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% do-vimlatex-onwrite
\documentclass[]{article}
\usepackage[utf8]{inputenc}
\usepackage{siunitx}
\usepackage[slovene]{babel}
\usepackage[inline]{enumitem}
\usepackage[a4paper, total={7in, 10in}]{geometry}
\usepackage{hologo}
\usepackage[hidelinks,unicode]{hyperref}
\usepackage{datetime}
\usepackage{tkz-euclide}
\usepackage{amssymb}
% \sisetup{output-decimal-marker = {,}, quotient-mode=fraction, per-mode=fraction} % frac način
% \sisetup{output-decimal-marker = {,}, quotient-mode=fraction, per-mode=symbol} % poševnica način
\sisetup{output-decimal-marker = {,}, quotient-mode=fraction} % na -1 način
\settimeformat{hhmmsstime}
\newcommand{\razhroscevanje}{1}
\newcommand{\razhroscevanjeg}{0} % grafično razhroščevanje
\makeatletter
\newcommand{\xslalph}[1]{\expandafter\@xslalph\csname c@#1\endcsname}
\newcommand{\@xslalph}[1]{%
\ifcase#1\or a\or b\or c\or \v{c}\or d\or e\or f\or g\or h\or i%
\or j\or k\or l\or m\or n\or o\or p\or r\or s\or \v{s}%
\or t\or u\or v\or z\or \v{z}
\else\@ctrerr\fi%
}
\AddEnumerateCounter{\xslalph}{\@xslalph}{m}
\makeatother
\title{Naloga: Sile na klancu in lepenje}
\author{Anton Luka Šijanec, 3. a}
\begin{document}
\maketitle
\begin{abstract}
Rešitev ponedeljkove naloge pri maturitetni fiziki.
\end{abstract}
\tableofcontents
\section{Naloga}
\newcommand{\njuton}{2}
\newcommand{\alfa}{25}
\newcommand{\koef}{0,8}
\subsection{Navodilo}
Na klancu z naklonom $\measuredangle{\alpha}=\ang{\alfa}$ je telo z maso $m=\SI{\njuton}{\kilo\gram}$. Gravitacijski pospešek je $g=\SI{10}{\meter\per\second\squared}$. Koeficient lepenja telesa s klancem je $k_l=\koef$. Kolikšna je največja sila $F_r$, s katero lahko telo vzporedno s klancem vlečemo navkreber, da le-to ne bo zdrsnilo? Kolikšen je največji kot $\measuredangle{\alpha}$, pri katerem brez $F_r$ telo ne bo zdrsnilo?
\subsection{Skice}
\subsubsection{Privzeti pogoji}
\begin{center}
\begin{tikzpicture}
\tkzDefPoint(0,0){A} \tkzDefPoint(1,0){B} \tkzDefPoint(\alfa:3){C}
\tkzInterLC(C,A)(C,A) \tkzGetPoints{D}{D2} \tkzInterLC(A,C)(D,C) \tkzGetPoints{E2}{E}
\tkzInterCC(E,C)(C,E) \tkzGetPoints{F2}{F} \tkzInterCC(A,D)(D,A) \tkzGetPoints{G}{G2}
\tkzDefMidPoint(C,G) \tkzGetPoint{H} \tkzDefMidPoint(D,F) \tkzGetPoint{I}
\tkzDefMidPoint(C,I) \tkzGetPoint{J} \tkzDefShiftPointCoord[J](-90:\njuton*2){K}
\tkzDefLine[parallel=through K](A,C) \tkzGetPoint{L}
\tkzDefMidPoint(C,D) \tkzGetPoint{M} \tkzInterLL(J,M)(K,L) \tkzGetPoint{N}
\tkzDefMidPoint(C,H) \tkzGetPoint{O} \tkzDefLine[parallel=through K](C,O)
\tkzGetPoint{P} \tkzInterLL(P,K)(O,J) \tkzGetPoint{R}
\tkzDefMidPoint(J,K) \tkzGetPoint{S} \tkzDefMidPoint(I,D) \tkzGetPoint{T}
\tkzDefShiftPointCoord[T](\alfa:\njuton*1){U}
\tkzDefMidPoint(J,M) \tkzGetPoint{V} \tkzInterLC(V,N)(V,N) \tkzGetPoints{Z}{Z2}
\tkzInterLC(M,D)(V,R) \tkzGetPoints{W}{W2} \tkzDefLine[parallel=through W](M,Z)
\tkzGetPoint{X} \tkzDefLine[parallel=through Z](C,D) \tkzGetPoint{Y}
\tkzInterLL(Z,Y)(W,X) \tkzGetPoint{Q} \tkzDefMidPoint(M,Q) \tkzGetPoint{1}
\tkzDrawLine[add = 5 and 6](A,B) \tkzDrawLine[add = 0 and 0](A,E)
\tkzDrawLine[dotted](K,N) \tkzDrawLine[dotted](R,K)
\tkzDrawLine[dotted](Q,W) \tkzDrawLine[dotted](Z,Q)
\tkzMarkRightAngles(R,J,N C,M,N)
\tkzDrawPolygon[fill=red!30, opacity=.3](C,D,I,H)
\tkzFillAngle[fill=green!30, opacity=.3, size=3](B,A,C) \tkzLabelAngle[pos=2](B,A,C){$\alpha$}
\tkzDrawSegments[->](J,K) \tkzLabelLine[left, pos=.69](J,K){$\vec{F_g}$}
\tkzDrawSegments[->, dashed](J,N) \tkzLabelLine[above right](J,N){$\vec{F_s}$}
\tkzDrawSegments[->, dashed](J,R) \tkzLabelLine[above](J,R){$\vec{F_d}$}
% \tkzDrawSegments[->](T,U) \tkzLabelLine[above](T,U){$\vec{F_r}$}
\tkzDrawSegments[->, dashed, color=blue](M,Z) \tkzLabelLine[above](M,Z){$\vec{F_0}$}
\tkzDrawSegments[->, dashed, color=cyan](M,W) \tkzLabelLine[above](M,W){$\vec{F_l}$}
\tkzDrawSegments[->](M,Q) \tkzLabelLine[left](M,Q){$\vec{F_p}$}
\tkzDrawPoint[shape=cross out, color=red, size=12](S)
\tkzDrawPoint[shape=cross out, color=red, size=12](1)
\if\razhroscevanjeg1
\tkzDrawPoints(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,R,S,T,U,V,Z,W,X,Y,Q,1)
\tkzLabelPoints(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,R,S,T,U,V,Z,W,X,Y,Q,1)
\fi
\end{tikzpicture}
\end{center}
\subsubsection{Vleka navkreber}
\begin{center}
\begin{tikzpicture}
\tkzDefPoint(0,0){A} \tkzDefPoint(1,0){B} \tkzDefPoint(\alfa:3){C}
\tkzInterLC(C,A)(C,A) \tkzGetPoints{D}{D2} \tkzInterLC(A,C)(D,C) \tkzGetPoints{E2}{E}
\tkzInterCC(E,C)(C,E) \tkzGetPoints{F2}{F} \tkzInterCC(A,D)(D,A) \tkzGetPoints{G}{G2}
\tkzDefMidPoint(C,G) \tkzGetPoint{H} \tkzDefMidPoint(D,F) \tkzGetPoint{I}
\tkzDefMidPoint(C,I) \tkzGetPoint{J} \tkzDefShiftPointCoord[J](-90:\njuton*2){K}
\tkzDefLine[parallel=through K](A,C) \tkzGetPoint{L}
\tkzDefMidPoint(C,D) \tkzGetPoint{M} \tkzInterLL(J,M)(K,L) \tkzGetPoint{N}
\tkzDefMidPoint(C,H) \tkzGetPoint{O} \tkzDefLine[parallel=through K](C,O)
\tkzGetPoint{P} \tkzInterLL(P,K)(O,J) \tkzGetPoint{R}
\tkzDefMidPoint(J,K) \tkzGetPoint{S} \tkzDefMidPoint(I,D) \tkzGetPoint{T}
\tkzDefShiftPointCoord[T](\alfa:\njuton*1.45){U}
\tkzDefMidPoint(J,M) \tkzGetPoint{V} \tkzInterLC(V,N)(V,N) \tkzGetPoints{Z}{Z2}
\tkzDefShiftPointCoord[M](\alfa:\njuton*-1.45){W}
\tkzDefLine[parallel=through W](M,Z)
\tkzGetPoint{X} \tkzDefLine[parallel=through Z](C,D) \tkzGetPoint{Y}
\tkzInterLL(Z,Y)(W,X) \tkzGetPoint{Q} \tkzDefMidPoint(M,Q) \tkzGetPoint{1}
\tkzDrawLine[add = 5 and 6](A,B) \tkzDrawLine[add = 0 and 0](A,E)
\tkzDrawLine[dotted](K,N) \tkzDrawLine[dotted](R,K)
\tkzDrawLine[dotted](Q,W) \tkzDrawLine[dotted](Z,Q)
\tkzMarkRightAngles(R,J,N C,M,N)
\tkzDrawPolygon[fill=red!30, opacity=.3](C,D,I,H)
\tkzFillAngle[fill=green!30, opacity=.3, size=3](B,A,C) \tkzLabelAngle[pos=2](B,A,C){$\alpha$}
\tkzDrawSegments[->](J,K) \tkzLabelLine[left, pos=.69](J,K){$\vec{F_g}$}
\tkzDrawSegments[->, dashed](J,N) \tkzLabelLine[above right](J,N){$\vec{F_s}$}
\tkzDrawSegments[->, dashed](J,R) \tkzLabelLine[above](J,R){$\vec{F_d}$}
\tkzDrawSegments[->](T,U) \tkzLabelLine[above](T,U){$\vec{F_r}$}
\tkzDrawSegments[->, dashed, color=blue](M,Z) \tkzLabelLine[above](M,Z){$\vec{F_0}$}
\tkzDrawSegments[->, dashed, color=cyan](M,W) \tkzLabelLine[above](M,W){$\vec{F_l}$}
\tkzDrawSegments[->](M,Q) \tkzLabelLine[left](M,Q){$\vec{F_p}$}
\tkzDrawPoint[shape=cross out, color=red, size=12](S)
\tkzDrawPoint[shape=cross out, color=red, size=12](1)
\if\razhroscevanjeg1
\tkzDrawPoints(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,R,S,T,U,V,Z,W,X,Y,Q,1)
\tkzLabelPoints(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,R,S,T,U,V,Z,W,X,Y,Q,1)
\fi
\end{tikzpicture}
\end{center}
\subsubsection{Največji kot}
\begin{center}
\begin{tikzpicture}
\tkzDefPoint(0,0){A} \tkzDefPoint(1,0){B} \tkzDefPoint(38.6598:3){C}
\tkzInterLC(C,A)(C,A) \tkzGetPoints{D}{D2} \tkzInterLC(A,C)(D,C) \tkzGetPoints{E2}{E}
\tkzInterCC(E,C)(C,E) \tkzGetPoints{F2}{F} \tkzInterCC(A,D)(D,A) \tkzGetPoints{G}{G2}
\tkzDefMidPoint(C,G) \tkzGetPoint{H} \tkzDefMidPoint(D,F) \tkzGetPoint{I}
\tkzDefMidPoint(C,I) \tkzGetPoint{J} \tkzDefShiftPointCoord[J](-90:\njuton*2){K}
\tkzDefLine[parallel=through K](A,C) \tkzGetPoint{L}
\tkzDefMidPoint(C,D) \tkzGetPoint{M} \tkzInterLL(J,M)(K,L) \tkzGetPoint{N}
\tkzDefMidPoint(C,H) \tkzGetPoint{O} \tkzDefLine[parallel=through K](C,O)
\tkzGetPoint{P} \tkzInterLL(P,K)(O,J) \tkzGetPoint{R}
\tkzDefMidPoint(J,K) \tkzGetPoint{S} \tkzDefMidPoint(I,D) \tkzGetPoint{T}
\tkzDefShiftPointCoord[T](38.6598:\njuton*1){U}
\tkzDefMidPoint(J,M) \tkzGetPoint{V} \tkzInterLC(V,N)(V,N) \tkzGetPoints{Z}{Z2}
\tkzInterLC(M,D)(V,R) \tkzGetPoints{W}{W2} \tkzDefLine[parallel=through W](M,Z)
\tkzGetPoint{X} \tkzDefLine[parallel=through Z](C,D) \tkzGetPoint{Y}
\tkzInterLL(Z,Y)(W,X) \tkzGetPoint{Q} \tkzDefMidPoint(M,Q) \tkzGetPoint{1}
\tkzDrawLine[add = 5 and 6](A,B) \tkzDrawLine[add = 0 and 0](A,E)
\tkzDrawLine[dotted](K,N) \tkzDrawLine[dotted](R,K)
\tkzDrawLine[dotted](Q,W) \tkzDrawLine[dotted](Z,Q)
\tkzMarkRightAngles(R,J,N C,M,N)
\tkzDrawPolygon[fill=red!30, opacity=.3](C,D,I,H)
\tkzFillAngle[fill=green!30, opacity=.3, size=3](B,A,C) \tkzLabelAngle[pos=2](B,A,C){$\alpha$}
\tkzDrawSegments[->](J,K) \tkzLabelLine[left, pos=.69](J,K){$\vec{F_g}$}
\tkzDrawSegments[->, dashed](J,N) \tkzLabelLine[above right](J,N){$\vec{F_s}$}
\tkzDrawSegments[->, dashed](J,R) \tkzLabelLine[above](J,R){$\vec{F_d}$}
% \tkzDrawSegments[->](T,U) \tkzLabelLine[above](T,U){$\vec{F_r}$}
\tkzDrawSegments[->, dashed, color=blue](M,Z) \tkzLabelLine[above](M,Z){$\vec{F_0}$}
\tkzDrawSegments[->, dashed, color=cyan](M,W) \tkzLabelLine[above](M,W){$\vec{F_l}$}
\tkzDrawSegments[->](M,Q) \tkzLabelLine[left](M,Q){$\vec{F_p}$}
\tkzDrawPoint[shape=cross out, color=red, size=12](S)
\tkzDrawPoint[shape=cross out, color=red, size=12](1)
\if\razhroscevanjeg1
\tkzDrawPoints(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,R,S,T,U,V,Z,W,X,Y,Q,1)
\tkzLabelPoints(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,R,S,T,U,V,Z,W,X,Y,Q,1)
\fi
\end{tikzpicture}
\end{center}
\section{Rešitev}
\subsection{Vleka navkreber}
$$F_r={F_l}_{maks}=k_l\cdot F_0=k_l\cdot F_g\cos\alpha=\koef\cdot\SI{\njuton}{\newton}\cos\ang{\alfa}=\SI{1,45}{\newton}$$
\subsection{Največji kot}
Telo ne bo zdrsnilo, če je $$F_d \text{ (dinamična komponenta sile teže)}\leq F_l\text{.}$$
$$k_l=\frac{{F_l}_{maks}}{F_0} \rightarrow F_g\sin\alpha=F_g\cos\alpha\cdot\koef\rightarrow\alpha=\ang{38,6598}$$
\section{Zaključek}
\hologo{LaTeX} izvorna koda dokumenta je objavljena na \url{https://git.sijanec.eu/sijanec/sola-gimb-3}. Za izdelavo dokumenta je potreben \texttt{TeXLive 2020}.
\if\razhroscevanje1
\vfill
\section*{Razhroščevalne informacije}
Konec generiranja dokumenta \today\ ob \currenttime.
Dokument se je generiral R0qK1KR2 \SI{}{\second}. % aaasecgeninsaaa
\fi
\end{document}
|
