427, 2.19) -% (621, 3.14) -% (918 , 4.52) -% (1322, 6.41) -% }; -% \addplot+[only marks ] plot[error bars/.cd, y dir=both, y explicit] -% coordinates { -% (0,0) -% (332,1.99) -% (466, 3.14) -% (659, 4.30) -% (927 , 5.80) -% (1322, 7.62) -% }; -% -% -% \addplot[blue, domain=0:1350]{0.042+0.0035*x}; -% \addplot[red, domain=0:1350]{0.0619+0.0048*x}; -% \addplot[brown, domain=0:1350]{0.0210+0.0063*x}; -% \legend{ -% $q_{max} = 0.2276 \frac{г}{с}$, $q_{1} = 0.123 \frac{г}{с};$ , $ q_{2} = 0.085 \frac{г}{с};$ -% }; -% \end{axis} -% \end{tikzpicture} -% \end{center} -% ${y_{max} = (3.5 \pm 0.3 )10^{-3}x+0.042 \pm 0.027}$,\\ -% $\; {y_{1} = (4.8 \pm 0.1)10^{-3}x+0.062 \pm 0.044}$,\\ -% $\; {y_{2} = (6.3 \pm 0.2)10^{-3}x+0.021 \pm 0.12}.$ -% $Чтобы \; найти \; k_{max},\; k_1, \; k_2$, необходимо перевернуть соотвествующие коэффициенты $k'$ наклонов прямых, а погрешность считать по формуле: $$\sigma_k = k^2 \sigma_{k'}.$$ -% $k_{max} = 0.286 \pm 0.025 \; \frac{Вт}{К}, \; k_{1} = 0.208 \pm 0.004 \; \frac{Вт}{К}, \; k_{2} = 0.159 \pm 0.005 \; \frac{Вт}{К} $ -% -% Прямая для расхода $q_2$ не идёт из нуля, вероятно, из-за поспешного начала третей серии измерений (начальная температура была несколько больше комнатной). Однако в целом из вида наших прямых, можно сделать вывод о том, что тепловые потери пропорциональны разности температур. -% -% -% Построим график зависимости $k(q)$ и по его наклону определим теплоёмкость воздуха при постоянном давлении $c_p$ -% -% \begin{center} -% \begin{tikzpicture}[scale = 1] -% \begin{axis}[ -% axis lines = left, -% legend style={at={(1,1)}}, -% xlabel = {$q, \; \frac{г}{с} \cdot 10^{-1}$}, -% ylabel = {$k, \frac{Вт}{К}$ -% }, -% xmin=0, xmax=3, -% ymin=0, ymax=0.5, -% ymajorgrids = true, -% xmajorgrids = true, -% minor tick num = 4 -% ] -% \addplot+[only marks ] plot[error bars/.cd, x dir=both, x explicit ,y dir=both, y explicit] -% coordinates { -% (2.276,0.286) +- (0.004,0.025) -% (1.23, 0.208 ) +- (0.02,0.004) -% ( 0.85,0.159) +- (0.01,0.005) -% }; -% \addplot [red, domain=0:3]{0.0928+0.0860*x}; -% -% \legend{ -% $$y=0.0928+0.0860x$$ -% }; -% \end{axis} -% \end{tikzpicture} -% \end{center} -% -%Итак, из графика найдём $c_p = (0.86 \pm 0.1) \; \frac{Дж}{К \cdot г} \cdot 29 \frac{г}{моль} = 24.94 \pm 2.29 \; \frac{Дж}{К моль}.$ A также $\alpha = 0.093 \pm 0.018 \; \frac{Вт}{К}.$ К сожалению, теплоёмкость значительно отличается от теоретического значения $29.09 \; \frac{Дж}{К моль}.$ Измерения необходимо было проводить более тщательно, ждать установления термодинамического равновесия более длительное время. Однако не смотря на это, результаты, полученные при обработке данных совпадают (почти) с теоретическими с учётом погрешностей. -% -%Посчитаем доли тепловых потерь в опытах : $\frac{N_{пот}}{N} =\frac{\alpha}{k}$ -% -% \begin{center} -% \begin{tabular}{|c|c|} -% \hline -% $q, \; \frac{г}{с}$ & $\frac{N_{пот}}{N}$ \\ -% \hline -% 0.085 & $0.59 \pm 0.11$ \\ -% \hline -% 0.123 & $0.45 \pm 0.09$\\ -% \hline -% 0.2276 & $0.33 \pm 0.06$ \\ -% \hline -% \end{tabular} -% \end{center} -% -%\end{enumerate} + \begin{enumerate} + \item Запишем показания комнатной температуры, давления и влажности воздуха. + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|} + \hline + & Значение & $ \sigma $ \\ + \hline + $ T_{0} $, К & 295.45 & 0.1 \\ + \hline + $ p $, Па & 99375.15 & 1 \\ + \hline + $ \phi $ & 64\% & 1\% \\ + \hline + + \end{tabular} + \end{center} + \end{table} + +Рассчитываем плотность воздуха $ \rho_{0} $: + $$ \rho_{0} = \frac{2.9 * 10^{-3}*99375.15}{8.31*295.45} \approx 1.17 \; кг/м^{3} $$ + Относительную погрешность вычисления плотности воздуха вычислим по формуле: + $$ \frac{\sigma_{\rho_{0}}}{\rho_{0}} = \sqrt{(\frac{\sigma_{P_{0}}}{P_{0}})^{2}+(\frac{\sigma_{\phi}}{\phi})^{2}+(\frac{\sigma_{T}}{T})^{2}} $$ + $$ \Rightarrow \sigma_{\rho_{0}} = 0.018 \; \frac{кг}{м^{3}} $$ + \item Рассчитываем теоретическую теплоемкость воздуха при постоянном давлении (в предположении, что воздух - смесь двух идеальных двухатомных газов): + $$ C_{p} = \frac{7}{2}R \approx 29.09 \; \frac{Дж}{моль*К} $$ + $$ C_{p}^{\mu} = \frac{C_{p}}{\mu} \approx 1 \; \frac{Дж}{г*К} $$ + + \item С помощью газового счетчика и секундомера измерим расход воздуха для двух случаев, пользуясь формулой \ref{eq5} ($ q_{1} $ и $ q_{2} $). Результаты измерений представлены в таблице. + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|c|c|} + \hline + $ \Delta V $, л & $ \Delta t $, с & $ \frac{\Delta V}{\Delta t} \; , \frac{л}{с} $ & $ q_{1} \; , \frac{г}{с}$ & $ \sigma_{q_{1}кос}, \; \frac{г}{с}*10^{-3} $\\ + \hline + 5 & 24.9 & 0.2 & 0.234 & 0.47 \\ + \hline + 5 & 25.6 & 0.2 & 0.234 & 0.47 \\ + \hline + 5 & 26.0 & 0.19 & 0.222 & 0.45 \\ + \hline + \end{tabular} + \caption{Измерение $ q_{1} $} + \end{center} + \end{table} + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|c|c|} + \hline + $ \Delta V $, л & $ \Delta t $, с & $ \frac{\Delta V}{\Delta t} \; , \frac{л}{с} $ & $ q_{2} \; , \frac{г}{с}$ & $ \sigma_{q_{2}кос}, \; \frac{г}{с}*10^{-3} $ \\ + \hline + 5 & 32.7 & 0.15 & 0.176 & 0.35 \\ + \hline + 5 & 33.1 & 0.15 & 0.176 & 0.35 \\ + \hline + 5 & 33.2 & 0.15 & 0.176 & 0.35 \\ + \hline + \end{tabular} + \caption{Измерение $ q_{2} $} + \end{center} + \end{table} + Итого, $ q_{1} = \overline{q_{1}} = 0.23 \; г/с $, $ q_{2} = \overline{q_{2}} = 0.176 \; г/с $. + + Вычислим случайную погрешность измерения $ q $: + $$ \sigma_{q} = \sqrt{\frac{1}{n(n-1)}\sum_{i=1}^{n}(q_{i}-\overline q)^{2}} $$ + Итак, $ \sigma_{q_{1}} = 4*10^{-3} \; \frac{г}{с} $, $ \sigma_{q_{2}} \approx 0\; \frac{г}{с} $ + + Вычислим величину относительной косвенной погрешности измерения $ q_{1} $ и $ q_{2} $: + $$ \frac{\sigma_{qкос}}{q} = \sqrt{(\frac{\sigma_{T_{0}}}{T_{0}})^{2} + (\frac{\sigma_{P_{0}}}{P_{0}})^{2} + (\frac{\sigma_{t}}{t})^{2}} $$ + Косвенная погрешность для среднего значения $ q $: $ \sigma_{q_{1}кос} = 0.00046 \; \frac{г}{с} $, $ \sigma_{q_{2}кос} = 0.00035 \; \frac{г}{с}$ + + Суммарная погрешность: + $$ \sigma_{\overline{q}} = \sqrt{(\sigma_{q})^{2}+(\sigma_{qкос})^2 } $$ + $$ \sigma_{\overline{q_{1}}} = 0.0006 \; \frac{г}{с} $$ + $$ \sigma_{\overline{q_{2}}} = 0.0004 \; \frac{г}{с} $$ + + \item Оценим величину тока нагревателя $ I_{0} $, требуемого для нагрева воздуха на $ \delta Q = 1К $. Оценим минимальную мощность $ N_{0} $, необходимую для нагрева газа при максимальном расходе $ N_{0} = c_{p}q\Delta T \approx 0.23 \; Вт $ в первом случае, $ N_{0} = c_{p}q\Delta T \approx 0.176 \; Вт $ во втором случае. + + Учитывая, что сопротивление проволоки нагревателя составляет приблизительно $ R_{н} \approx 35 \; Ом $ и в процессе опыта практически не меняется, искомое значение тока: $ I_{0}=\sqrt{\frac{N_{0}}{R_{н}}} \approx 0.08 \; А $ (в первом случае), $ I_{0}=\sqrt{\frac{N_{0}}{R_{н}}} \approx 0.07 \; А $(во втором случае). + \item Зафиксируем $ \Delta T(N) $ для двух значений $ q $ (между измерениями калориметр надлежит остудить): + $ q_{1} = 0.236 \; г/с $, $ q_{2} = 0.179 \; г/с $. + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|c|c|} + \hline + $U, В$ & $I, мА$ & $ N $, Вт & $ \varepsilon, мкВ $ & $ \Delta T, К $ \\ + \hline + 6.105 & 169.81 & 1.0367 & 154 & 3.78 \\ + \hline + 5.576 & 155.02 & 0.8644 & 127 & 3.12 \\ + \hline + 5.334 & 148.46 & 0.7919 & 120 & 2.95 \\ + \hline + 4.947 & 133.77 & 0.6618 & 102 & 2.51 \\ + \hline + + \end{tabular} + \end{center} + \caption{Зависимость $ \Delta T(N)$ для $ q_{1} $} + \end{table} + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|c|c|} + \hline + $U, В$ & $I, мА$ & $ N $, Вт & $ \varepsilon, мкВ $ & $ \Delta T, К $ \\ + \hline + 5.934 & 165.82 & 0.984 & 186 & 4.57 \\ + \hline + 5.021 & 140.4 & 0.7049 &131 & 3.22 \\ + \hline + 4.572 & 127.83 & 0.5844 & 108 & 2.65 \\ + \hline + 4.231 & 118.3 & 0.5005 & 93 & 2.29 \\ + \hline + + \end{tabular} + \end{center} + \caption{Зависимость $ \Delta T(N)$ для $ q_{2} $} + \end{table} + +\end{enumerate} +\subsection{Обработка данных} +Построим на одном графике зависимости $ \Delta T(N) $ при $ q_{1} $ и $ q_{2} $. +Коэффициент аппроксимирующей прямой найдем по формуле $$ k = \frac{\overline{\Delta T N}}{\overline{N^{2}}} $$ + +Итого, $ k_{1} = 3.68 $, $ k_{2} = 4.6 $. +$$ \sigma_{k} = \sqrt{\frac{1}{n-1}(\frac{\overline{\Delta T^{2}}}{\overline{N^{2}}}-k^{2})} $$ +$$ \sigma_{k_{1}} \approx 0.023 \; \frac{К}{Вт} $$, $$ \sigma_{k_{2}} \approx 0.0188 \; \frac{К}{Вт} $$ +\begin{figure}[h!] + \centering + \begin{gnuplot}[terminal=epslatex] + set grid +set xlabel '$ N $, Вт' +set ylabel '$ \Delta T $, C' + +set multiplot +set yrange [0:1500] +set xrange [0:7] +set key spacing 2 +set key bottom right +set key off + +set style line 1 lt 1 pt 7 ps 0.5 lc rgb "red" +set style line 2 lt 1 pt 7 ps 0.5 lc rgb "blue" +set style line 4 lc black + +set key box linestyle 4 +set key opaque Left + +plot "2.1.3(22.8).data" using 1:2 notitle linestyle 1 + + \end{gnuplot} +\end{figure} + +Воспользуемся соотношениями $ C_{p}q_{1} + \alpha = \frac{1}{k_{1}}, \; C_{p}q_{2} + \alpha = \frac{1}{k_{2}} $, откуда имеем +$$ C_{p} = \frac{k_{2}-k_{1}}{k_{2}k_{1}(q_{1}-q_{2})} $$ +$$ \alpha = \frac{1}{k_{1}} - C_{p}q_{1} $$ +Отсюда имеем: $ C_{p} \approx 1.006 \; \frac{Дж}{г*К} $, $ \alpha \approx 0.042 \; \frac{Вт}{К} $ + +Определим долю тепловых потерь: $ \frac{N_{пот}}{N} = \frac{\alpha}{C_{p}q + \alpha}$. +$$ \frac{N_{пот}}{N_{1}} \approx 0.153$$ +$$ \frac{N_{пот}}{N_{2}} \approx 0.189$$ + +Определим погрешность измерения $ C_{p} $: +$$ \sigma_{C_{p}} = \sqrt{(\frac{\sigma_{k_{1}}}{k_{1}^{2}(q_{1}-q_{2})})^{2}+(\frac{\sigma_{k_{2}}}{k_{2}^{2}(q_{1}-q_{2})})^{2}+ (\frac{k_{2}-k_{1}}{k_{2}k_{1}(q_{1}-q_{2})^{2}})^{2}(\sigma_{q_{1}}^{2}+\sigma_{q_{2}}^{2})^{2}} $$ +Итого, $\sigma_{C_{p}} \approx 0.165 \;\frac{Дж}{г*К}$ + +$$ \sigma_{\alpha} = \sqrt{(\frac{\sigma_{k_{1}}}{k_{1}^{2}})^{2} + (q_{1}\sigma_{C_{p}})^{2} + (C_{p}\sigma_{q_{1}})^{2} } \approx 0.002 \; \frac{Вт}{К} $$ + +$$ \frac{\sigma_{N_{пот}}}{N} = \frac{1}{(C_{p}q+\alpha)^{2}}\sqrt{(C_{p}q\sigma_{\alpha})^{2}+(\alpha q \sigma_{C_{p}})^{2} + (\alpha C_{p} \sigma_{q})^{2} }$$ +$$ \frac{\sigma_{N_{пот}}}{N_{1}} \approx 0.0217, \; \frac{\sigma_{N_{пот}}}{N_{2}} \approx 0.0257 $$ +\subsection{Обсуждение результатов} +Полученное значение $ C_{p} = 1.006 \pm 0.165 \; \frac{Дж}{г*К} $ практически совпадает с табличным $ C_{p}^{табл} = 1.004 \; \frac{Дж}{г*К} $ +Предположение о линейной зависимости $ \Delta T (N) $ подтвердилось. +\subsection{Вывод} +Экспериментальным путем мы смогли определить крайне близкое к реальному значение удельной теплоемкости воздуха при постоянном давлении. \end{document} diff --git a/2.1.1/lab.tex1 b/2.1.1/lab.tex1 new file mode 100644 index 0000000..8c61076 --- /dev/null +++ b/2.1.1/lab.tex1 @@ -0,0 +1,275 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{cmap} +\usepackage[utf8]{inputenc} +\usepackage[warn]{mathtext} +\usepackage{epsf,amsmath,amsfonts,amssymb,amsbsy} +\usepackage[mathscr]{eucal} +\usepackage[english, russian]{babel} +\usepackage{gnuplottex} + +\author{Гришаев Григорий С01-119} +\title{Отчёт о выполнении лабораторной работы 2.1.1} +\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} +\usepackage{graphicx} +\usepackage{indentfirst} +\graphicspath{{images/}} +\DeclareGraphicsExtensions{.pdf,.png,.jpg} +\usepackage{pgfplots} +\begin{document} + \maketitle + \begin{center} + {\Large Измерение удельной теплоёмкости воздуха при постоянном давлении} + \end{center} + \paragraph*{Цель работы:}измерить повышение температуры воздуха в зависимости от мощности подводимого тепла и расхода при стационарном течении через трубу; исключив тепловые потери, по результатам измерений определить теплоёмкость воздуха при постоянном давлении + \paragraph*{В работе используются:}теплоизолированная стеклянная трубка; электронагреватель; источник питания постоянного тока; амперметр, вольтметр (цифровые мультиметры); термопара, подключенная к микровольтметру; компрессор; газовый счётчик; секундомер. + + + \section{Теоретические сведения} + Определение теплоёмкости обычно производится в калориметрах. При этом регистрируется количество тепла {Q}, полученное телом, и изменение температуры этого тела $ {\Delta T} $. + Теплоёмкость тела определяется как их отношение: + \begin{equation} + \label{eq1} + C = \frac{\delta Q}{dT} \; + \end{equation} + Необходимо, чтобы количество тепла, затрачиваемое на нагревание исследуемого тела, существенно превосходило тепло, расходуемое на нагревание самого калориметра, а также на потери тепла из установки. + Для увеличения количества нагреваемого газа при неизменных размерах установки в нашей работе исследуемый газ (воздух) продувается через калориметр, внутри которого установлен нагреватель. При этом измеряются мощность нагревателя, масса воздуха, протекающего в единицу времени (расход), и приращение его температуры. + Рассмотрим газ, протекающий стационарно слева направо через трубу постоянного сечения, в которой установлен нагревательный элемент (см. рис. 1). Пусть за некоторое время $ dT $ через калориметр прошла малая порция газа массой $ dm = qdt $, где $ q $ [кг/с] — массовый расход газа в трубе. Если мощность нагрева равна $ N $, мощность тепловых потерь на обмен с окружающей средой $ N_{пот} $, то порция получила тепло $ \delta Q = (N - N_{пот})dt $. С другой стороны, по определению теплоёмкости \ref{eq1}, $ \delta Q = c dm \Delta T$, где $ \Delta T = T_{2} - T_{1}$ - приращение температуры газа, и $ c $ — удельная (на единицу массы) теплоёмкость газа в рассматриваемом процессе. При малых расходах газа и достаточно большом диаметре трубы перепад давления на её концах мал, поэтому можно принять, что $ P_{1} \approx P_{2} = P_{0} $, где $ P_{0} $ — атмосферное давление. Следовательно, в условиях опыта измеряется удельная теплоёмкость при постоянном давлении $ c_{p} $. Таким образом, получаем + \begin{equation} + \label{eq2} + c_{p} = \frac{N - N_{пот}}{q \Delta T}; + \end{equation} + \begin{figure}[h!] + \center{\includegraphics[width=0.5\linewidth]{1}} + \caption{Нагрев газа при течении по трубе} + \end{figure} + + + \subsection{Экспериментальная установка:} + + Схема установки изображена на рис.2. Воздух, нагнетаемый компрессором, прокачивается через калориметр. Калориметр представляет собой стеклянную цилиндрическую трубку с двойными стенками, запаянными с торцов. + + \begin{figure}[h] + \center{\includegraphics{3}} + \caption{Схема экспериментальной установки} + \end{figure} + + Нагреватель в виде намотанной на пенопласт нихромовой проволоки расположен внутри калориметра непосредственно в воздушном потоке. Нагрев проволоки производится от регулируемого источника постоянного тока (ИП). + Напряжение $U$ на нагревателе и ток $I$ через него регистрируются цифровыми мультиметрами. Таким образом, мощность нагрева равна + \begin{equation} + \label{eq3} + N= UI \; + \end{equation} + Для измерения разности температур $\Delta T$ служит медно-константановая + термопара. Один спай термопары расположен в струе воздуха, входящего в + калориметр, и находится при комнатной температуре, а второй — в струе выходящего нагретого воздуха. Константановая проволока термопары расположена внутри калориметра, а медные проводники подключены к цифровому вольтметру. Возникающая в термопаре ЭДС $\varepsilon$ пропорциональна разности температур $\Delta T$ спаев: + \begin{equation} + \label{eq4} + \varepsilon =\beta \Delta T \; + \end{equation} + где $\beta = 40.7 \frac{мкВ}{^\circ C}$ — чувствительность медно-константановой термопары в рабочем диапазоне температур (20–30 $^\circ C$ ). ЭДС регистрируется с помощью микровольтметра. + + Объём воздуха, прошедшего через калориметр, измеряется газовым счётчиком ГС. Для регулировки расхода служит кран К. Время $ \Delta t $ прохождения некоторого объема $ \Delta V $ воздуха измеряется секундомером. Объёмный расход равен $ \Delta V/\Delta T $, массовый расход может быть найден как + \begin{equation} + \label{eq5} + q = \rho_{o}\frac{\Delta V}{\Delta t} \; + \end{equation} + где $ \rho_{0} $ — плотность воздуха при комнатной температуре, которая в свою очередь может быть получена из уравнения Менделеева–Клапейрона: $ \rho_{0} = \frac{\mu P_{0}}{R T_{0}} $, где $ P_{0} $ — атмосферное давление, $ T_{0} $ — комнатная температура (в Кельвинах), $ \mu $= 29,0 г/моль — средняя молярная масса (сухого) воздуха. + Учитывая особенности устройства калориметра, следует ожидать, что мощность нагревателя расходуется не только на нагрев массы прокачиваемого воздуха, но и частично теряется за счет нагрева внутренних стенок термостата и рассеяния тепла через торцы термостата. Можно предположить, что при небольшом нагреве ($ \Delta T << T_{0} $) мощность потерь тепла $ N_{пот} $ прямо +пропорциональна разности температур: +\begin{equation} + \label{eq6} + N_{пот} = \alpha \Delta T \; +\end{equation} +где $ \alpha $ — некоторая константа. При этом условии основное соотношение \ref{eq2} принимает вид +\begin{equation} + \label{eq7} + N = (c_{p}q + \alpha)\Delta T \; +\end{equation} +Следовательно, при фиксированном расходе воздуха ($ q $ = const) подводимая мощность и разность температур связаны прямой пропорциональностью ($ \Delta T{N} $ — линейная функция). + \subsection{Оборудование и инструментальные погрешности} + Для измерения $ \varepsilon $ использовался вольтметр универсальный В7-78/1, его абсолютная погрешность равна $ \pm (0.005 \frac{\varepsilon}{100} + 3.5) \; мкВ$. + + Для измерения $ U $ использовался цифровой универсальный вольтметр GDM-8145, абсолютная погрешность которого равна $ \pm (0.0003U \pm 4 \; мВ) $. + Его же мы использовали для измерения $ I $, абсолютная погрешность: $ \pm (0.002I+20 \; мкА) $. + \subsection{Ход работы} + \begin{enumerate} + \item Запишем показания комнатной температуры, давления и влажности воздуха. + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|} + \hline + & Значение & $ \sigma $ \\ + \hline + $ T_{0} $, К & 295.45 & 0.1 \\ + \hline + $ p $, Па & 99375.15 & 1 \\ + \hline + $ \phi $ & 64\% & 1\% \\ + \hline + + \end{tabular} + \end{center} + \end{table} + +Рассчитываем плотность воздуха $ \rho_{0} $: + $$ \rho_{0} = \frac{2.9 * 10^{-3}*99375.15}{8.31*295.45} \approx 1.17 \; кг/м^{3} $$ + Относительную погрешность вычисления плотности воздуха вычислим по формуле: + $$ \frac{\sigma_{\rho_{0}}}{\rho_{0}} = \sqrt{(\frac{\sigma_{P_{0}}}{P_{0}})^{2}+(\frac{\sigma_{\phi}}{\phi})^{2}+(\frac{\sigma_{T}}{T})^{2}} $$ + $$ \Rightarrow \sigma_{\rho_{0}} = 0.018 \; \frac{кг}{м^{3}} $$ + \item Рассчитываем теоретическую теплоемкость воздуха при постоянном давлении (в предположении, что воздух - смесь двух идеальных двухатомных газов): + $$ C_{p} = \frac{7}{2}R \approx 29.09 \; \frac{Дж}{моль*К} $$ + $$ C_{p}^{\mu} = \frac{C_{p}}{\mu} \approx 1 \; \frac{Дж}{г*К} $$ + + \item С помощью газового счетчика и секундомера измерим расход воздуха для двух случаев, пользуясь формулой \ref{eq5} ($ q_{1} $ и $ q_{2} $). Результаты измерений представлены в таблице. + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|c|c|} + \hline + $ \Delta V $, л & $ \Delta t $, с & $ \frac{\Delta V}{\Delta t} \; , \frac{л}{с} $ & $ q_{1} \; , \frac{г}{с}$ & $ \sigma_{q_{1}кос}, \; \frac{г}{с}*10^{-3} $\\ + \hline + 5 & 24.9 & 0.2 & 0.234 & 0.47 \\ + \hline + 5 & 25.6 & 0.2 & 0.234 & 0.47 \\ + \hline + 5 & 26.0 & 0.19 & 0.222 & 0.45 \\ + \hline + \end{tabular} + \caption{Измерение $ q_{1} $} + \end{center} + \end{table} + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|c|c|} + \hline + $ \Delta V $, л & $ \Delta t $, с & $ \frac{\Delta V}{\Delta t} \; , \frac{л}{с} $ & $ q_{2} \; , \frac{г}{с}$ & $ \sigma_{q_{2}кос}, \; \frac{г}{с}*10^{-3} $ \\ + \hline + 5 & 32.7 & 0.15 & 0.176 & 0.35 \\ + \hline + 5 & 33.1 & 0.15 & 0.176 & 0.35 \\ + \hline + 5 & 33.2 & 0.15 & 0.176 & 0.35 \\ + \hline + \end{tabular} + \caption{Измерение $ q_{2} $} + \end{center} + \end{table} + Итого, $ q_{1} = \overline{q_{1}} = 0.23 \; г/с $, $ q_{2} = \overline{q_{2}} = 0.176 \; г/с $. + + Вычислим случайную погрешность измерения $ q $: + $$ \sigma_{q} = \sqrt{\frac{1}{n(n-1)}\sum_{i=1}^{n}(q_{i}-\overline q)^{2}} $$ + Итак, $ \sigma_{q_{1}} = 4*10^{-3} \; \frac{г}{с} $, $ \sigma_{q_{2}} \approx 0\; \frac{г}{с} $ + + Вычислим величину относительной косвенной погрешности измерения $ q_{1} $ и $ q_{2} $: + $$ \frac{\sigma_{qкос}}{q} = \sqrt{(\frac{\sigma_{T_{0}}}{T_{0}})^{2} + (\frac{\sigma_{P_{0}}}{P_{0}})^{2} + (\frac{\sigma_{t}}{t})^{2}} $$ + Косвенная погрешность для среднего значения $ q $: $ \sigma_{q_{1}кос} = 0.00046 \; \frac{г}{с} $, $ \sigma_{q_{2}кос} = 0.00035 \; \frac{г}{с}$ + + Суммарная погрешность: + $$ \sigma_{\overline{q}} = \sqrt{(\sigma_{q})^{2}+(\sigma_{qкос})^2 } $$ + $$ \sigma_{\overline{q_{1}}} = 0.0006 \; \frac{г}{с} $$ + $$ \sigma_{\overline{q_{2}}} = 0.0004 \; \frac{г}{с} $$ + + \item Оценим величину тока нагревателя $ I_{0} $, требуемого для нагрева воздуха на $ \delta Q = 1К $. Оценим минимальную мощность $ N_{0} $, необходимую для нагрева газа при максимальном расходе $ N_{0} = c_{p}q\Delta T \approx 0.23 \; Вт $ в первом случае, $ N_{0} = c_{p}q\Delta T \approx 0.176 \; Вт $ во втором случае. + + Учитывая, что сопротивление проволоки нагревателя составляет приблизительно $ R_{н} \approx 35 \; Ом $ и в процессе опыта практически не меняется, искомое значение тока: $ I_{0}=\sqrt{\frac{N_{0}}{R_{н}}} \approx 0.08 \; А $ (в первом случае), $ I_{0}=\sqrt{\frac{N_{0}}{R_{н}}} \approx 0.07 \; А $(во втором случае). + \item Зафиксируем $ \Delta T(N) $ для двух значений $ q $ (между измерениями калориметр надлежит остудить): + $ q_{1} = 0.236 \; г/с $, $ q_{2} = 0.179 \; г/с $. + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|c|c|} + \hline + $U, В$ & $I, мА$ & $ N $, Вт & $ \varepsilon, мкВ $ & $ \Delta T, К $ \\ + \hline + 6.105 & 169.81 & 1.0367 & 154 & 3.78 \\ + \hline + 5.576 & 155.02 & 0.8644 & 127 & 3.12 \\ + \hline + 5.334 & 148.46 & 0.7919 & 120 & 2.95 \\ + \hline + 4.947 & 133.77 & 0.6618 & 102 & 2.51 \\ + \hline + + \end{tabular} + \end{center} + \caption{Зависимость $ \Delta T(N)$ для $ q_{1} $} + \end{table} + \begin{table}[h] + \begin{center} + \begin{tabular}{|c|c|c|c|c|} + \hline + $U, В$ & $I, мА$ & $ N $, Вт & $ \varepsilon, мкВ $ & $ \Delta T, К $ \\ + \hline + 5.934 & 165.82 & 0.984 & 186 & 4.57 \\ + \hline + 5.021 & 140.4 & 0.7049 &131 & 3.22 \\ + \hline + 4.572 & 127.83 & 0.5844 & 108 & 2.65 \\ + \hline + 4.231 & 118.3 & 0.5005 & 93 & 2.29 \\ + \hline + + \end{tabular} + \end{center} + \caption{Зависимость $ \Delta T(N)$ для $ q_{2} $} + \end{table} + +\end{enumerate} +\subsection{Обработка данных} +Построим на одном графике зависимости $ \Delta T(N) $ при $ q_{1} $ и $ q_{2} $. +Коэффициент аппроксимирующей прямой найдем по формуле $$ k = \frac{\overline{\Delta T N}}{\overline{N^{2}}} $$ + +Итого, $ k_{1} = 3.68 $, $ k_{2} = 4.6 $. +$$ \sigma_{k} = \sqrt{\frac{1}{n-1}(\frac{\overline{\Delta T^{2}}}{\overline{N^{2}}}-k^{2})} $$ +$$ \sigma_{k_{1}} \approx 0.023 \; \frac{К}{Вт} $$, $$ \sigma_{k_{2}} \approx 0.0188 \; \frac{К}{Вт} $$ +\begin{figure}[h!] + \centering + \begin{gnuplot}[terminal=epslatex] + set grid +set xlabel '$ N $, Вт' +set ylabel '$ \Delta T $, C' + +set multiplot +set yrange [0:8] +set xrange [0:1.5] +set key spacing 2 +set key bottom right +set key off + +set style line 1 lt 1 pt 7 ps 0.5 lc rgb "red" +set style line 2 lt 1 pt 7 ps 0.5 lc rgb "blue" +set style line 4 lc black + +set key box linestyle 4 +set key opaque Left + +plot "2.1.1(q1).txt" using 1:2 notitle linestyle 1 +plot "2.1.1(q2).txt" using 1:2 notitle linestyle 2 + +plot 3.68*x smooth csplines title "$ q_{1} $" linestyle 1, \ + 4.6*x smooth csplines title "$ q_{2} $" linestyle 2 + \end{gnuplot} +\end{figure} + +Воспользуемся соотношениями $ C_{p}q_{1} + \alpha = \frac{1}{k_{1}}, \; C_{p}q_{2} + \alpha = \frac{1}{k_{2}} $, откуда имеем +$$ C_{p} = \frac{k_{2}-k_{1}}{k_{2}k_{1}(q_{1}-q_{2})} $$ +$$ \alpha = \frac{1}{k_{1}} - C_{p}q_{1} $$ +Отсюда имеем: $ C_{p} \approx 1.006 \; \frac{Дж}{г*К} $, $ \alpha \approx 0.042 \; \frac{Вт}{К} $ + +Определим долю тепловых потерь: $ \frac{N_{пот}}{N} = \frac{\alpha}{C_{p}q + \alpha}$. +$$ \frac{N_{пот}}{N_{1}} \approx 0.153$$ +$$ \frac{N_{пот}}{N_{2}} \approx 0.189$$ + +Определим погрешность измерения $ C_{p} $: +$$ \sigma_{C_{p}} = \sqrt{(\frac{\sigma_{k_{1}}}{k_{1}^{2}(q_{1}-q_{2})})^{2}+(\frac{\sigma_{k_{2}}}{k_{2}^{2}(q_{1}-q_{2})})^{2}+ (\frac{k_{2}-k_{1}}{k_{2}k_{1}(q_{1}-q_{2})^{2}})^{2}(\sigma_{q_{1}}^{2}+\sigma_{q_{2}}^{2})^{2}} $$ +Итого, $\sigma_{C_{p}} \approx 0.165 \;\frac{Дж}{г*К}$ + +$$ \sigma_{\alpha} = \sqrt{(\frac{\sigma_{k_{1}}}{k_{1}^{2}})^{2} + (q_{1}\sigma_{C_{p}})^{2} + (C_{p}\sigma_{q_{1}})^{2} } \approx 0.002 \; \frac{Вт}{К} $$ + +$$ \frac{\sigma_{N_{пот}}}{N} = \frac{1}{(C_{p}q+\alpha)^{2}}\sqrt{(C_{p}q\sigma_{\alpha})^{2}+(\alpha q \sigma_{C_{p}})^{2} + (\alpha C_{p} \sigma_{q})^{2} }$$ +$$ \frac{\sigma_{N_{пот}}}{N_{1}} \approx 0.0217, \; \frac{\sigma_{N_{пот}}}{N_{2}} \approx 0.0257 $$ +\subsection{Обсуждение результатов} +Полученное значение $ C_{p} = 1.006 \pm 0.165 \; \frac{Дж}{г*К} $ практически совпадает с табличным $ C_{p}^{табл} = 1.004 \; \frac{Дж}{г*К} $ +Предположение о линейной зависимости $ \Delta T (N) $ подтвердилось. + +\subsection{Вывод} +Экспериментальным путем мы смогли определить крайне близкое к реальному значение удельной теплоемкости воздуха при постоянном давлении. +\end{document} diff --git a/2.1.1/texput.log b/2.1.1/texput.log new file mode 100644 index 0000000..26621ff --- /dev/null +++ b/2.1.1/texput.log @@ -0,0 +1,21 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.1.14) 13 MAR 2022 16:28 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**shell-escape lab.tex + +! 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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 +/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def +/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def +/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def +/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 +/DiaE {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke} def +/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke} def +/TriUE {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 closepath stroke} def +/TriDE {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 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 closepath gsave fill grestore stroke} 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 +% +/PatternFill {gsave /PFa [ 9 2 roll ] def + 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 + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse + 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 +dup 2 lt + {/InterpretLevel1 true def + /InterpretLevel3 false def} + {/InterpretLevel1 Level1 def + 2 gt + {/InterpretLevel3 Level3 def} + {/InterpretLevel3 false def} + ifelse } + 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 +% +Level1 SuppressPDFMark or +{} { +/SDict 10 dict def +systemdict /pdfmark known not { + userdict /pdfmark systemdict /cleartomark get put +} if +SDict begin [ + /Title (./lab-gnuplottex-fig1.tex) + /Subject (gnuplot plot) + /Creator (gnuplot 5.4 patchlevel 3) +% /Producer (gnuplot) +% /Keywords () + /CreationDate (Mon Apr 11 10:40:37 2022) + /DOCINFO pdfmark +end +} ifelse +% +% Support for boxed text - Ethan A Merritt Sep 2016 +% +/InitTextBox { userdict /TBy2 3 -1 roll put userdict /TBx2 3 -1 roll put + userdict /TBy1 3 -1 roll put userdict /TBx1 3 -1 roll put + /Boxing true def } def +/ExtendTextBox { dup type /stringtype eq + { Boxing { gsave dup false charpath pathbbox + dup TBy2 gt {userdict /TBy2 3 -1 roll put} {pop} ifelse + dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse + dup TBy1 lt {userdict /TBy1 3 -1 roll put} {pop} ifelse + dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse + grestore } if } + {} ifelse} def +/PopTextBox { newpath TBx1 TBxmargin sub TBy1 TBymargin sub M + TBx1 TBxmargin sub TBy2 TBymargin add L + TBx2 TBxmargin add TBy2 TBymargin add L + TBx2 TBxmargin add TBy1 TBymargin sub L closepath } def +/DrawTextBox { PL PopTextBox stroke /Boxing false def} def +/FillTextBox { gsave PopTextBox fill grestore /Boxing false def} def +0 0 0 0 InitTextBox +/TBxmargin 20 def +/TBymargin 20 def +/Boxing false def +/textshow { ExtendTextBox Gshow } def +% +end +%%EndProlog +%%Page: 1 1 +gnudict begin +gsave +doclip +50 50 translate +0.050 0.050 scale +0 setgray +newpath +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +0.500 UL +LTa +LCa setrgbcolor +858 440 M +5945 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +858 440 M +63 0 V +5882 0 R +-63 0 V +stroke +LTb +0.500 UL +LTa +LCa setrgbcolor +858 1024 M +5945 0 V +stroke +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +858 1024 M +63 0 V +5882 0 R +-63 0 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explanation.% + }{Either use 'blacktext' in gnuplot or load the package + color.sty in LaTeX.}% + \renewcommand\color[2][]{}% + }% + \providecommand\includegraphics[2][]{% + \GenericError{(gnuplot) \space\space\space\@spaces}{% + Package graphicx or graphics not loaded% + }{See the gnuplot documentation for explanation.% + }{The gnuplot epslatex terminal needs graphicx.sty or graphics.sty.}% + \renewcommand\includegraphics[2][]{}% + }% + \providecommand\rotatebox[2]{#2}% + \@ifundefined{ifGPcolor}{% + \newif\ifGPcolor + \GPcolorfalse + }{}% + \@ifundefined{ifGPblacktext}{% + \newif\ifGPblacktext + \GPblacktexttrue + }{}% + % define a \g@addto@macro without @ in the name: + \let\gplgaddtomacro\g@addto@macro + % define empty templates for all commands taking text: + \gdef\gplbacktext{}% + \gdef\gplfronttext{}% + \makeatother + \ifGPblacktext + % no textcolor at all + \def\colorrgb#1{}% + \def\colorgray#1{}% + \else + % gray or color? + \ifGPcolor + \def\colorrgb#1{\color[rgb]{#1}}% + 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b/2.1.2/lab.tex @@ -0,0 +1,170 @@ +% pdflatex -shell-escape -enable-write18 lab.tex +%\documentclass[a4paper,12pt]{article} + +\documentclass[a4paper,12pt]{article} + +%%% Работа с русским языком + +%\usepackage{cmap} +%\usepackage[utf8]{inputenc} +%\usepackage[warn]{mathtext} +\usepackage{epsf,amsmath,amsfonts,amssymb,amsbsy} +\usepackage[mathscr]{eucal} +\usepackage[english, russian]{babel} +\usepackage{gnuplottex} + + +\usepackage{cmap} % поиск в PDF +\usepackage{mathtext} % русские буквы в формулах +%\usepackage[T2A]{fontenc} % кодировка +\usepackage[utf8]{inputenc} % кодировка исходного текста +%\usepackage[english,russian]{babel} % локализация и переносы +%\usepackage{gnuplottex} +%Матеша +%\usepackage{amsmath,amsfonts,amssymb,amsthm,mathtools} % AMS +%\usepackage{icomma} % "Умная" запятая +\usepackage{xcolor} +%\mathtoolsset{showonlyrefs=true} % Показывать номера только у тех формул, на которые есть \eqref{} в тексте. + +%% Шрифты +%\usepackage{euscript} % Шрифт Евклид +%\usepackage{mathrsfs} % Красивый матшрифт + +%% Свои команды +%\DeclareMathOperator{\sgn}{\mathop{sgn}} + +%% Перенос знаков в формулах (по Львовскому) +%\newcommand*{\hm}[1]{#1\nobreak\discretionary{} +%{\hbox{$\mathsurround=0pt #1$}}{}} + +%%% Заголовок +\author{Гришаев Григорий С01-119} +\title{Лабораторная работа 2.1.3} +\date{\today} + +\begin{document} + +\maketitle + + +\paragraph*{Цель работы:}определение отношения $C_p / C_v$ углекислого газа по измерения давления в стеклянном сосуде. Измерения производятся сначала после адиабатического расширения газа а затем после нагревания сосуда и газа до комнатной температуры. +\paragraph*{В работе используются:}стеклянный сосуд: U-образный жидкостный манометр; резиновая груша; газгольдер с углекислым газом. + +\begin{figure}[b!] \label{plan2} + + \center{\includegraphics[width=1 \linewidth]{1.jpg}} + \caption{Установка для определения $C_p / C_v$ методом адиабатического расширения газа} + +\end{figure} + + + +\subparagraph*{Экспериментальная установка.} Используемая для опытов экспериментальня установка состоит из стеклянного сосуда А (объёмом около 20 л), снабженного краном К, и U-образного жидкостного манометра, измеряющего избыточное давление газа в сосуде. Схема установки показана на Рис. 1. + +Избыточное давление создаётся с помощью резиновой груши, сосединённой с сосудом трубкой с краном $К_1$. + +В начале опыта в стеклянном сосуде А находится исследуемый газ при комнатной температуре $T_1$ и давлении $P_1$, несколько превышающем атмосферное давление $P_0$. После открытия крана К, соединяющего сосуд А с атмосферой, давление и температура газа будут понижаться. Это уменьшение температуры приближённо можно считать адиабатическим. + +Для адиабатического процесса можно записать следующее уравнение: + +\begin{equation}\label{mk} +\left(\dfrac{P_1}{P_2}\right)^{\gamma - 1} = \left(\dfrac{T_1}{T_2}\right)^\gamma , +\end{equation} + +где индексом "1" обозначено состояние после повышения давления в сосуде и выравнивания температуры с комнатной, а индексом "2" $-$ сразу после открытия крана и выравнивания давления с атмосферным. + +После того, как кран К вновь отсоединит сосуд от атмосферы , происходит медленное изохорическое нагревание газа со скоростью, определяемой теплопроводностью стеклянных стенок сосуда. Вместе с ростом температуры растёт и давление газа. З время порядка $\Delta t_T$  (время установления температуры) система достигает равновесия, и установившаяся температура газа $T_3$ становится равной комнатной температуре $T_1$. + +Тогда используя закон Гей-Люссака для изохорического процесса и уравнение \eqref{mk} найдём $\gamma$: + +\begin{equation}\label{acc} +\gamma = \dfrac{\ln(P_1 / P_0)}{\ln (P_1 / P_3)}. +\end{equation} + +С учётом того, что $P_i = P_0 + \rho g h_i$ и пренебрегая членами второго порядка малости получим из \eqref{acc}: + +\begin{equation}\label{r} +\gamma \approx \dfrac{h_1}{h_1 - h_2}. +\end{equation} + + +\newpage + +\section*{Ход работы} + +\subparagraph*{1.} Перед началом работы убедимся в том, что краны и места сочленений трубок достаточно герметичны. Для этого нужно наполнить баллон углекислым газом до давления, превышающего атмосферное и перекроем кран $К_1$. По U-образному манометру снимем зависимость давления $h$ в баллоне от времени $t$ и построим график зависимости $h = f(t)$. Из графика определим время установления термодинамического равновесия $\Delta t_T$. Стабильное избыточное давление воздуха $h_1$ в баллоне должно быть тщательно измерено. + +\subparagraph*{2.} Откроем кран К на короткое время и закроем его снова. Подождём, пока уровень жидкости в манометре перестанет изменяться. Это произойдёт, когда температура газа в сосуде сравняется с комнатной, примерно через время $\Delta t_T$. Запишем разность уровней жидкости в манометре $h_2$. Проведём серию из 5--8 измерений сначала для времени открытия крана $\Delta t = 0,5 с$, а затем для $\Delta t \approx 1,0 с и \Delta t \approx 1,5 с$. По полученным данным вычислим используя формулу \eqref{r} вычислим $\gamma$ и построим график зависимости $\gamma(\Delta t)$. + +\begin{table}[h!] + \caption{Экспериментальные данные для $\Delta t = 0,5$} + \begin{center} + \begin{tabular}{|*{4}{l|}} + \hline + № & $h_1$ , см & $h_2$, см & $\gamma$ \\ \hline + 1& & & \\ \hline + 2 & & & \\ \hline + 3 & & & \\ \hline + 4 & & & \\ \hline + 5 & & & \\ \hline + 6 & & & \\ \hline + 7 & & & \\ \hline + & & $\gamma_{ср} = $ & $\sigma_{с, \gamma} = $ \\ \hline + \end{tabular} + \end{center} +\end{table} + +\begin{table}[h!] + \caption{Экспериментальные данные для $\Delta t = 1,0$} + \begin{center} + \begin{tabular}{|*{4}{l|}} + \hline + № & $h_1$ , см& $h_2$, см & $\gamma$ \\ \hline + 1 & & & \\ \hline + 2 & & & \\ \hline + 3 & & & \\ \hline + 4 & & & \\ \hline + 5 & & & \\ \hline + & & $\gamma_{ср} = 1.155 $ & $\sigma_{с, \gamma} = 0.003$ \\ \hline + \end{tabular} + \end{center} +\end{table} + + + +\begin{table}[h!] + \caption{Экспериментальные данные для $\Delta t = 2,0$} + \begin{center} + \begin{tabular}{|*{4}{l|}} + \hline + № & $h_1$, см & $h_2$, см & $\gamma$ \\ \hline + 1 & & & \\ \hline + 2 & & & \\ \hline + 3 & & & \\ \hline + 4 & & & \\ \hline + 5 & & & \\ \hline + 6 & & & \\ \hline + & & $\gamma_{ср} = 1.124 $ & $\sigma_{с, \gamma} = 0.007$ \\ \hline + \end{tabular} + \end{center} +\end{table} +\begin{figure}[ht] + \centering + \begin{gnuplot}[terminal=epslatex] +set grid +set multiplot +set yrange [0:1500] +set xrange [0:7] + + set style line 1 lt 1 pt 7 ps 0.5 lc rgb "red" + set style line 2 lt 1 pt 7 ps 0.5 lc rgb "blue" + set style line 3 lt 1 pt 7 ps 0.5 lc rgb "green" + set style line 4 lt 1 pt 7 ps 0.5 lc rgb "gray" + + plot "2.1.3(22.8).data" using 1:2 notitle linestyle 1 + plot "2.1.3(30.1).data" using 1:2 notitle linestyle 2 + plot "2.1.3(40.0).data" using 1:2 notitle linestyle 3 + plot "2.1.3(50.0).data" using 1:2 notitle linestyle 4 + \end{gnuplot} +\end{figure} +\end{document} diff --git a/2.1.2/notes.md b/2.1.2/notes.md new file mode 100644 index 0000000..19de648 --- /dev/null +++ b/2.1.2/notes.md @@ -0,0 +1,2 @@ +Длина трубы - 800 pm 1 мм +T(комн) = 22.8 diff --git a/2.1.2/texput.log b/2.1.2/texput.log new file mode 100644 index 0000000..9b6fb94 --- /dev/null +++ b/2.1.2/texput.log @@ -0,0 +1,21 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.3.24) 17 APR 2022 16:06 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**main.tex + +! 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z9JfkO7yB+n?$ck96fCaVZskh1o^DEEF$A+ig}lO@g|}vqCq@eQrdC+Do=y!beMgm_ k5G;-=ssH^!c5u|QcXa)Ut)Lm%*?$5VXc7`(SrO>}3x19$4gdfE literal 0 HcmV?d00001 diff --git a/2.1.3/lab2.1.3.tex b/2.1.3/lab2.1.3.tex new file mode 100644 index 0000000..e5fa0d4 --- /dev/null +++ b/2.1.3/lab2.1.3.tex @@ -0,0 +1,344 @@ +\documentclass[a4paper,12pt]{article} % добавить leqno в [] для нумерации слева +\usepackage[a4paper,top=1.3cm,bottom=2cm,left=1.5cm,right=1.5cm,marginparwidth=0.75cm]{geometry} +%%% Работа с русским языком +\usepackage{cmap} % поиск в PDF +\usepackage[warn]{mathtext} % русские буквы в фомулах +\usepackage[T2A]{fontenc} % кодировка +\usepackage[utf8]{inputenc} % кодировка исходного текста +\usepackage[english,russian]{babel} % локализация и переносы +%\usepackage{physics} +\usepackage{multirow} +\usepackage{longtable} + +%%% Нормальное размещение таблиц (писать [H] в окружении таблицы) +\usepackage{float} +\restylefloat{table} + + + +\usepackage{graphicx} + +\usepackage{wrapfig} +\usepackage{tabularx} + +\usepackage{hyperref} +\usepackage[rgb]{xcolor} +\hypersetup{ + colorlinks=true,urlcolor=blue +} + +\usepackage{pgfplots} +\pgfplotsset{compat=1.9} + +%%% Дополнительная работа с математикой +\usepackage{amsmath,amsfonts,amssymb,amsthm,mathtools} % AMS +\usepackage{icomma} % "Умная" запятая: $0,2$ --- число, $0, 2$ --- перечисление + +%% Номера формул +\mathtoolsset{showonlyrefs=true} % Показывать номера только у тех формул, на которые есть \eqref{} в тексте. + +%% Шрифты +\usepackage{euscript} % Шрифт Евклид +\usepackage{mathrsfs} % Красивый матшрифт + +%% Свои команды +\DeclareMathOperator{\sgn}{\mathop{sgn}} + +%% Перенос знаков в формулах (по Львовскому) +\newcommand*{\hm}[1]{#1\nobreak\discretionary{} + {\hbox{$\mathsurround=0pt #1$}}{}} + +\date{\today} + +\usepackage{gensymb} + +\begin{document} + +\begin{titlepage} + + \vspace{4.5cm} + {\huge + \begin{center} + {\bf Отчёт о выполнении лабораторной работы 2.1.3}\\ + Определение $ C_p/C_v $ по скорости звука в газе + \end{center} + } + \vspace{2cm} + \begin{flushright} + {\LARGE Автор:\\ Гришаев Григорий Павлович \\ + \vspace{0.2cm} + С01-119} + \end{flushright} +\end{titlepage} + + +\section{Введение} +\textbf{Цель работы:} \begin{enumerate} +\item измерение частоты колебаний и длины волны при резонансе звуковых колебаний в газе, заполняющем трубу; +\item определение показателя адиабаты с помощью уравнения состояния идеального газа. +\end{enumerate} + +\textbf{В работе используются:} звуковой генератор ГЗ; электронный осциллограф ЭО; микрофон; телефон; раздвижная труба; теплоизолированная труба, обогреваемая водой из термостата; баллон со сжатым углекислым газом; газгольдер. + +\section{Теоретические сведения} + +Скорость распространения звуковой волны в газах зависит от показателя адиабаты $ \gamma $. На измерении скорости звука основан один из наиболее точных методов определения показателя адиабаты. + +Скорость звука в газах определяется формулой: + +\begin{equation}\label{velocity} +c=\sqrt{\gamma\frac{RT}{\mu}}. +\end{equation} +где $ R $ -- газовая постоянная, $ T $ -- температура газа, а $ \mu $ -- его молярная масса. Преобразуя эту формулу, найдем +\begin{equation}\label{gamma} +\boxed{\gamma = \frac{\mu}{RT}c^2}. +\end{equation} + +Таким образом, для определения показателя адиабаты достаточно измерить температуру газа и скорость распространения звука (молярная масса газа предполагается известной). + +Звуковая волна, распространяющаяся вдоль трубы, испытывает многократные отражения от торцов. Звуковые колебания в трубе являются наложением всех отраженных волн и очень сложны. Картина упрощается, если длина трубы $ L $ равна целому числу полуволн, то есть когда \[ L=n\lambda/2, \] где $ \lambda $ -- длина волны звука в трубе, а $ n $ -- любое целое число. Если это условие выполнено, то волна, отраженная от торца трубы, вернувшаяся к ее началу и вновь отраженная, совпадает по фазе с падающей. Совпадающие по фазе волны усиливают друг друга. Амплитуда звуковых колебаний при этом резко возрастает -- наступает резонанс. + +При звуковых колебаниях слои газа, прилегающие к торцам трубы, не испытывают смещения. Узлы смещения повторяются по всей длине трубы через $ \lambda/2 $. Между узлами находятся максимумы смещения. + +Скорость звука c связана с его частотой $ f $ и длиной волны $ \lambda $ соотношением + +\begin{equation}\label{lambda_f} +c=\lambda f. +\end{equation} + +Подбор условий, при которых возникает резонанс, можно производить двояко: +\begin{enumerate} + \item При неизменной частоте $ f $ звукового генератора (а следовательно, и неизменной длине звуковой волны $ \lambda $) можно изменять длину трубы $ L $. Для этого применяется раздвижная труба. Длина раздвижной трубы постепенно увеличивается, и наблюдается ряд последовательных резонансов. Возникновение резонанса легко наблюдать на осциллографе по резкому увеличению амплитуды колебаний. Для последовательных резонансов имеем \begin{equation}\label{first} + L_n=n\frac{\lambda}{2}, \quad L_{n+1}=(n+1)\frac{\lambda}{2}, \quad \dots, \quad L_{n+k} = n\frac{\lambda}{2}+k\frac{\lambda}{2}, + \end{equation} т. е. $ \lambda/2 $ равно угловому коэффициенту графика, изображающего зависимость длины трубы $ L $ от номера резонанса $ k $. Скорость звука находится по формуле \eqref{lambda_f}. + \item При постоянной длине трубы можно изменять частоту звуковых колебаний. В этом случае следует плавно изменять частоту $ f $ звукового генератора, а следовательно, и длину звуковой волны $ \lambda $. Для последовательных резонансов получим + \begin{equation}\label{4} + L=\frac{\lambda_1}{2}n=\frac{\lambda_2}{2}(n+1)=\dots=\frac{\lambda_{k+1}}{2}(n+k). + \end{equation} + + Из \eqref{lambda_f} и \eqref{4} имеем: + \[ f_1=\frac{c}{\lambda_1}=\frac{c}{2L}n, \quad f_2=\frac{c}{\lambda_2}=\frac{c}{2L}(n+1)=f_1+\frac{c}{2L},\quad \dots, \] + \begin{equation}\label{5} + f_{k+1}=\frac{c}{\lambda_{k+1}}=\frac{c}{2L}(n+k)=f_1+\frac{c}{2L}k. + \end{equation} + Скорость звука, деленная на $ 2L $, определяется, таким образом, по угловому коэффициенту графика зависимости частоты от номера резонанса. +\end{enumerate} + +\section{Экспериментальная установка} + +Соответственно двум методам измерения скорости звука в работе имеются две установки (рис. \ref{img1} и \ref{img2}). В обеих установках звуковые колебания в трубе возбуждаются телефоном Т и улавливаются микрофоном М. Мембрана телефона приводится в движение переменным током звуковой частоты; в качестве источника переменной ЭДС используется звуковой генератор ГЗ. Возникающий в микрофоне сигнал наблюдается на осциллографе ЭО. + +Микрофон и телефон присоединены к установке через тонкие резиновые трубки. Такая связь достаточна для возбуждения и обнаружения звуковых колебаний в трубе и в то же время мало возмущает эти колебания: при расчетах оба торца трубы можно считать неподвижными, а влиянием соединительных отверстий пренебречь. + +Первая установка (рис. \ref{img1}) содержит раздвижную трубу с миллиметровой шкалой. Через патрубок (на рисунке не показан) труба может наполняться воздухом или углекислым газом из газгольдера. На этой установке производятся измерения $ \gamma $ для воздуха и для $ CO_2 $. Вторая установка (рис. \ref{img2}) содержит теплоизолированную трубу постоянной длины. Воздух в трубе нагревается водой из термостата. Температура газа принимается равной температуре омывающей трубу воды. На этой установке измеряется зависимость скорости звука от температуры. + +\begin{figure}[H] + \begin{center} + \includegraphics[width=12cm]{ust1.jpg} + \end{center} + \caption{\textit{Установка для измерения скорости звука при помощи раздвижной трубы}} + \label{img1} +\end{figure} + +\begin{figure}[H] + \begin{center} + \includegraphics[width=12cm]{ust2.jpg} + \end{center} + \caption{\textit{Установка для изучения зависимости скорости звука от температуры}} + \label{img2} +\end{figure} + +\section{Ход работы} + +\subsection{Измерение $ C_p/C_v $ для воздуха при различных температурах} + +Проведём измерения $ C_p/C_v $ для воздуха при различных температурах. Для этого будем использовать трубу постоянного размера $ L = (740 \pm 1) $ мм. Для фиксированной температуры будем изменять частоту звукового сигнала, тем самым изменяя и длину волны, так, чтобы мы могли наблюдать последовательные резонансы. Для каждого резонанса будем фиксировать частоту, при которой он возник. Полученные измерения занесём в таблицу \ref{tab:constL}. + +\begin{table}[H] + \centering + \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} + \hline + $ T $, К & \multicolumn{2}{c|}{\textbf{303}} & \multicolumn{2}{c|}{\textbf{310}} & \multicolumn{2}{c|}{\textbf{317}} & \multicolumn{2}{c|}{\textbf{323}} & \multicolumn{2}{c|}{\textbf{328}} \\ \hline + k & $ \hat{f_k} $, Гц & $ f_k $, Гц & $ \hat{f_k} $, Гц & $ f_k $, Гц & $ \hat{f_k} $, Гц & $ f_k $, Гц & $ \hat{f_k} $, Гц & $ f_k $, Гц & $ \hat{f_k} $, Гц & $ f_k $, Гц \\ \hline + 0 & 712 & 0 & 255 & 0 & 253 & 0 & 258 & 0 & 260 & 0 \\ \hline + 1 & 941 & 229 & 489 & 234 & 493 & 240 & 488 & 230 & 496 & 236 \\ \hline + 2 & 1171 & 459 & 718 & 463 & 725 & 472 & 732 & 474 & 731 & 471 \\ \hline + 3 & 1410 & 698 & 944 & 689 & 963 & 710 & 971 & 713 & 977 & 717 \\ \hline + 4 & 1640 & 928 & 1185 & 930 & 1201 & 948 & 1203 & 945 & 1216 & 956 \\ \hline + 5 & 1875 & 1163 & 1423 & 1168 & 1436 & 1183 & 1450 & 1192 & 1463 & 1203 \\ \hline + 6 & 2106 & 1394 & 1657 & 1402 & 1677 & 1424 & 1697 & 1439 & 1697 & 1437 \\ \hline + 7 & 2337 & 1625 & 1897 & 1642 & 1917 & 1664 & 1937 & 1679 & 1950 & 1690 \\ \hline + \end{tabular} + \caption{Результаты измерений при разных температурах для воздуха} + \label{tab:constL} +\end{table} + +Также занесём в таблицу величину $ f_k = \hat{f_k} - \hat{f_0} $. Погрешность измерения такой величины составит $ \sigma_{f_k} = \sigma_{\hat{f_k}}\sqrt{2} \approx 2,82 $ Гц. + +По полученным экспериментальным данным построим графики зависимости $ f_k(k) $. + + +\begin{center} + \begin{tikzpicture} + \begin{axis}[ + title={График зависимости $f_k(k)$ для воздуха}, + xlabel={Номера резонанса $ k $}, + ylabel={Резонансная частота $ f $, Гц}, + legend pos=north west, + xmajorgrids=true, + ymajorgrids=true, + grid style=dashed, + /pgf/number format/.cd,% + set thousands separator={}, + set decimal separator={,}, + xmin = 0, + %xmax = 4.3, + ymin = 0, + %ymax = 5500, + width = 510, + height = 670, + ] + \legend{ + $ T = 303 $ К,, + $ T = 310 $ К,, + $ T = 317 $ К,, + $ T = 323 $ К,, + $ T = 328 $ К,, + }; + + \addplot+ [black, only marks, mark size = 3pt, + mark=*, + mark options = { + fill = red, + draw = black}, + error bars/.cd, + y dir=both, y explicit, + ] table [x = k, y = L, y error = dL,] { + k L dL + 0 0 3 + 1 229 3 + 2 459 3 + 3 698 3 + 4 928 3 + 5 1163 3 + 6 1394 3 + 7 1625 3 + }; + \addplot [red, domain=0:7.1, line width = 2.2pt] { 232.192857085818 * x}; + + \addplot+ [black, only marks, mark size = 3pt, + mark options = { + fill = blue, + draw = black}, + error bars/.cd, + y dir=both, y explicit, + ] table [x = k, y = L, y error = dL,] { + k L dL + 0 0 3 + 1 234 3 + 2 463 3 + 3 689 3 + 4 930 3 + 5 1168 3 + 6 1402 3 + 7 1642 3 + }; + \addplot [blue, domain=0:7.1, line width = 2.2pt] { 233.521428412963 * x}; + + \addplot+ [black, only marks, mark size = 3pt, + mark options = { + fill = green, + draw = black}, + error bars/.cd, + y dir=both, y explicit, + ] table [x = k, y = L, y error = dL,] { + k L dL + 0 0 3 + 1 240 3 + 2 472 3 + 3 710 3 + 4 948 3 + 5 1183 3 + 6 1424 3 + 7 1664 3 + }; + \addplot [green, domain=0:7.1, line width = 2.2pt] { 237.235714303741 * x}; + + \addplot+ [black, only marks, mark size = 3pt, + mark options = { + fill = orange, + draw = black}, + error bars/.cd, + y dir=both, y explicit, + ] table [x = k, y = L, y error = dL,] { + k L dL + 0 0 3 + 1 230 3 + 2 474 3 + 3 713 3 + 4 945 3 + 5 1192 3 + 6 1439 3 + 7 1679 3 + }; + \addplot [orange, domain=0:7.1, line width = 2.2pt] { 238.885714210443 * x}; + + \addplot+ [black, mark=halfcircle*, only marks, mark size = 3pt, + mark options = { + fill = violet, + draw = black}, + error bars/.cd, + y dir=both, y explicit, + ] table [x = k, y = L, y error = dL,] { + k L dL + 0 0 3 + 1 236 3 + 2 471 3 + 3 717 3 + 4 956 3 + 5 1203 3 + 6 1437 3 + 7 1690 3 + }; + \addplot [violet, domain=0:7.1, line width = 2.2pt] { 240.142856845106 * x}; + + \end{axis} + \end{tikzpicture} +\end{center} + +Аппроксимируем полученные зависимости прямыми $ y=ax $ используя метод наименьших квадратов. Коэффициент $ a $ и погрешности его определения находим согласно формулам \eqref{mnk:a}, \eqref{mnk:sigma_a} и \eqref{mnk:full_sigma}. Результаты вычислений для каждой температуры заносим в таблицу \ref{tab:resConstL}. + +\begin{table}[H] + \centering + \begin{tabular}{|c|c|c|c|c|c|c|} + \hline + $ T $, К & $ a $, с$ ^{-1} $ & $ \sigma_a $, с$ ^{-1} $ & $ c $, м/с & $ \sigma_c $, м/с & $ \gamma $ & $ \sigma_\gamma $ \\ \hline + 303 & 232,2 & 0,3 & 343,6 & 0,6 & 1,358 & 0,005 \\ \hline + 310 & 233,5 & 0,3 & 345,6 & 0,6 & 1,343 & 0,005 \\ \hline + 317 & 237,2 & 0,3 & 351,1 & 0,6 & 1,355 & 0,005 \\ \hline + 323 & 238,9 & 0,3 & 353,6 & 0,6 & 1,349 & 0,005 \\ \hline + 328 & 240,1 & 0,3 & 355,4 & 0,6 & 1,342 & 0,005 \\ \hline + \end{tabular} + \caption{Результаты вычислений при различных температурах} + \label{tab:resConstL} +\end{table} + +Также, согласно формуле \eqref{5}, коэффициент наклона $ \displaystyle a = \frac{c}{2L}$. Тогда вычислим скорость звука $ c $ при фиксированной температуре и её погрешность, результаты вычислений занесём в таблицу \ref{tab:resConstL}. + +Кроме того, по формуле \eqref{gamma} вычислим $ \gamma $ при фиксированной температуре и погрешность этого вычисления. Результаты занесём в таблицу $ \ref{tab:resConstL} $. + +Согласно полученным данным, можно утверждать, что $ \gamma $ остаётся постоянной в исследуемом диапазоне температур. Поэтому усредним результаты, полученные при различных значениях температуры и получим для воздуха: + +\[ \boxed{\gamma = 1,350 \pm 0,004}\quad (\varepsilon=0,3\%) \] + +\section{Обсуждение результатов и выводы} + +В ходе выполнения работы мы измерили частоту колебаний и длину волны при резонансе звуковых колебаний в газе, заполняющем экспериментальную установку. + +Измерения проводились на установке, на которой длина трубы оставалась постоянной на протяжении всего опыта, а резонанса мы добивались при помощи изменения частоты звукового сигнала. В ходе этих измерений также исследовалась зависимость коэффициента адиабаты $ \gamma $ от температуры газа. Было получено, что показатель адиабаты не зависит от температуры в диапазоне температур $ 20-60 $ $ ^\circ C $ и равняется: + +\[ \boxed{\gamma_L = 1,350 \pm 0,004}\quad (\varepsilon=0,5\%) \] + +Сравним полученные данные с табличными. Согласно справочнику, показатель адиабаты для воздуха при нормальных условиях равен \underline{$ \gamma = 1,4 $}. Таким образом, можно утверждать, что результаты измерения незначительно отличаются от табличных. Это может быть связано с большой неточностью определения резонансных частот. Чтобы этого избежать, необходимо использовать генератор частоты с возможностью более точной настройки для возможности чёткого отслеживания резонансов. + +\end{document} diff --git a/2.1.3/ust1.jpg b/2.1.3/ust1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..7c744b9ff154f9c3f65f185b6c0237dfbb4aec54 GIT binary patch literal 23898 zcmeHv2|QG7|M!tZqzRFTsgTN6B85yFl1hpul&O?8xmzsNm}5znER|?cMp241$(m&B zk|fC?1i)x{1ODZYZw46SEJ2~!!XltS#yn^SA3y($nf$y_gVx^Q zduXPZfcTQtn`TLv9~NACa=xbTmAgU;n^OxU_f*pr*Bm+JH(OY0f%HNdrDe*?SFF_1 zUaPZiz3!H+hDOHQOt$Z}u-u2=Z)I(F^q9Saqm%RLGwvRqUT4qwUkJDu7!(|GHS$_i z^z|DzZ{3Sexc?yW;iIHyY3Ui6S=rBX3X6(MO3PlAzkXlyp|-BRp|PpGqw`bO=kA^_ zy^P_J(J|&YGBL@U7azp`^R#}O*_3&SfqBiCIg@{;Aa7oLGd#h?FE&$P$?93+o6H3d zpOjdt=_@pU^Od`)1+x{_?4e5@IaMtzrKr`e#NbWs>&*Vz#QgrPnf*4gf6c22n#0cr zHjiHn!aEl8^RGCp+lXAO307In6BPSsG(&Mw}R+$7Rxx$IP z<@|0Ip61cPK}GUHJzkK!ui^aNe1G|pqzkeQC~bipVcI+#{h%EeD!;}z&iT=8xTqHu zg5RYu__>fk9gcl-BNnbEK+|p;|I^{mYslb2`2=L8U}060az|+ZeZ1(LV5n+Fie1&P 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0.5 lc rgb "blue" +set style line 4 lc black + +set key box linestyle 4 +set key opaque Left + +plot "PT.txt" using 1:2 notitle linestyle 1 +plot "PT2.txt" using 1:2 notitle linestyle 2 + +#plot "2.1.1(q2).txt" using 1:2 notitle linestyle 2 + +#plot 3.68*x smooth csplines title "$ q_{1} $" linestyle 1, \ +# 4.6*x smooth csplines title "$ q_{2} $" linestyle 2 diff --git a/2.4.1/lab2.4.1-gnuplottex-fig1.tex b/2.4.1/lab2.4.1-gnuplottex-fig1.tex new file mode 100644 index 0000000..782cc4a --- /dev/null +++ b/2.4.1/lab2.4.1-gnuplottex-fig1.tex @@ -0,0 +1,160 @@ +% GNUPLOT: LaTeX picture with Postscript +\begingroup + \makeatletter + \providecommand\color[2][]{% + \GenericError{(gnuplot) \space\space\space\@spaces}{% + Package color not loaded in conjunction with + terminal option `colourtext'% + }{See the gnuplot documentation for explanation.% + }{Either use 'blacktext' in gnuplot or load the package + color.sty in LaTeX.}% + \renewcommand\color[2][]{}% + }% + 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Такой метод, однако, не позволяет получить точных результатов из-за неконтролируемых потерь тепла, которые трудно сделать малыми. В настоящей работе для определения теплоты испарения применен +косвенный метод, основанный на формуле Клапейрона–Клаузиуса: + +$$\frac{dP}{dT} = \frac{L}{T(V_2 - V_1)}\;(1).$$ + +Здесь $P$ — давление насыщенного пара жидкости при температуре $T$, $T$ — абсолютная температура жидкости и пара, $L$ — теплота испарения жидкости, $V_2$ — объем пара, $V_1$ — объем жидкости. +Найдя из опыта $\frac{dP}{dT},\; T,\; V_2$ и $V_1$, можно определить $L$ путем расчета. Величины $L, \;V_2$ и $V_1$ в формуле (1) должны относиться к одному и тому же количеству вещества; мы будем относить их к одному молю. + +В нашем приборе измерения производятся при давлениях ниже атмосферного. В этом случае задача существенно упрощается. + +С помощью уравнения Ван-дер-Ваальса можно получить зависимость $P(T)$, с помощью которой определить искомую величину: + +$$(P+\frac{a}{V^2})(V-b)=RT \; (2)$$ +В таблице ниже приведены все значения параметров различных жидкостей уранения Ван-дер-Ваальса в условиях данного опыта. +\begin{figure}[h] + \center{\includegraphics[scale=1]{tabl}} +\end{figure} +Откуда видно, что $\frac{V_1}{V_2} < 0.005$, a $\frac{a}{PV^2}<0.03$, ошибка метода измерений равна 4\%, тогда записав уравнение Клапейрона-Менделеева для насыщенного пара, получим: +$V=\frac{RT}{P}\;.$ +Пренебрегая $V_1$ (который не превосходит $0,5\%$ от $V_2$), запишем: +$$L=\frac{RT^2}{P} \frac{dP}{dT} = -R\frac{d(lnP)}{d(1/T)}\;(3).$$ +Эта формула является окончательной. + +\section{Экспериментальная установка:} + +Схема установки изображена на рисунке 1. Установка включает термостат +\begin{figure}[h] + \center{\includegraphics[scale=0.5]{2}} + \caption{Схема установки для определения теплоты испарения} +\end{figure} + +\section{Ход работы} + +\begin{enumerate} +\item Измерим разность уровней в ртутном U-образном манометре с помощью микроскопа и температуру по термометру. $H$ - высота высокого колена, $h$ -низкого. При этом будем настраивать микроскоп, так, чтобы каждый раз основание мениска было у метки прибора (в дальнейшем считаем, что высота мениска не меняется, не смотря на то что поверхностное натяжение ртути на самом деле зависит от температуры и высота немного должна меняется). Результаты представлены в таблицах 1 и 2. Под $P_0$ подразумевается давление 1 мм рт.ст. + +Погрешность определения температуры возьмём учитывая точность прибора и тот факт, что во время измерений уровней температура могла немного изменяться $$\sigma_{T} = 0.2 \; K.$$ Соответсвенно $$\sigma_{\frac{1}{T}} = \frac{\sigma_T}{T^2} \; K^{-1}.$$ + +Исследуемая жидкость - вода. + +\begin{center} + \begin{table}[h] + \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} + \hline + $h ,\; {мм}$ & 83.3 & 83.05 & 82.75 & 81.95 & 81.25 & 80.5 & 79.9 & 78.7 & 78.05 & 77.2 & 76.55 & 76.00 & 74.7 & 73.75 & 72.45 & 71.15& 70 & 68.7 & 67.85 \\ + \hline + $H, \; {мм}$ & 109.1 & 109.35 & 110.2 & 110.8 & 111.65 & 112.4 & 113.5 & 114.2 & 115.3 & 116.0 & 117.1 & 117.7 & 118.8 & 120 & 121.35 & 122.45 & 123.7 & 125.15 & 126.4 \\ + \hline + $t,\; ^\circ С$ & 22.04 & 23.01 & 24.00 & 25.04 & 26.00 & 27.03 & 28.02 & 29.05 & 30.03 & 31.04 & 32.00 & 33.04 & 34.03 & 35.03 & 36.05 & 37.04 & 38.03 & 39.04 & 40.02 \\ + \hline +% $P, {кПа}$ & & & & & & & & & & & \\ +% \hline +% $\sigma_{P}, {Па}$ & \multicolumn{11}{|c|}{} \\ +% \hline +% $\ln(\frac {P}{P_0})$ & & & & & & & & & & & \\ +% \hline +% $\sigma_{\ln(\frac {P}{P_0})}$ & & & & & & & & & & & \\ +% \hline +% $T,\; {K}$ & & & & & & & & & & & \\ +% \hline +% $\sigma_{T},\; {K}$ &\multicolumn{11}{|c|}{} \\ +% \hline +% $\frac{1}{T}$ $\cdot 10^{-3},{K}^{-1}$ & & & & & & & & & & & \\ +% \hline +% $\sigma_{\frac{1}{T}} \cdot 10^{-6},\; {K}^{-1}$ & & & & & & & & & & & \\ +% \hline +\end{tabular} + \caption{При нагреве} +\end{table} +\end{center} + +\begin{center} +\begin{table}[h] +\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} + \hline + $h ,\; {мм}$ & 67.85 & 69.6 & 71.45 & 73.15 & 73.15 & 76.05 & 78 & & & & \\ + \hline + $H, \; {мм}$ & 126.4 & 124.5 & 122.3& 119.9 & 119.9 & 117.8 & 115.45 & & & & \\ + \hline + $t,\; ^\circ С$ & 40.02 & 38.04 & 36.05 & 34.04 & 32.03 & 30.03 & 28. & & & & \\ + \hline +% $P, {кПа}$ & & & & & & & & & & & \\ +% \hline +% $\sigma_{P}, {Па}$ & \multicolumn{11}{|c|}{} \\ +% \hline +% $\ln(\frac {P}{P_0})$ & & & & & & & & & & & \\ +% \hline +% $\sigma_{\ln(\frac {P}{P_0})}$ & & & & & & & & & & & \\ +% \hline +% $T,\; {K}$ & & & & & & & & & & & \\ +% \hline +% $\sigma_{T},\; {K}$ &\multicolumn{11}{|c|}{} \\ +% \hline +% $\frac{1}{T}$ $\cdot 10^{-3},{K}^{-1}$ & & & & & & & & & & & \\ +% \hline +% $\sigma_{\frac{1}{T}} \cdot 10^{-6},\; {K}^{-1}$ & & & & & & & & & & & \\ +% \hline +\end{tabular} + \caption{При остывании} +\end{table} +\end{center} + +\begin{figure}[h!] + \centering + \begin{gnuplot}[terminal=epslatex] + set grid + #set xlabel '$ N $, Вт' + #set ylabel '$ \Delta T $, C' + + set multiplot + set yrange [20:65] + set xrange [22:50] +set key spacing 2 +set key bottom right +set key off + +set style line 1 lt 1 pt 7 ps 0.5 lc rgb "red" +set style line 2 lt 1 pt 7 ps 0.5 lc rgb "blue" +set style line 4 lc black + +set key box linestyle 4 +set key opaque Left + +plot "PT.txt" using 1:2 notitle linestyle 1 +plot "PT2.txt" using 1:2 notitle linestyle 2 + + #plot "2.1.1(q2).txt" using 1:2 notitle linestyle 2 + + #plot 3.68*x smooth csplines title "$ q_{1} $" linestyle 1, \ + # 4.6*x smooth csplines title "$ q_{2} $" linestyle 2 + \end{gnuplot} +\end{figure} + +\end{enumerate} + + +\end{document} diff --git a/2.5.1/data.txt b/2.5.1/data.txt new file mode 100644 index 0000000..707de4d --- /dev/null +++ b/2.5.1/data.txt @@ -0,0 +1,61 @@ +l = 0.05 мм - цена деления микроскопа +19 делений - игла + +Спирт +_____ +51 Дел +49 +48 +49 +49 + +K = 0.2 +9,80665 Па +______ +Анилин +22 мм - h1 +23 С + +117 Дел +116 +118 +116 +117 + +5 мм - h2 +23 С +211 Дел +212 +212 +211 +210 + +28 С +210 Дел +209 +209 +208 +208 + +33 C +207 Дел +207 +207 +207 +206 + +38 С +205 Дел +206 +205 +205 +204 + +43 С + + +48 С + +53 С + +58 С diff --git a/2.5.1/images/1.jpg b/2.5.1/images/1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..b7759deb1272250e02f404e3de8d58ad8b300822 GIT binary patch literal 217605 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zM|FEWzhg6NR==+o`Eg`SQ+`bDMRVdRp|SqeDygc$ZRKmR{rLN^^Eb79jPq2ze)UgOW7x;HHDIdmW-gR`A&TJJ;)nUAJ5sia#2I^86^$oCg1-&dti`rKu}1)<)Iaz Q;9H6cw4$PVX8N@M0oUIRE&u=k literal 0 HcmV?d00001 diff --git a/Malinovskii_V2.5.1/main.py b/Malinovskii_V2.5.1/main.py new file mode 100644 index 0000000..6468a93 --- /dev/null +++ b/Malinovskii_V2.5.1/main.py @@ -0,0 +1,42 @@ +from math import * + +s = [[float(k) for k in x.split('\t') if k != '--'] for x in '23 176 176 176 176 175 -- -- -- -- -- -- -- -- -- --\n\ +30 174 174 175 175 174 175 -- -- -- -- -- -- -- -- --\n\ +35 174 174 174 174 174 174 173 173 174 174 174 174 174 173 174\n\ +40 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172\n\ +45 171 171 171 171 171 171 171 171 171 171 171 171 171 171 171\n\ +50 170 170 169 169 170 170 170 169 169 170 170 169 170 170 170\n\ +55 169 169 169 169 169 169 169 170 170 169 169 170 170 169 169\n\ +60 168 169 168 168 168 168 168 168 169 169 168 168 168 168 168'.split('\n')] + + +def avg(list_): + m = 0 + ms = 0 + for v in list_: + m += v + ms += v ** 2 + n = len(list_) + + m /= n + ms /= n + return (m, ((ms - m ** 2) / (n * (n - 1))) ** 0.5) + +dsdt = 112 / 10**6 +ddsdt = 8 / 10 ** 6 +for line in s: + T = line[0] + p, dp = avg(line[1:]) + #print(p, sqrt(dp ** 2 + 0.5 ** 2)) + #print(T, '{0:.1f}'.format(p), '{0:.2f}'.format(p * 1.05 / 1000 * 0.2 / 4 * 1000), sep='$&$', end='$\\\\ \\hline\n$') + #print(T, 0.1, p * 9.8 * 1.05 / 1000 * 0.2 / 4 * 1000, p * 9.8 * 1.05 / 1000 * 0.2 / 4 * 1000 * (0.025 / 1.05 + 0.5 / p)) + #print(T, 0.1, - T * dsdt * 1000, 1000 * T * dsdt * (ddsdt / dsdt + 0.1 / T)) + #print(T, 0.1, ) + f = - T * dsdt * 1000 + s = p * 9.8 * 1.05 / 1000 * 0.2 / 4 * 1000 + df = 1000 * T * dsdt * (ddsdt / dsdt + 0.1 / T) + ds = p * 9.8 * 1.05 / 1000 * 0.2 / 4 * 1000 * (0.025 / 1.05 + 0.5 / p) + #print(T, '{0:.1f}'.format(p), '{0:.0f}'.format(s), sep='$&$', end='$\\\\ \\hline\n$') + #print(T, 0.1, '{0:.1f}'.format(f), '{0:.1f}'.format(df), '{0:.0f}'.format(f + s), '{0:.0f}'.format(df + ds), sep='$&$', end='$\\\\ \\hline\n$') + print(T, 0.1, s - f, ds + df) + #print(T, 0.1, p * 9.8 * 1.05 / 1000 * 0.2 / 4 * 1000 - T * dsdt * 1000, 1000 * T * dsdt * (ddsdt / dsdt + 0.1 / T) + p * 9.8 * 1.05 / 1000 * 0.2 / 4 * 1000 * (0.025 / 1.05 + 0.5 / p)) \ No newline at end of file diff --git a/Malinovskii_V2.5.1/main.tex b/Malinovskii_V2.5.1/main.tex new file mode 100644 index 0000000..44ef57c --- /dev/null +++ b/Malinovskii_V2.5.1/main.tex @@ -0,0 +1,182 @@ +\include{head} +\usepackage{booktabs} + +\begin{document} + +\begin{center} + \LARGE{Работа 2.5.1}\\[0.2cm] + \LARGE{Измерение коэффициента поверхностного натяжения жидкости}\\[0.2cm] + \large{Гришаев Григорий С01-119}\\[0.2cm] + +\end{center} + +\textbf{Цель работы:} 1) измерение коэффициента поверхностного натяжения исследуемой жидкости при разной температуре с использованием известного коэффициента поверхностного натяжениядругой жидкости 2) определение полной поверхностной энергиии теплоты, необходимой для изотермического образования единицы поверхности жидкости. + +\textbf{В работе используются:} прибор Ребиндера с термостатом, исследуемые жидкости, стаканы. + +\section*{Описание работы} + +Наличие поверхностного слоя приводит к различию давлений поразные стороны от искривленной границы раздела двух сред. Для сферического пузырька внутри жидкости избыточное давление дается формулой Лапласа +$$\Delta P = P_\text{внутри}-P_\text{снаружи}=2\sigma/r.$$ +Эта формула лежит в основе предлагаемого метода определения коэффициента поверхностного натяжения жидкости. Измеряется давление, необходимое для выталкивания в жидкость пузырька газа.\\ + +Наличие поверхностного слоя приводит к различию давлений по разные стороны от искривленной границы раздела двух сред. Для сферического пузырька внутри жидкости избыточное давление дается формулой Лапласа $\Delta P = P_\text{внутри}-P_\text{снаружи}=2\sigma/r$. Эта формула лежит в основе предлагаемого метода определения коэффициента поверхностного натяжения жидкости. Измеряется давление, необходимое для выталкивания в жидкость пузырька газа. + +Исследуемая жидкость наливается в сосуд $B$. Дистиллированная вода наливается в сосуд $E$. Сосуды закрыты пробками. Через пробку сосуда, в котором проводятся измерения, проходит полая металлическая игла $С$, нижний конец которой погружен в жидкость, а верхний открыт в атмосферу. Если другой сосуд герметично закрыт, то в сосуде с иглой создается разрежение, и пузырьки воздуха начинают пробулькивать через жидкость. Поверхностное натяжение можно найти по величине разрежения, необходимого для прохождения пузырьков. При приоткрытом кране $\text{К}_1$ из аспиратора $A$ по каплям вытекает вода, создавая разрежение, которое измеряется наклонным спиртовым манометром $М$. Показания манометра, умноженные на зависящий от наклона коэффициент ($0.2$), дают давление в $\text{кгс}/\text{м}^2$. Чтобы пополнить запас воды, достаточно при помощи крана $\text{К}_2$ соединить нижнюю часть аспиратора с атмосферой и предварительно заполненной водой верхней частью. Через рубашку $D$ непрерывно прогоняется вода из термостата для стабилизации температуры исследуемой жидкости. + +\newpage +Схема установки представлена на рис. 1: +\begin{center} +\includegraphics[width=0.95\textwidth]{equip.png}. +\end{center} +\newpage + +В начале эксперимента зальем аспиратор $A$ водой, и поместим чистую иглу в сосуд со спиртом $B$ так, чтобы кончик иглы лишь касался поверхности.\\ +Откроем кран $K_1$, в следствие этому давление в установке упадет, из-за чего показания манометра вырастут. Проверим установку на наличие утечки, закрыв кран $K_1$. Если показания манометра не меняются со временем, то все в порядке.\\ +При открытом кране так, что период падения капель равна $\approx 5 \text{с}$, давление прекращает расти при значении, равном $2\sigma/(r\,\sin(\alpha))$, поскольку при нем образуется пузырек газа в шприце, и давление падает. На манометре получаются давления:\\ + +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} +\hline +$p,\,\text{кгс}/\text{м}^2$&$55$&$54$&$55$&$54$&$55$&$55$&$55$&$54$&$55$&$54$&$55$&$55$&$54$&$54$&$54$&$54$\\ +\hline +\end{tabular}, +\end{center} +$$\Delta p = 0.5\,\text{кгс}/\text{м}^2, \sin(\alpha) = 0.2.$$ +Из измерений следует, что +$$p = (54.5 \pm 0.6) \text{кгс}/\text{м}^2.$$ +Взяв табличное значение вязкости спитра $\sigma = (22\pm2) \text{мН}/\text{м}$, получим диаметр, равный +$$d_\text{косв} = \frac{4\sigma}{p\sin(\alpha)} = (0.82\pm0.08) \text{мм}.$$ +Измерив диаметр иголки под микроскопом, получаем значение диаметра, равное +$$d_\text{прям} = (1.05\pm0.03) \text{мм}.$$ +Промыв и просушив иглоку, переставим ее в сосуд с водой. Сначала измерим давления появления пузырьков при касании иголкой воды (при высоте иголки над дном сосуда $h_1$), а потом измерим то же самое, но при максимальном погружении иглы -- $h_2$. +\begin{center} + \begin{tabular}{|c|c|c|c|c|c|c|c|} + \hline + $h,\,\text{мм}$ & \multicolumn{5}{c|}{$p,\,\text{кгс}/\text{м}^2$} & $

      ,\,\text{кгс}/\text{м}^2$ & $\Delta

      ,\,\text{кгс}/\text{м}^2$\\ + \hline + $37.5$&$134$&$133$&$133$&$133$&$133$&$133.2$&$0.5$\\ + \hline + $30.5$&$176$&$176$&$176$&$176$&$175$&$175.8$&$0.5$\\ + \hline + \end{tabular} +\end{center} +$$\Delta h = 0.25\,\text{мм},\,\Delta p = 0.5\,\text{кгс}/\text{м}^2, \sin(\alpha) = 0.2.$$ +Прямое измерение $h_1 - h_2$ дает результат: +$$h_1 - h_2 = 7 \pm 1 \text{мм}.$$ +Из давлений следует, что разница высот: +$$\frac{p_2 - p_1}{\rho\,g}sin(\alpha)=8.5\pm0.2\text{мм}.$$ +Далее проведем опыт при самой большой глубине погружения иглы и разной температуте. Для того, чтобы достичь равномерного прогрева воды в установке, после смены температуры термостата, подождем $5$ минут перед измерениями давления. +Результаты представлены ниже: +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} +\hline +$T,\,^\circ C$&\multicolumn{15}{c|}{$p,\,\text{кгс}/\text{м}^2$}\\ +\hline +$23$&$176$&$176$&$176$&$176$&$175$&$--$&$--$&$--$&$--$&$--$&$--$&$--$&$--$&$--$&$--$\\ +\hline +$30$&$174$&$174$&$175$&$175$&$174$&$175$&$--$&$--$&$--$&$--$&$--$&$--$&$--$&$--$&$--$\\ +\hline +$35$&$174$&$174$&$174$&$174$&$174$&$174$&$173$&$173$&$174$&$174$&$174$&$174$&$174$&$173$&$174$\\ +\hline +$40$&$172$&$172$&$172$&$172$&$172$&$172$&$172$&$172$&$172$&$172$&$172$&$172$&$172$&$172$&$172$\\ +\hline +$45$&$171$&$171$&$171$&$171$&$171$&$171$&$171$&$171$&$171$&$171$&$171$&$171$&$171$&$171$&$171$\\ +\hline +$50$&$170$&$170$&$169$&$169$&$170$&$170$&$170$&$169$&$169$&$170$&$170$&$169$&$170$&$170$&$170$\\ +\hline +$55$&$169$&$169$&$169$&$169$&$169$&$169$&$169$&$170$&$170$&$169$&$169$&$170$&$170$&$169$&$169$\\ +\hline +$60$&$168$&$169$&$168$&$168$&$168$&$168$&$168$&$168$&$169$&$169$&$168$&$168$&$168$&$168$&$168$\\ +\hline +\end{tabular} +\end{center} +Из данных выше можно найти $

      $ и $\sigma = \frac{pr}{2} \sin(\alpha)$. +\begin{center} +\begin{tabular}{|c|c|c|} +\hline +$T,\,^\circ C$&$

      ,\,\text{кгс}/\text{м}^2$&$\sigma, \text{мН}/\text{м}$\\\hline +$23.0$&$175.8$&$90$\\ \hline +$30.0$&$174.5$&$90$\\ \hline +$35.0$&$173.8$&$89$\\ \hline +$40.0$&$172.0$&$88$\\ \hline +$45.0$&$171.0$&$88$\\ \hline +$50.0$&$169.7$&$87$\\ \hline +$55.0$&$169.3$&$87$\\ \hline +$60.0$&$168.2$&$87$\\ \hline +\end{tabular} +\end{center} +$$\Delta p = \Delta

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+\usepackage[warn]{mathtext} % русские буквы в фомулах +\usepackage[T2A]{fontenc} % кодировка +\usepackage[utf8]{inputenc} % кодировка исходного текста +\usepackage[english,russian]{babel} % локализация и переносы +%\usepackage{physics} +\usepackage{multirow} + +%%% Нормальное размещение таблиц (писать [H] в окружении таблицы) +\usepackage{float} +\restylefloat{table} + + + +\usepackage{graphicx} + +\usepackage{wrapfig} +\usepackage{tabularx} + +\usepackage{hyperref} +\usepackage[rgb]{xcolor} +\hypersetup{ + colorlinks=true,urlcolor=blue +} + +\usepackage{pgfplots} +\pgfplotsset{compat=1.9} + +%%% Дополнительная работа с математикой +\usepackage{amsmath,amsfonts,amssymb,amsthm,mathtools} % AMS +\usepackage{icomma} % "Умная" запятая: $0,2$ --- число, $0, 2$ --- перечисление + +%% Номера формул +\mathtoolsset{showonlyrefs=true} % Показывать номера только у тех формул, на которые есть \eqref{} в тексте. + +%% Шрифты +\usepackage{euscript} % Шрифт Евклид +\usepackage{mathrsfs} % Красивый матшрифт + +%% Свои команды +\DeclareMathOperator{\sgn}{\mathop{sgn}} + +%% Перенос знаков в формулах (по Львовскому) +\newcommand*{\hm}[1]{#1\nobreak\discretionary{} + {\hbox{$\mathsurround=0pt #1$}}{}} + +\date{\today} + +\usepackage{gensymb} + +\author{Гришаев Григорий С01-119} +\title{Лабораторная работа 2.5.1} +\date{\today} + +\begin{document} + +\maketitle + +\section{Введение} +\textbf{Цель работы:} \begin{enumerate} + \item измерение температурной зависимости коэффициента поверхностного натяжения дистиллированной воды с использованием известного коэффициента поверхностного натяжения спирта; + \item определение полной поверхностной энергии и теплоты, необходимой для изотермического образования единицы поверхности жидкости при различной температуре. +\end{enumerate} + +\textbf{В работе используются:} прибор Ребиндера с термостатом и микроманометром; исследуемые жидкости; стаканы; микроскоп. +\section{Теоретические сведения} + +Наличие поверхностного слоя приводит к различию давлений по разные стороны от искривленной границы раздела двух сред. Для сферического пузырька с воздухом внутри жидкости избыточное давление даётся формулой Лапласа: + +\begin{equation}\label{key} +\Delta P = P_{int} - P_{ext} = \frac{2\sigma}{r}, +\end{equation} +где $ \sigma $ -- коэффициент поверхностного натяжения, $ P_{int} $ и $ P_{ext} $ -- давление внутри пузырька и снаружи, $ r $ -- радиус кривизны поверхности раздела двух фаз. Эта формула лежит в основе предлагаемого метода определения коэффициента поверхностного натяжения жидкости. Измеряется давление $ \Delta P $, необходимое для выталкивания в жидкость пузырька воздуха. + +\section{Экспериментальная установка} + +\begin{figure}[H] + \begin{center} + \includegraphics[width=15cm]{ust.jpg} + \caption{Рисунок экспериментальной установки}\label{img:ust} + \end{center} +\end{figure} + +Исследуемая жидкость (дистиллированная вода) наливается в сосуд (колбу) $ B $ (рис. \ref{img:ust}). Тестовая жидкость (этиловый спирт) наливается в сосуд $ E $. При измерениях колбы герметично закрываются пробками. Через одну из двух пробок проходит полая металлическая игла $ С $. Этой пробкой закрывается сосуд, в котором проводятся измерения. Верхний конец иглы открыт в атмосферу, а нижний погружен в жидкость. Другой сосуд герметично закрывается второй пробкой. При создании достаточного разряжения воздуха в колбе с иглой пузырьки воздуха начинают пробулькивать через жидкость. Поверхностное натяжение можно определить по величине разряжения $ \Delta P $ \eqref{key}, необходимого для прохождения пузырьков (при известном радиусе иглы). + +Разряжение в системе создается с помощью аспиратора $ A $. Кран $ K_2 $ разделяет две полости аспиратора. Верхняя полость при закрытом кране $ K_2 $ заполняется водой. Затем кран $ K_2 $ открывают и заполняют водой нижнюю полость аспиратора. Разряжение воздуха создается в нижней полости при открывании крана $ K_1 $, когда вода вытекает из неё по каплям. В колбах $ В $ и $ С $, соединённых трубками с нижней полостью аспиратора, создается такое же пониженное давление. Разность давлений в полостях с разряженным воздухом и атмосферой измеряется спиртовым микроманометром. + +Для стабилизации температуры исследуемой жидкости через рубашку $ D $ колбы $ В $ непрерывно прогоняется вода из термостата. + +Обычно кончик иглы лишь касается поверхности жидкости, чтобы исключить влияние гидростатического давления столба жидкости. Однако при измерении температурной зависимости коэффициента поверхностного натяжения возникает ряд сложностей. Во-первых, большая теплопроводность металлической трубки приводит к тому, что температура на конце трубки заметно ниже, чем в глубине жидкости. Во-вторых, тепловое расширение поднимает уровень жидкости при увеличении температуры. + +Обе погрешности можно устранить, погрузив кончик трубки до самого дна. Полное давление, измеренное при этом микроманометром, равно \[ P = \Delta P + \rho g h.\] Заметим, что $ \rho gh $ от температуры практически не зависит, так как подъём уровня жидкости компенсируется уменьшением её плотности (произведение $ \rho g $ определяется массой всей жидкости и поэтому постоянно). Величину $ \rho g h $ следует измерить двумя способами. + +Во-первых, замерить величину $ P_1= \Delta P' $, когда кончик трубки только касается поверхности жидкости. Затем при этой же температуре опустить иглу до дна и замерить $ P_2= \rho gh + \Delta P'' $ ($ \Delta P' $, $ \Delta P'' $ -- давление Лапласа). Из-за несжимаемости жидкости можно положить $ \Delta P' = \Delta P'' $ и тогда \[ \rho gh= P_2 - P_1. \] + +Во-вторых, при измерениях $ P_1 $ и $ P_2 $ замерить линейкой глубину погружения иглы $ h $. Это можно сделать, замеряя расстояние между верхним концом иглы и любой неподвижной частью прибора при положении иглы на поверхности и в глубине колбы. +\newpage +\section{Ход работы} + +\subsection{Измерение диаметра иглы} + +Измерим максимальное давление $ \Delta P_{alc} $ при пробулькивании пузырьков воздуха через спирт. Результаты измерений занесём в таблицу \ref{tab:alcohol}. + +\begin{table}[H] + \centering + \begin{tabular}{|c|c|c|c|c|} + \hline + № & $ P' $, дел. & $ P $, Па & $ \langle P \rangle $, Па & $ \sigma_{P} $, Па \\ \hline + 1 & 47 & 92,2 & \multirow{5}{*}{93,1} & \multirow{5}{*}{2,0} \\ \cline{1-3} + 2 & 48 & 94,2 & & \\ \cline{1-3} + 3 & 48 & 94,2 & & \\ \cline{1-3} + 4 & 47 & 92,2 & & \\ \cline{1-3} + 5 & 47 & 92,2 & & \\ \hline + \end{tabular} + \caption{Результаты измерений в спирте} + \label{tab:alcohol} +\end{table} + +Учтём, что показания микроманометра связаны с давлением следующим соотношением: \[ P=P' \cdot 0,2 \cdot 9,81, \] где $ P' $ -- колличество делений шкалы, а константы остаются постоянными во время всей работы и определяются исходя из паспорта устройства. + +Вычислим среднее значение измеренного давления. Для этого воспользуемся следующей формулой: + +\begin{equation}\label{mid} +\langle P \rangle = \frac{\sum\limits_{k=1}^{N} P_k}{N} \approx 93,1 \text{ Па}, +\end{equation} +где $ N $ -- число проведённых измерений. + +Также вычислим случайную погрешность измерений по формуле + +\begin{equation}\label{occasion} +\sigma_{P}^{\text{случ}} = \sqrt{\frac{1}{N(N-1)}\sum\limits_{k=1}^N\left(P_k-\langle P \rangle\right)^2} \approx 0,5 \text{ Па}. +\end{equation} + +Систематическую погрешность определим из расчёта, что погрешность измерения составила $ 1 $ деление прибора, или же \underline{$ \sigma_{P}^\text{сист} \approx 1,9 $ Па}. + +Полная погрешность измерений определяется по формуле: + +\begin{equation}\label{full_pogr} +\sigma_{P}=\sqrt{(\sigma_{P}^\text{сист})^2 + (\sigma_{P}^\text{случ})^2} \approx 2,0 \text{ Па}. +\end{equation} + +Итого получаем \underline{ $ \Delta P_{alc} = (93,1 \pm 2,0) \text{ Па},$} \quad $(\varepsilon = 2,2 \%) $. + +\medskip + +Согласно ГОСТ 8.428-81 коэффициент поверхностного натяжения этилового спирта при комнатной температуре равен $ \sigma_{alc} = 22,4 $ мН/м. По полученным результатам измерения и при помощи \eqref{key} вычисляем диаметр иглы по формуле: + +\label{diametr} + +\begin{equation}\label{igla} +d=\frac{4\sigma_{alc}}{\Delta P_{alc}} \approx 0,96 \text{ мм}. +\end{equation} + +Также вычисляем погрешность полученного результата: + +\begin{equation}\label{igla_pogr} +\sigma_d=d\cdot\varepsilon_{\Delta P_{alc}} \approx 0,02 \text{ мм}. +\end{equation} + +Таким образом, получаем окончательный результат измерения диаметра иглы косвенным способом: +\begin{itemize} + \item $\underline{ d = (0,96 \pm 0,02) \text{ мм},} \: (\varepsilon = 2,2\%). $ +\end{itemize} + +Также проведём измерение диаметра иглы при помощи оптического микроскопа. + +По результатам прямого измерения получаем $ \underline{d = (1,00 \pm 0,05) \text{ мм}}, \: (\varepsilon = 5\%) $. + +\medskip + +Таким образом, диаметр иглы, измеренный двумя различными способами, совпадает в пределах погрешности, что может говорить о справедливости формулы, представленной в теоретических сведениях, а также об исправной работе экспериментальной установки. + +\subsection{Определения поправки при измерении давления для погруженной в воду иглы} + +Перенесём предварительно промытую и просушенную от спирта иглу в колбу с дистиллированной водой. Измерим максимальное давление $ P_1 $ при пробулькивании пузырьков, когда игла лишь касается поверхности воды. Измерите расстояние между верхним концом иглы и любой неподвижной часть прибора $ h_1 $. + +Утопим иглу в воду. Измерим $ h_2 $. Также измерим максимальное давление в пузырьках $ P_2 $. Полученные результаты заносим в таблицу \ref{tab:popravka}. + +% Please add the following required packages to your document preamble: +% \usepackage{multirow} +\begin{table}[H] + \centering + \begin{tabular}{ccccccc} + \hline + \multicolumn{1}{|c|}{№} & + \multicolumn{1}{c|}{$ P_1 $, дел.} & + \multicolumn{1}{c|}{$ P_1 $, Па} & + \multicolumn{1}{c|}{$ \langle P_1 \rangle $, Па} & + \multicolumn{1}{c|}{$ \sigma_{P_1} $, Па} & + \multicolumn{1}{c|}{$ h_1 $, мм} & + \multicolumn{1}{c|}{$ \sigma_{h_1} $, мм} \\ \hline + \multicolumn{1}{|c|}{1} & + \multicolumn{1}{c|}{131} & + \multicolumn{1}{c|}{257,0} & + \multicolumn{1}{c|}{\multirow{6}{*}{257,0}} & + \multicolumn{1}{c|}{\multirow{6}{*}{1,9}} & + \multicolumn{1}{c|}{\multirow{6}{*}{54,0}} & + \multicolumn{1}{c|}{\multirow{6}{*}{0,5}} \\ \cline{1-3} + \multicolumn{1}{|c|}{2} & + \multicolumn{1}{c|}{131} & + \multicolumn{1}{c|}{257,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{1-3} + \multicolumn{1}{|c|}{3} & + \multicolumn{1}{c|}{131} & + \multicolumn{1}{c|}{257,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{1-3} + \multicolumn{1}{|c|}{4} & + \multicolumn{1}{c|}{131} & + \multicolumn{1}{c|}{257,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{1-3} + \multicolumn{1}{|c|}{5} & + \multicolumn{1}{c|}{131} & + \multicolumn{1}{c|}{257,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{1-3} + \multicolumn{1}{|c|}{6} & + \multicolumn{1}{c|}{131} & + \multicolumn{1}{c|}{257,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \hline + & + & + & + & + & + & + \\ \hline + \multicolumn{1}{|c|}{№} & + \multicolumn{1}{c|}{$ P_2 $, дел.} & + \multicolumn{1}{c|}{$ P_2 $, Па} & + \multicolumn{1}{c|}{$ \langle P_2 \rangle $, Па} & + \multicolumn{1}{c|}{$ \sigma_{P_2} $, Па} & + \multicolumn{1}{c|}{$ h_2 $, мм} & + \multicolumn{1}{c|}{$ \sigma_{h_2} $, мм} \\ \hline + \multicolumn{1}{|c|}{1} & + \multicolumn{1}{c|}{190} & + \multicolumn{1}{c|}{372,2} & + \multicolumn{1}{c|}{\multirow{6}{*}{374,4}} & + \multicolumn{1}{c|}{\multirow{6}{*}{1,9}} & + \multicolumn{1}{c|}{\multirow{6}{*}{42,0}} & + \multicolumn{1}{c|}{\multirow{6}{*}{0,5}} \\ \cline{1-3} + \multicolumn{1}{|c|}{2} & + \multicolumn{1}{c|}{191} & + \multicolumn{1}{c|}{374,7} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{1-3} + \multicolumn{1}{|c|}{3} & + \multicolumn{1}{c|}{191} & + \multicolumn{1}{c|}{374,7} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{1-3} + \multicolumn{1}{|c|}{4} & + \multicolumn{1}{c|}{191} & + \multicolumn{1}{c|}{374,7} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{1-3} + \multicolumn{1}{|c|}{5} & + \multicolumn{1}{c|}{191} & + \multicolumn{1}{c|}{374,7} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{1-3} + \multicolumn{1}{|c|}{6} & + \multicolumn{1}{c|}{191} & + \multicolumn{1}{c|}{374,7} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \hline + \end{tabular} + \caption{Определение поправки к давлению} + \label{tab:popravka} +\end{table} + +Исходя из экспериментальных данных, определяем среднее значение давления $ \langle P \rangle $ и погрешность измерения $ \sigma_{P} $ по формулам \eqref{mid}, \eqref{occasion} и \eqref{full_pogr}. + +По полученным данным определяем \[ P_2-P_1 = 117,4 \text{ Па}. \] + +Также вычисляем погрешность: \begin{equation}\label{pogr_sum} +\sigma_{\Delta P} = \sqrt{\sigma^2_{P_1}+\sigma^2_{P_2}} \approx 2,8 \text{ Па}. +\end{equation} + +Таким образом, получаем $ \underline{\Delta P = (117,4 \pm 2,8) \text{ Па},} \: (\varepsilon = 2,4\%).$ + +По полученному значению $ \Delta P $ можем рассчитать $ \Delta h $ по следующей формуле: \[ \Delta h = \frac{\Delta P}{\rho g} \approx 11,9 \text{ мм}, \] где $ \rho = 1000 $ кг/$ \text{м}^3 $ -- плотность воды и $ g = 9,81 $ м/$ \text{с}^2 $ -- ускорение свободного падения. + +\medskip + +При этом погрешность нашего измерения равна \[ \sigma_{\Delta h} = \Delta h \cdot \varepsilon_{\Delta P} \approx 0,3 \text{ мм}. \] + +Таким образом, получаем $ \underline{\Delta h = (11,9 \pm 0,3) \text{ мм}}, \: (\varepsilon = 2,4\%). $ + +\medskip + +Заметим, что полученный результат в пределах погрешности совпадает с результатом, полученном прямым измерением $ \underline{\Delta h' = (12,0 \pm 0,7) \text{ мм},} \: (\varepsilon = 5,9\%). $ + +Значит, в ходе дальнейших измерений мы будем делать поправку \label{popravka} $ \underline{\Delta P = (117,4 \pm 2,8) \text{ Па},}$ на~добавочное давление со стороны столба жидкости. + +\subsection{Измерение температурной зависимости коэффициента поверхностного натяжения} + +Снимем температурную зависимость $ \sigma(T) $ дистиллированной воды. Для этого включим термостат и подождём, пока нужная нам температура не стабилизируется. После этого проведём измерение давления. Для уменьшения погрешности опыта замер давления при фиксированной температуре проведём несколько раз. Результаты измерений занесём в таблицу \ref{tab:pov}. + +% Please add the following required packages to your document preamble: +% \usepackage{multirow} +\begin{table}[H] + \centering + \begin{tabular}{ccccccc} + \hline + \multicolumn{1}{|c|}{$ T $, К} & + \multicolumn{1}{c|}{$ P' $, дел} & + \multicolumn{1}{c|}{$ P' $, Па} & + \multicolumn{1}{c|}{$ \langle P' \rangle $, Па} & + \multicolumn{1}{c|}{$ \sigma_{P'} $, Па} & + \multicolumn{1}{c|}{$ P $, Па} & + \multicolumn{1}{c|}{$ \sigma_P $, Па} \\ \hline + \multicolumn{1}{|c|}{\multirow{5}{*}{302}} & + \multicolumn{1}{c|}{190} & + \multicolumn{1}{c|}{372,8} & + \multicolumn{1}{c|}{\multirow{5}{*}{374,3}} & + \multicolumn{1}{c|}{\multirow{5}{*}{2,0}} & + \multicolumn{1}{c|}{\multirow{5}{*}{257,0}} & + \multicolumn{1}{c|}{\multirow{5}{*}{3,4}} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{191} & + \multicolumn{1}{c|}{374,7} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{191} & + \multicolumn{1}{c|}{374,7} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{191} & + \multicolumn{1}{c|}{374,7} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{191} & + \multicolumn{1}{c|}{374,7} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \hline + & + & + & + & + & + & + \\ \hline + \multicolumn{1}{|c|}{$ T $, К} & + \multicolumn{1}{c|}{$ P' $, дел} & + \multicolumn{1}{c|}{$ P' $, Па} & + \multicolumn{1}{c|}{$ \langle P' \rangle $, Па} & + \multicolumn{1}{c|}{$ \sigma_{P'} $, Па} & + \multicolumn{1}{c|}{$ P $, Па} & + \multicolumn{1}{c|}{$ \sigma_P $, Па} \\ \hline + \multicolumn{1}{|c|}{\multirow{5}{*}{307}} & + \multicolumn{1}{c|}{189} & + \multicolumn{1}{c|}{370,8} & + \multicolumn{1}{c|}{\multirow{5}{*}{370,8}} & + \multicolumn{1}{c|}{\multirow{5}{*}{2,0}} & + \multicolumn{1}{c|}{\multirow{5}{*}{253,4}} & + \multicolumn{1}{c|}{\multirow{5}{*}{3,4}} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{189} & + \multicolumn{1}{c|}{370,8} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{189} & + \multicolumn{1}{c|}{370,8} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{189} & + \multicolumn{1}{c|}{370,8} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{189} & + \multicolumn{1}{c|}{370,8} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \hline + & + & + & + & + & + & + \\ \hline + \multicolumn{1}{|c|}{$ T $, К} & + \multicolumn{1}{c|}{$ P' $, дел} & + \multicolumn{1}{c|}{$ P' $, Па} & + \multicolumn{1}{c|}{$ \langle P' \rangle $, Па} & + \multicolumn{1}{c|}{$ \sigma_{P'} $, Па} & + \multicolumn{1}{c|}{$ P $, Па} & + \multicolumn{1}{c|}{$ \sigma_P $, Па} \\ \hline + \multicolumn{1}{|c|}{\multirow{5}{*}{312}} & + \multicolumn{1}{c|}{187} & + \multicolumn{1}{c|}{366,9} & + \multicolumn{1}{c|}{\multirow{5}{*}{366,9}} & + \multicolumn{1}{c|}{\multirow{5}{*}{2,0}} & + \multicolumn{1}{c|}{\multirow{5}{*}{249,5}} & + \multicolumn{1}{c|}{\multirow{5}{*}{3,4}} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{187} & + \multicolumn{1}{c|}{366,9} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{187} & + \multicolumn{1}{c|}{366,9} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{187} & + \multicolumn{1}{c|}{366,9} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{187} & + \multicolumn{1}{c|}{366,9} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \hline + & + & + & + & + & + & + \\ \hline + \multicolumn{1}{|c|}{$ T $, К} & + \multicolumn{1}{c|}{$ P' $, дел} & + \multicolumn{1}{c|}{$ P' $, Па} & + \multicolumn{1}{c|}{$ \langle P' \rangle $, Па} & + \multicolumn{1}{c|}{$ \sigma_{P'} $, Па} & + \multicolumn{1}{c|}{$ P $, Па} & + \multicolumn{1}{c|}{$ \sigma_P $, Па} \\ \hline + \multicolumn{1}{|c|}{\multirow{5}{*}{317}} & + \multicolumn{1}{c|}{185} & + \multicolumn{1}{c|}{363,0} & + \multicolumn{1}{c|}{\multirow{5}{*}{363,4}} & + \multicolumn{1}{c|}{\multirow{5}{*}{2,0}} & + \multicolumn{1}{c|}{\multirow{5}{*}{246,0}} & + \multicolumn{1}{c|}{\multirow{5}{*}{3,4}} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{185} & + \multicolumn{1}{c|}{363,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{186} & + \multicolumn{1}{c|}{364,9} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{185} & + \multicolumn{1}{c|}{363,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{185} & + \multicolumn{1}{c|}{363,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \hline + & + & + & + & + & + & + \\ \hline + \multicolumn{1}{|c|}{$ T $, К} & + \multicolumn{1}{c|}{$ P' $, дел} & + \multicolumn{1}{c|}{$ P' $, Па} & + \multicolumn{1}{c|}{$ \langle P' \rangle $, Па} & + \multicolumn{1}{c|}{$ \sigma_{P'} $, Па} & + \multicolumn{1}{c|}{$ P $, Па} & + \multicolumn{1}{c|}{$ \sigma_P $, Па} \\ \hline + \multicolumn{1}{|c|}{\multirow{5}{*}{322}} & + \multicolumn{1}{c|}{184} & + \multicolumn{1}{c|}{361,0} & + \multicolumn{1}{c|}{\multirow{5}{*}{360,6}} & + \multicolumn{1}{c|}{\multirow{5}{*}{2,0}} & + \multicolumn{1}{c|}{\multirow{5}{*}{243,2}} & + \multicolumn{1}{c|}{\multirow{5}{*}{3,4}} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{184} & + \multicolumn{1}{c|}{361,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{183} & + \multicolumn{1}{c|}{359,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{184} & + \multicolumn{1}{c|}{361,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{184} & + \multicolumn{1}{c|}{361,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \hline + & + & + & + & + & + & + \\ \hline + \multicolumn{1}{|c|}{$ T $, К} & + \multicolumn{1}{c|}{$ P' $, дел} & + \multicolumn{1}{c|}{$ P' $, Па} & + \multicolumn{1}{c|}{$ \langle P' \rangle $, Па} & + \multicolumn{1}{c|}{$ \sigma_{P'} $, Па} & + \multicolumn{1}{c|}{$ P $, Па} & + \multicolumn{1}{c|}{$ \sigma_P $, Па} \\ \hline + \multicolumn{1}{|c|}{\multirow{5}{*}{327}} & + \multicolumn{1}{c|}{183} & + \multicolumn{1}{c|}{359,0} & + \multicolumn{1}{c|}{\multirow{5}{*}{358,3}} & + \multicolumn{1}{c|}{\multirow{5}{*}{2,0}} & + \multicolumn{1}{c|}{\multirow{5}{*}{240,9}} & + \multicolumn{1}{c|}{\multirow{5}{*}{3,4}} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{182} & + \multicolumn{1}{c|}{357,1} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{183} & + \multicolumn{1}{c|}{359,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{183} & + \multicolumn{1}{c|}{359,0} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{182} & + \multicolumn{1}{c|}{357,1} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \hline + & + & + & + & + & + & + \\ \hline + \multicolumn{1}{|c|}{$ T $, К} & + \multicolumn{1}{c|}{$ P' $, дел} & + \multicolumn{1}{c|}{$ P' $, Па} & + \multicolumn{1}{c|}{$ \langle P' \rangle $, Па} & + \multicolumn{1}{c|}{$ \sigma_{P'} $, Па} & + \multicolumn{1}{c|}{$ P $, Па} & + \multicolumn{1}{c|}{$ \sigma_P $, Па} \\ \hline + \multicolumn{1}{|c|}{\multirow{5}{*}{332}} & + \multicolumn{1}{c|}{181} & + \multicolumn{1}{c|}{355,1} & + \multicolumn{1}{c|}{\multirow{5}{*}{354,7}} & + \multicolumn{1}{c|}{\multirow{5}{*}{2,0}} & + \multicolumn{1}{c|}{\multirow{5}{*}{237,3}} & + \multicolumn{1}{c|}{\multirow{5}{*}{3,4}} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{181} & + \multicolumn{1}{c|}{355,1} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{181} & + \multicolumn{1}{c|}{355,1} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{181} & + \multicolumn{1}{c|}{355,1} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \cline{2-3} + \multicolumn{1}{|c|}{} & + \multicolumn{1}{c|}{180} & + \multicolumn{1}{c|}{353,2} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} & + \multicolumn{1}{c|}{} \\ \hline + \end{tabular} + \caption{Измерение температурной зависимости коэффициента поверхностного натяжения} + \label{tab:pov} +\end{table} +\newpage +Исходя из экспериментальных данных, определяем среднее значение давления $ \langle P' \rangle $ и погрешность измерения $ \sigma_{P'} $ (для давления без учёта поправки) по формулам \eqref{mid}, \eqref{occasion} и \eqref{full_pogr}. + +Также учитываем поправку к измеренному давлению, которая была вычислена в \ref{popravka}. Погрешность после её учёта вычисляем по формуле \eqref{pogr_sum}. Полученные результаты также заносим в таблицу \ref{tab:pov}. + +По полученным данным вычислим коэффициент поверхностного натяжения для каждой из температур по формуле + +\begin{equation}\label{sigma} +\sigma = \frac{Pd}{4}, +\end{equation} +где $ d $ -- диаметр иглы, вычисленный в \ref{diametr}. Погрешность такого результата вычисляется по следующей формуле: + +\begin{equation}\label{otn_pogr} +\sigma_\sigma = \sigma\sqrt{\varepsilon^2_P + \varepsilon^2_d}. +\end{equation} + +Полученные результаты заносим в таблицу \ref{tab:temp}. + +\begin{table}[H] + \centering + \begin{tabular}{|c|c|c|c|c|} + \hline + № & $ T $, К & $ \sigma_T, К $ & $ \sigma $, мН/м & $ \sigma_\sigma $, мН/м \\ \hline + 1 & 302,0 & 0,1 & 61,9 & 1,6 \\ \hline + 2 & 307,0 & 0,1 & 61,0 & 1,6 \\ \hline + 3 & 312,0 & 0,1 & 60,1 & 1,5 \\ \hline + 4 & 317,0 & 0,1 & 59,2 & 1,5 \\ \hline + 5 & 322,0 & 0,1 & 58,6 & 1,5 \\ \hline + 6 & 327,0 & 0,1 & 58,0 & 1,5 \\ \hline + 7 & 332,0 & 0,1 & 57,2 & 1,5 \\ \hline + \end{tabular} + \caption{Зависимость коэффициента пов. натяжения от $T$} + \label{tab:temp} +\end{table} + +Полученную зависимость наносим на график (рис. ). Вычислим коэффициенты аппроксимирующей прямой $ \sigma = kT + b $, где $ \displaystyle k = \frac{d\sigma}{dT} $, используя метод наименьших квадратов: + +\[ k = \frac{\langle T\sigma \rangle - \langle T \rangle \langle \sigma \rangle}{\langle T^2 \rangle - \langle T \rangle ^2} \approx -0,155\text{ } \frac{\text{мН}}{\text{м}\cdot\text{К}},\] + +\[ b = \langle \sigma \rangle - k\langle T \rangle \approx 108,6\text{ } \frac{\text{мН}}{\text{м}}. \] + +Случайные погрешности определения этих коэффициентов вычислим по следующим формулам: + +\[ \sigma^\text{случ}_k = \sqrt{\frac{1}{N-2} \left(\frac{\left\langle\left(\sigma - \langle \sigma\right\rangle\right)^2 \rangle}{\left\langle\left(T - \langle T\right\rangle\right)^2 \rangle}\right)-k^2} \approx 0,005 \text{ } \frac{\text{мН}}{\text{м}\cdot\text{К}},\] + +\[ \sigma^\text{случ}_b=\sigma^\text{случ}_k\sqrt{\left\langle x^2 \right\rangle} \approx 1,7\text{ } \frac{\text{мН}}{\text{м}}. \] + +Систематические погрешности оценим по следующим формулам: + +\[ \sigma^\text{сист}_k = |k|\sqrt{\varepsilon^2_T+\varepsilon^2_\sigma} \approx 0,004 \text{ } \frac{\text{мН}}{\text{м}\cdot\text{К}}, \] +\[ \sigma^\text{сист}_b = |b|\sqrt{\varepsilon^2_T+\varepsilon^2_\sigma} \approx 2,8 \text{ } \frac{\text{мН}}{\text{м}}. \] + +\newpage + +Таким образом, полные погрешности измерений определяются следующими соотношениями: +\[ \sigma_k = \sqrt{(\sigma_k^\text{сист})^2 + (\sigma_k^\text{случ})^2} \approx 0,006 \text{ } \frac{\text{мН}}{\text{м}\cdot\text{К}},\] +\[ \sigma_b = \sqrt{(\sigma_b^\text{сист})^2 + (\sigma_b^\text{случ})^2} \approx 3,3 \text{ } \frac{\text{мН}}{\text{м}}. \] + +Таким образом, окончательно получаем: +\begin{itemize} + \item $ \displaystyle \underline{k = \frac{d\sigma}{dT} = (-0,155 \pm 0,006) \text{ } \frac{\text{мН}}{\text{м}\cdot\text{К}}, \: (\varepsilon = 4,2\%);} $ + \item $ \displaystyle \underline{b = (108,6 \pm 3,3) \text{ } \frac{\text{мН}}{\text{м}}, \: (\varepsilon = 2,9\%).}$ +\end{itemize} + +\begin{center} + \begin{tikzpicture} + \begin{axis}[ + title={Графики зависимостей}, + xlabel={$ T $, \textdegree К}, + ylabel={$\sigma$, мН/м}, + legend pos=north east, + xmajorgrids=true, + ymajorgrids=true, + grid style=dashed, + width = 520, + height = 350, + xmin = 300, + xmax = 335, + ymin =40, + ymax =135, + ] + \legend{ + ,,, + $ \sigma(T) $, + $ q = - T \frac{d\sigma}{dT} $, + $ \frac{U}{F} = (\sigma - T\frac{d\sigma}{dT}) $ + }; + \addplot+ [blue, only marks, mark size = 4pt, + error bars/.cd, + x dir=both, x explicit, + y dir=both, y explicit, + ] table [x = T, y = sigma, x error = dT, y error = ds,] { + T sigma dT ds + 302 61.89142053 0.1 1.578711731 + 307 61.04078762 0.1 1.560148221 + 312 60.09563994 0.1 1.542739875 + 317 59.24500703 0.1 1.530060624 + 322 58.58340366 0.1 1.517994324 + 327 58.01631505 0.1 1.509164465 + 332 57.16568214 0.1 1.492275431 + }; + \addplot+ [blue, only marks, mark size = 4pt, + ] table [x = T, y = sigma,] { + T sigma + 302 46.892827 + 307 47.66919831 + 312 48.44556962 + 317 49.22194093 + 322 49.99831224 + 327 50.77468354 + 332 51.55105485 + }; + \addplot+ [blue, only marks, mark size = 4pt, + ] table [x = T, y = sigma,] { + T sigma + 302 108.7842475 + 307 108.7099859 + 312 108.5412096 + 317 108.466948 + 322 108.5817159 + 327 108.7909986 + 332 108.716737 + }; + \addplot [red, domain=297:350, line width =3.2pt] {108.6 -0.155*x}; + \addplot [pink, domain=297:350, line width =3.2pt] {x * 0.155}; + \addplot [black, domain=297:350, line width =3.2pt] {108.6}; + \end{axis} + \end{tikzpicture} +\end{center} + +По полученным данным можно нанести на график зависимость от температуры коэффициента поверхностного натяжения $ \underline{\sigma(T)} $, теплоты образования единицы поверхности жидкости $ \displaystyle \underline{q = -T\frac{d\sigma}{dT}} $ и поверхностной энергии $ U $ единицы площади $ F $: $ \displaystyle \underline{\frac{U}{F} = \left(\sigma - T \frac{d\sigma}{dT}\right).} $ + +\section{Обсуждение результатов и выводы} + +В ходе работы были выполнены следующие задачи: + +\begin{itemize} + \item Был измерен диаметр иглы при помощи известного коэффициента поверхностного натяжения спирта. Полученный результат $\underline{ d = (0,96 \pm 0,02) \text{ мм},} \: (\varepsilon = 2,2\%)$ с хорошей точностью совпадает с диаметром, измеренным при помощи светового микроскопа. + + \item Было определено добавочное давление $ \underline{\Delta P = (117,4 \pm 2,8) \text{ Па},} \: (\varepsilon = 2,4\%)$, создаваемое столбом жидкости при опускании иглы на $ \underline{\Delta h' = (12,0 \pm 0,7) \text{ мм},} \: (\varepsilon = 5,9\%). $ Полученное экспериментально значение $ \Delta h $ в пределах погрешности совпало с прямым измерением $ \Delta h' $. Полученная поправка к давлению была использована в дальнейшем в основной части работы. + + \item Был экспериментально получен коэффициент поверхностного натяжения воды при различных её температурах. Так, например, при температуре $ 29 ^\circ C $ коэффициент $ \sigma \approx (61,9 \pm 1,6) $ мН/м. А при $ 59 ^\circ C $ -- $ \sigma \approx (57,2 \pm 1,5) $ мН/м. + + \item Была экспериментально получена зависимость коэффициента поверхностного натяжения дистиллированной воды от температуры. Был вычислен коэффициент пропорциональности $ \displaystyle \underline{k = \frac{d\sigma}{dT} = (-0,155 \pm 0,006) \text{ } \frac{\text{мН}}{\text{м}\cdot\text{К}},} \: (\varepsilon = 4,2\%). $ + + \item Были построены графики зависимости от температуры теплоты образования единицы поверхности жидкости и поверхностной энергии единицы площади. +\end{itemize} + +Полученные результаты дают основание полагать, что теоретические данные довольно точно описывают наблюдаемые зависимости. При этом численные результаты значительно отличаются от табличных данных, что может говорить о низкой точности метода измерений, которая, вероятно, связана с необходимостью учёта сложных не квазистатических процессов, происходящих при пробулькивании пузырька. Также большую погрешность мог внести тот факт, что <<дистиллированная>> вода, используемая в ходе эксперимента имела примеси спирта, которые попали туда в результате некачественной очистки иглы от остатков этанола после первой части работы. Более того, в колбе с водой были обнаружены волокна тряпки, которая использовалась для протирки иглы от спирта, что тоже могло повлиять на точность полученных результатов. + + + + + + + + + + + +\end{document} diff --git a/Senokosov_A/2.5.1.xlsx b/Senokosov_A/2.5.1.xlsx new file mode 100644 index 0000000000000000000000000000000000000000..efeb1ebc30963c973747b5e704a09dae8b29015c GIT binary patch literal 20774 zcmeFYb98M_wDwrywU#P$hKj1${VPHfw@ZQIV5-+TLZzdi2lKfb@a*BDiM zu3EKst+lG2XV#wMnTj%?U}!)PKu|zHKtw=e*=_(&U?89p2p}L-ASe(W5j$IFfUUEE zs)s$mNsr#$#`;Gd7zkw!5Xg7`|9AXf{07Dox8!>n(FU*Nyg~|_g3+h3Ot5rw^r^$1 zfg1QsVD;AS(E^*E%HPA7P=z&#IjkBVy|$_-E}DAG)lN*I31qC87iHRLfk`T;CumIw z=aMHuWf-CI0~;Zj8zD^dcODiDTV4I2)o{l(k$?Z%#{&Q-cF%J&vc~Hegb7yp3}(hS zF@bJgxn8x@llwtuDq2J|$3~qzQ4SX@@Uoa^NS<6I zDjaQwh?w1hH)7m8R-Agbsjm2yzxb>y#Ei_}9v;s4EPcmtc9gw1IoC%FpTIsP13fw_ zCWbFL!*0a8f8HyUAM=-O^8VnEm8%D@$^-C7LEE0vR~JS8rl$W0Ba5D-n)Cx zM+QXUfx%h9FF(43<_%@Q?g({2DugX?cPm_w;3=DP(I1~St_OBp68VnChczKkY4K$s 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      = 0.5\,\text{кгс}/\text{м}^2, \Delta T = 0.1\,^\circ C, \Delta \sigma = \sigma (\frac{\Delta p}{p} + \frac{\Delta d}{d}) = 2 \text{мН}/\text{м}.$$ +Статистическая погрешность $p$ получилась сильно меньше приборной. +\begin{center} +\includegraphics[width=0.90\textwidth]{plot0.png} +\end{center} +Из МНК получаются коэффициенты: +$$\sigma = (93.0\pm0.2) \text{мН}/\text{м}^2 - (110 \pm 5) \text{$\mu$Н}/(\text{м}^2\,^\circ C).$$ +Понятно, что погрешности этих величин неправильные. Намного удобнее считать МНК графика температуры от давления: +\begin{center} +\includegraphics[width=0.90\textwidth]{plot1.png} +\end{center} +Так происходит из-за относительно высокого значения погрешности диаметра иголки. +Параметры этого графика: +$$T = (8.4\pm 0.4)\cdot10^2\,^\circ C - (4.6 \pm 0.2)\,\frac{^\circ C}{\text{кгс}}\cdot p.$$ +Этой погрешности уже можно верить. Из нее получим +$$\frac{\delta \sigma}{\delta T} = \frac{d\sin(\alpha)}{4}\frac{\delta p}{\delta T} = (112\pm8)\frac{\text{$\mu$Н}}{\text{м}\,^\circ C}.$$ +Также мы можем найти графики теплоты образования единицы поверхности жидкости $q$ и поверхнострую энергию $U$ площади $F$. +$$q = - T \frac{\delta \sigma}{\delta T}$$ +$$U/F = \sigma - T \frac{\delta \sigma}{\delta T}$$ +\begin{center} +\begin{tabular}{cc} +\includegraphics[width=0.48\textwidth]{plot2.png}& +\includegraphics[width=0.48\textwidth]{plot3.png}\\ +\end{tabular} +\end{center} + +В этих графиках я брал полученное значение $\frac{\delta \sigma}{\delta T}$. Пересчет значений: +\begin{center} +\begin{tabular}{|c|c|c|c|c|c|} +\hline +$T\,^\circ C$&$\Delta T\,^\circ C$&$q,\,\frac{\text{мН}}{\text{м}}$&$\Delta q,\,\frac{\text{мН}}{\text{м}}$&$U/F,\,\frac{\text{мН}}{\text{м}}$&$\Delta (U/F),\,\frac{\text{мН}}{\text{м}}$\\ +\hline +$23.0$&$0.1$&$-2.6$&$0.2$&$93$&$3$\\ \hline +$30.0$&$0.1$&$-3.4$&$0.3$&$93$&$3$\\ \hline +$35.0$&$0.1$&$-3.9$&$0.3$&$93$&$3$\\ \hline +$40.0$&$0.1$&$-4.5$&$0.3$&$93$&$3$\\ \hline +$45.0$&$0.1$&$-5.0$&$0.4$&$93$&$3$\\ \hline +$50.0$&$0.1$&$-5.6$&$0.4$&$93$&$3$\\ \hline +$55.0$&$0.1$&$-6.2$&$0.5$&$93$&$3$\\ \hline +$60.0$&$0.1$&$-6.7$&$0.5$&$93$&$3$\\ \hline +\end{tabular}. +\end{center} + +\section*{Вывод} +Поверхностное натяжение линейно меняется от температуры, и мы смогли это пронаблюдать. Характеристики этой зависимости не совпали с табличными значениями, но не сильно (для задач на поверхностное натяжение) -- меньше, чем на 1 порядок. Главным источником погрешности в этом эксперименте являлась неточность измерения диаметра. Если бы я улучшал этот эксперимент, я бы двигался в направлении уменьшения этой погрешности -- например, давал бы табличную. Я научился измерять поверхностное натяжение границы жидкость-газ при помощи иглы и того факта, что пузырьки воздуха выходят из этой иглы только при определенном давлении, зависящем от $\sigma$. +\end{document} + + + + + + + + +\lipsum[1-4] +\begin{wrapfigure}{R}{5cm} +\centering +\includegraphics[width=0.20\textwidth]{rd.png} +\caption{1} +\end{wrapfigure} +\lipsum[1-6] + + +\begin{figure}[h] +\begin{center}$ +\begin{array}{cccc} +\includegraphics[width=0.20\textwidth]{rd.png}& +\includegraphics[width=0.20\textwidth]{rd.png}& +\includegraphics[width=0.20\textwidth]{rd.png}& +\includegraphics[width=0.20\textwidth]{rd.png}\\ +(1) & (2) & (3) & (4) +\end{array}$ +\end{center} +\end{figure} diff --git a/Malinovskii_V2.5.1/main2.py b/Malinovskii_V2.5.1/main2.py new file mode 100644 index 0000000..3d4ddb7 --- /dev/null +++ b/Malinovskii_V2.5.1/main2.py @@ -0,0 +1,18 @@ +from math import * + +s = [float(x) for x in '176 176 176 176 175'.split('\t')] + + +def avg(list_): + m = 0 + ms = 0 + for v in list_: + m += v + ms += v ** 2 + n = len(list_) + + m /= n + ms /= n + return (m, ((ms - m ** 2) / (n * (n - 1))) ** 0.5) + +print(avg(s)) \ No newline at end of file diff --git a/Malinovskii_V2.5.1/makefile b/Malinovskii_V2.5.1/makefile new file mode 100644 index 0000000..6bcfceb --- /dev/null +++ b/Malinovskii_V2.5.1/makefile @@ -0,0 +1,8 @@ +all: + pdflatex main.tex interaction=batchmode + +clear: + rm *.log *.aux *.out + +read: + xreader main.pdf \ No newline at end of file diff --git a/Malinovskii_V2.5.1/out b/Malinovskii_V2.5.1/out new file mode 100644 index 0000000..4b53793 --- /dev/null +++ b/Malinovskii_V2.5.1/out @@ -0,0 +1,9 @@ +23.0$&$175.8$&$90$\\ \hline +$30.0$&$174.5$&$90$\\ \hline +$35.0$&$173.8$&$89$\\ \hline +$40.0$&$172.0$&$88$\\ \hline +$45.0$&$171.0$&$88$\\ \hline +$50.0$&$169.7$&$87$\\ \hline +$55.0$&$169.3$&$87$\\ \hline +$60.0$&$168.2$&$87$\\ \hline +$ \ No newline at end of file diff --git a/Malinovskii_V2.5.1/out.png b/Malinovskii_V2.5.1/out.png new file 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