From d4422f27c424cce84d8d0a1f40ab65ef07d8376e Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Anton=20Luka=20=C5=A0ijanec?= <anton@sijanec.eu>
Date: Sat, 20 Nov 2021 20:25:59 +0100
Subject: f31 naloga

---
 fiz/matura/klanec.tex | 126 +++++++++++++++++++++++++++++++++++++++++++++-----
 1 file changed, 115 insertions(+), 11 deletions(-)

(limited to 'fiz/matura/klanec.tex')

diff --git a/fiz/matura/klanec.tex b/fiz/matura/klanec.tex
index fef48b0..cb02959 100644
--- a/fiz/matura/klanec.tex
+++ b/fiz/matura/klanec.tex
@@ -9,9 +9,13 @@
 \usepackage[hidelinks,unicode]{hyperref}
 \usepackage{datetime}
 \usepackage{tkz-euclide}
-\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, per-mode=fraction} % per-mode=symbol
+\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]{%
@@ -31,41 +35,141 @@ 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(25:3){C}
+	\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](-90:4){K}
+	\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(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[->](J,N)			\tkzLabelLine[above right](J,N){$\vec{F_s}$}
-	\tkzDrawSegments[->](J,R)			\tkzLabelLine[above](J,R){$\vec{F_d}$}
+	\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)
-\if\razhroscevanje1
-	%\tkzDrawPoints(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,R,S)
-	%\tkzLabelPoints(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,R,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}
-Te informacije so generirane, ker je omogočeno razhroščevanje. Pred objavo dokumenta izklopite razhroščevanje. To naredite tako, da nastavite ukaz \texttt{razhroščevanje} na 0 v začetku dokumenta.
-
 Konec generiranje dokumenta \today\ ob \currenttime.
 
 Dokument se je generiral R0qK1KR2 \SI{}{\second}. % aaasecgeninsaaa
-- 
cgit v1.2.3

