-
Notifications
You must be signed in to change notification settings - Fork 0
/
formul.tex
152 lines (137 loc) · 5.11 KB
/
formul.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
\documentclass[a4paper,12pt]{article}
\usepackage{cmap}
\usepackage[T2A]{fontenc}
\usepackage[utf8x]{inputenc}
\usepackage[english, russian]{babel}
\usepackage{misccorr} % в заголовках появляется точка, но при ссылке на них ее нет
\usepackage{amssymb,amsfonts,amsmath,amsthm}
\usepackage{indentfirst}
\usepackage[usenames,dvipsnames]{color}
\usepackage[unicode, colorlinks, urlcolor=magenta, linkcolor=black, pagecolor=black]{hyperref}
\usepackage{makecell,multirow}
\usepackage{ulem}
\usepackage{graphicx}
\graphicspath{{img/}}
\usepackage{geometry}
\geometry{left=3cm,right=2cm,top=3cm,bottom=3cm,bindingoffset=0cm}
\usepackage{fancyhdr}
\linespread{1.3}
\frenchspacing
\renewcommand{\labelenumii}{\theenumii)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\labauthor{Сарафанов Ф.Г.}
\def\labauthors{Сарафанов Ф.Г.}
\def\labnumber{22}
\def\labtheme{Определение коэффициента внутреннего трения (вязкости) жидкости}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input{addons/lab-kol} % колинтулы на страницах
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{float}
\usepackage[mode=buildnew]{standalone}
\usepackage{tikz}
\usepackage{tikz,csvsimple}
\usetikzlibrary{scopes}
\usetikzlibrary{%
decorations.pathreplacing,%
decorations.pathmorphing,%
patterns,%
calc,%
scopes,%
arrows,%
% arrows.spaced,%
}
\makeatletter
\newif\if@gather@prefix
\preto\place@tag@gather{%
\if@gather@prefix\iftagsleft@
\kern-\gdisplaywidth@
\rlap{\gather@prefix}%
\kern\gdisplaywidth@
\fi\fi
}
\appto\place@tag@gather{%
\if@gather@prefix\iftagsleft@\else
\kern-\displaywidth
\rlap{\gather@prefix}%
\kern\displaywidth
\fi\fi
\global\@gather@prefixfalse
}
\preto\place@tag{%
\if@gather@prefix\iftagsleft@
\kern-\gdisplaywidth@
\rlap{\gather@prefix}%
\kern\displaywidth@
\fi\fi
}
\appto\place@tag{%
\if@gather@prefix\iftagsleft@\else
\kern-\displaywidth
\rlap{\gather@prefix}%
\kern\displaywidth
\fi\fi
\global\@gather@prefixfalse
}
\newcommand*{\beforetext}[1]{%
\ifmeasuring@\else
\gdef\gather@prefix{#1}%
\global\@gather@prefixtrue
\fi
}
\makeatother
\begin{document}
\input{addons/lab-titlepage}
% \tableofcontents
% \newpage
% \section{Снятая зависимость}
\def\rhos{\rho_\text{шар}}
\def\rhoz{\rho_\text{жид}}
\def\Q{g\frac{\rhos-\rhoz}{\rhos}}
\def\K{\frac{2}{9}gr^2\frac{\rhos-\rhoz}{\eta}}
\def\C{\Q-kv}%
\begin{gather}
\beforetext{Запишем II закон Ньютона} m\vec{a}=\vec{F}_\text{тр}+m\vec{g}+\vec{F}_\text{арх}\\
\beforetext{Где сила Стокса} F_\text{тр}=6\pi\eta{r}v\\
\label{eq:ox}\beforetext{В проекции на $x$:}ma=mg-6\pi\eta{r}v-\rhoz{}gV\\
\beforetext{}V=\frac{4}{3}\pi{r^3}, m=\rhos{}V\\
\beforetext{Перепишем (\ref{eq:ox}):}ma=mg-6\pi\eta{r}v-\rhoz{}gV%
%
\\
\beforetext{Введем константу $k$:}k=6\pi\eta{r}\cdot \frac{1}{V\rhos}=\frac{6\pi\eta{r}}{m}\\
\rhos{V}\frac{dv}{dt}=gV(\rhos-\rhoz)-kv \cdot V{\rhos}\\
\frac{dv}{\C}=dt\\
\beforetext{Замена переменной: } \ \ c=\C\\
dc=-k\,dv\\
-\frac{1}{k}\int^{\Q-kv(t)}_{\Q-kv_0} \frac{dc}{c} = \int_{0}^{t} dt\\
\ln\left(\frac{\Q-kv(t)}{\Q-kv_0}\right)=-kt\\
\frac{\Q-kv(t)}{\Q-kv_0}=e^{-kt}\\
{\Q-kv(t)}=e^{-kt}({\Q-kv_0})\\
kv(t)=e^{-kt}kv_0-e^{-kt}\Q+\Q\\
kv(t)=\Q(1-e^{-kt})+e^{-kt}kv_0\\
v(t)=\Q\frac{1-e^{-kt}}{k}+e^{-kt}v_0\\
v(t)=Vg\frac{\rhos-\rhoz}{6\pi\eta{r}}({1-e^{-kt}})+e^{-kt}v_0\\
v(t)=\frac{2}{9}gr^2\frac{\rhos-\rhoz}{\eta}\cdot({1-e^{-kt}})+e^{-kt}v_0, \text{ где } k=\frac{6\pi\eta{r}}{m}.\\%
%
%
s(t)=\int_0^{s(t)}v(t)\ dt = \K\cdot (\frac{e^{-kt}}{k}+t)-\frac{e^{-kt}v_0}{k}
%
\end{gather}
Для движения без начальной скорости
\begin{gather}
v(t)=\frac{2}{9}gr^2\frac{\rhos-\rhoz}{\eta}\cdot({1-e^{-kt}})\\
s(t)=\K\cdot (\frac{e^{-kt}}{k}+t)\\
1-e^{-kt^*}=\alpha=0.95\\
e^{-kt^*}=\beta=0.05\\
-kt^*=\ln{\beta}\\
t^*=-\frac{1}{k}\ln{\beta}=-\frac{m}{6\pi\eta{r}}\ln{\beta}\\
s^*=\K\cdot(\frac{\beta-\ln{\beta}}{k})=%
\frac{m}{27\pi}gr\frac{\rhos-\rhoz}{\eta^2}(\beta-\ln{\beta})
\end{gather}
$t^*$ -- время установления, когда скорость будет отличаться от установившейся $v_\text{уст}$ на бесконечности не более чем в $\alpha$ раз.
$s^*$ -- соответственно путь установления, когда скорость отличается от установившейся $v_\text{уст}$ на бесконечности не более чем в $\alpha$ раз.
\begin{gather}
v_\text{уст}=\lim_{t \to \infty}{\left[\K\cdot(1-e^{-kt})\right]}=\K
\end{gather}
Путь установления для стали 3.778 см, для стали 0.72 см.
\end{document}