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\begin_body
\begin_layout Title
Dušeno nihanje nitnega nihala
\end_layout
\begin_layout Author
\noun on
Anton Luka Šijanec
\end_layout
\begin_layout Date
16.
november 2022
\end_layout
\begin_layout Abstract
Poročilo četrte vaje pri predmetu
\noun on
F41
\noun default
na Gimnaziji Bežigrad v 4.
letniku.
Vaja je potekala 13.
oktobra 2022.
\end_layout
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_inset
\end_layout
\begin_layout Section
Zveza med amplitudo hitrosti in amplitudo odmika
\end_layout
\begin_layout Subsection
Naloga
\end_layout
\begin_layout Standard
Razišči zvezo med amplitudo hitrosti in amplitudo odmika pri nihanju nitnega
nihala.
\end_layout
\begin_layout Subsection
Potek dela
\end_layout
\begin_layout Enumerate
Ko je nihalo v ravnovesni legi, nastavi ničlo slednika gibanja.
\end_layout
\begin_layout Enumerate
S pomočjo slednika gibanja zajemi podatke za
\begin_inset Formula $\SI{60}{\second}$
\end_inset
dušenega nihanja nitnega nihala.
\end_layout
\begin_layout Enumerate
Na vsake štiri nihaje odčitaj amplitudo odmika in amplitudo hitrosti in
podatke zapiši v tabelo.
Ker nihalo nima največje hitrosti, ko je odmik največji, odčitavaj največjo
hitrost vedno četrt nihaja prej kot največji odmik.
\end_layout
\begin_layout Enumerate
Nariši graf amplitude hitrosti
\begin_inset Formula $v_{0}$
\end_inset
v odvisnosti od amplitude odmika
\begin_inset Formula $x_{0}$
\end_inset
! Določi naklon grafa in oceni nenatančnost te vrednosti.
\begin_inset Newline newline
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status open
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\backslash
begin{lstlisting}
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fit (x*p) "v_od_x.tsv" using 1:2 via p
\end_layout
\begin_layout Plain Layout
\backslash
end{lstlisting}
\end_layout
\end_inset
\begin_inset Formula
\[
p=\SI{4,52127}{\per\second}\pm\SI{0,1111}{\per\second}
\]
\end_inset
\end_layout
\begin_layout Enumerate
Iz grafa
\begin_inset Formula $x(t)$
\end_inset
določi
\begin_inset Formula $t_{0}$
\end_inset
in določi nenatančnost te vrednosti.
Iz tako določenega
\begin_inset Formula $t_{0}$
\end_inset
in iz naklona grafa
\begin_inset Formula $v_{0}(x_{0})$
\end_inset
preveri zvezo
\begin_inset Formula $v_{0}=x_{0}2\pi/t_{0}$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
t_{0_{\text{{odčitan}}}}=\frac{{t}}{n}=\SI{1,2806}{\second}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
t_{0_{\text{{izračunan}}}}=\frac{{2\pi}}{p}=\SI{1,38969}{\second}
\]
\end_inset
\end_layout
\begin_layout Standard
Red velikosti je enak, vendar se rešitvi razlikujeta.
Najverjetneje so bili med zaznavanjem prehodov čez nič za odčitanje
\begin_inset Formula $t_{0}$
\end_inset
iz grafa
\begin_inset Formula $x(t)$
\end_inset
lažni pozitivni odčitki.
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begin{tikzpicture}
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begin{axis}[width=0.75
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textwidth, scale only axis, ylabel=$v$, domain=-0.1:0.1, ytick pos=left, samples=2
56, xlabel=$x$, xtick pos=bottom]
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addplot[red, only marks] table [x=x, y=v] {v_od_x.tsv};
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\backslash
addplot[green] (x, {x*(4.52127+0.1111)});
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addplot[green] (x, {x*(4.52127-0.1111)});
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addplot[red] (x, {x*(4.52127)});
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end{axis}
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end{tikzpicture}
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\begin_inset Caption Standard
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\begin_inset CommandInset label
LatexCommand label
name "fig:graf"
\end_inset
Graf amplitude hitrosti nihanja v odvisnosti od amplitude odmika nihala.
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Section
Zmanjševanje amplitude pri dušenem nihanju
\end_layout
\begin_layout Subsection
Naloga
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\begin_layout Standard
Potrdi, da se amplituda
\begin_inset Formula $x_{0}$
\end_inset
pri dušenem nihanju zmanjšuje eksponentno kot
\begin_inset Formula $x_{0}(t)=x_{00}e^{-\beta t}$
\end_inset
, kjer je
\begin_inset Formula $x_{00}$
\end_inset
amplituda nihala ob času
\series bold
\begin_inset Formula $t=\SI{0}{\second}$
\end_inset
\series default
,
\begin_inset Formula $\beta$
\end_inset
pa koeficient dušenja.
\end_layout
\begin_layout Subsection
Potek dela
\end_layout
\begin_layout Enumerate
Nariši graf amplitude hitrosti
\begin_inset Formula $x_{0}$
\end_inset
v odvisnosti od časa! Skozi točke na grafu
\series bold
\begin_inset Formula $x_{0}(t)$
\end_inset
\series default
nariši ustrezno eksponentno krivuljo!
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status open
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begin{lstlisting}
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n=0.1
\end_layout
\begin_layout Plain Layout
b=0.01
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fit n*exp(-b*x) "v_od_x.tsv" using 3:4 via n, b
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fit (x*ln) "v_od_x.tsv" using 3:5 via ln
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end{lstlisting}
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\end_inset
\begin_inset Formula
\[
n=x_{00}=\SI{0,0943867}{\meter}\pm\SI{0,001524}{\meter}
\]
\end_inset
\begin_inset Formula
\[
b=\SI{0,0441715}{\per\second}\pm\SI{0,001038}{\per\second}=\beta
\]
\end_inset
\begin_inset Formula
\[
ln=-\beta=\SI{-0,0467447}{\per\second}\pm\SI{0,001239}{\per\second}
\]
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset Formula $b$
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in
\begin_inset Formula $ln$
\end_inset
, torej
\begin_inset Formula $\beta$
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, pridobljeni pred linearizacijo in po linearizaciji sta skoraj enaki.
Razlika med njima se pojavi samo zaradi namerne nenatančnosti računalnika.
\end_layout
\begin_layout Enumerate
Nariši graf
\begin_inset Formula $\ln\left(\frac{x_{0}}{x_{00}}\right)$
\end_inset
v odvisnosti od časa in iz grafa določi koeficient dušenja
\begin_inset Formula $\beta$
\end_inset
.
Določitev napak v
\begin_inset Formula $\ref{enu:Nari=00003D000161i-graf-}.$
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delu vaje ni potrebna.
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name "enu:Nariši-graf-"
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begin{axis}[width=0.75
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textwidth, scale only axis, domain=0:60, ytick pos=left, samples=256, xlabel=$t$
, xtick pos=bottom, ylabel=$
\backslash
sqrt{x^2}$]
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\backslash
addplot[only marks] table [x=t, y=absx] {v_od_x.tsv};
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addplot[green] (x, {(0.0943867+0.001524)*exp((-0.0441715+ 0.001038)*x)});
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addplot[green] (x, {(0.0943867-0.001524)*exp((-0.0441715- 0.001038)*x)});
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addplot[black] (x, {0.0943867*exp(-0.0441715*x)});
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end{axis}
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begin{axis}[width=0.75
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textwidth, scale only axis, domain=0:60, xlabel near ticks, samples=16,
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ln
\backslash
frac{x_0}{x_{00}}$]
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addplot[red, only marks] table [x=t, y=ln] {v_od_x.tsv};
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addplot[green] (x, {x*(-0.0467447+0.001239)});
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addplot[green] (x, {x*(-0.0467447-0.001239)});
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addplot[red] (x, {x*-0.0467447});
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end{axis}
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end{tikzpicture}
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\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:graf-1"
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Graf amplitude nihanja (največjega odmika od osi) v odvisnosti od časa.
\end_layout
\end_inset
\end_layout
\end_inset
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\begin_layout Section
Uporabljen program
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Za zaznavanje prehoda ničel in generiranje tabel sem spisal majhen program:
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lstinputlisting[language=Python]{preseki.py}
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\end_body
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