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\begin_body
\begin_layout Title
Kotaljenje kroglice po klancu
\end_layout
\begin_layout Author
\noun on
Anton Luka Šijanec
\end_layout
\begin_layout Date
3.
oktober 2022
\end_layout
\begin_layout Abstract
Poročilo druge vaje pri predmetu
\noun on
F41
\noun default
na Gimnaziji Bežigrad v 4.
letniku.
Vaja je potekala 15.
septembra 2022.
\end_layout
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_inset
\end_layout
\begin_layout Section
Naloga
\end_layout
\begin_layout Enumerate
Dokaži, da je kotaljenje kroglice po klancu enakomerno pospešeno gibanje.
\end_layout
\begin_layout Enumerate
Za gibanje kroglice po klancu navzdol določi povprečno hitrost in pospešek
pri kotaljenju.
Določi tudi napako obeh količin.
\end_layout
\begin_layout Enumerate
Dobljeni pospešek primerjaj s pospeškom za telo, ki bi brez trenja drselo
po klancu z enakom naklonom.
\end_layout
\begin_layout Section
Potrebščine
\end_layout
\begin_layout Itemize
aluminijast profil dolžine
\begin_inset Formula $\SI{1}{\meter}$
\end_inset
\end_layout
\begin_layout Itemize
kovinska kroglica
\end_layout
\begin_layout Itemize
merilna ura
\end_layout
\begin_layout Itemize
ravnilo ali merilni trak
\end_layout
\begin_layout Section
Potek dela
\end_layout
\begin_layout Subsection
Spreminjanje lege po klancu kotaleče se kroglice s časom
\begin_inset CommandInset label
LatexCommand label
name "subsec:Spreminjanje-lege-po"
\end_inset
\end_layout
\begin_layout Enumerate
Kroglico spusti po klancu in izmeri, koliko časa kroglica potuje prvih
\begin_inset Formula $\SI{4,0}{\centi\meter}$
\end_inset
,
\begin_inset Formula $\SI{16}{\centi\meter}$
\end_inset
,
\begin_inset Formula $\SI{36}{\centi\meter}$
\end_inset
in
\begin_inset Formula $\SI{100}{\centi\meter}$
\end_inset
.
Vse meritve zapiši v ustrezno tabelo.
Nariši graf
\begin_inset Formula $x\left(t\right)$
\end_inset
.
Za kakšno gibanje gre? Kako lahko to dokažeš?
\begin_inset Newline newline
\end_inset
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $d\text{\left[\si{\meter}\right]}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $t_{1}\left[\si{\second}\right]$
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\end_inset
|
\begin_inset Text
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\begin_inset Formula $t_{2}\left[\si{\second}\right]$
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|
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\begin_inset Formula $t_{3}\left[\si{\second}\right]$
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|
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\begin_inset Formula $t_{4}\left[\si{\second}\right]$
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|
\begin_inset Text
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\begin_inset Formula $\overline{{t}}$
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|
\begin_inset Text
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\begin_inset Formula $t^{2}\left[\si{\second\squared}\right]$
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Čas, potreben za dosego določenih točk
\end_layout
\end_inset
\end_layout
\end_inset
Gre za enakomerno pospešeno gibanje, kar je razvidno iz dejstva, da
\begin_inset Formula $d$
\end_inset
raste s kvadratom
\begin_inset Formula $t$
\end_inset
.
Če torej lineariziramo,
\begin_inset Formula $d$
\end_inset
raste linearno s
\begin_inset Formula $t^{2}.$
\end_inset
\end_layout
\begin_layout Enumerate
Nariši graf, iz kategega lahko razbereš pospešek gibanja, ter določi pospešek.
Kako lahko iz grafa oceniš napako pospeška?
\begin_inset Newline newline
\end_inset
S programom
\family typewriter
gnuplot
\family default
izračunamo enačbi krivulj, ki se točkam najbolj prilegajo:
\begin_inset Newline newline
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{lstlisting}
\end_layout
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fit sqrt(x*q) "podatki.tsv" using 1:6 via q
\end_layout
\begin_layout Plain Layout
fit (x*p)
\begin_inset Quotes gld
\end_inset
podatki.tsv
\begin_inset Quotes grd
\end_inset
using 1:7 via p
\end_layout
\begin_layout Plain Layout
\backslash
end{lstlisting}
\end_layout
\end_inset
in dobimo vrednosti parametrov
\begin_inset Formula $q=12,9273$
\end_inset
s standardno napako
\begin_inset Foot
status open
\begin_layout Plain Layout
Standardni odklon se izračuna po enačbi
\begin_inset Formula
\[
\sigma_{\overline{{x}}}=\sqrt{{\frac{{\mathop{\sum_{i=0}^{n}\left(x_{i}-\overline{{x}}\right)^{2}}}}{{n}}}}\text{{.}}
\]
\end_inset
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\begin_inset Formula $\pm0,6204$
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oziroma
\begin_inset Formula $4,799\%$
\end_inset
in
\begin_inset Formula $p=\SI{13,3468}{\second\squared\per\meter}$
\end_inset
s standardno napako
\begin_inset Formula $\pm0,2407$
\end_inset
oziroma
\begin_inset Formula $1,803\%$
\end_inset
.
\begin_inset Formula $p$
\end_inset
je koeficient premice na grafu
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:graf"
plural "false"
caps "false"
noprefix "false"
\end_inset
, torej obratna vrednost iskanega pospeška.
Pospešek je tedaj
\begin_inset Formula
\[
a=p^{-1}=\SI{0,0749243264}{\meter\per\second\squared}
\]
\end_inset
Standardna napaka pospeška je
\begin_inset Formula
\[
\pm0,01803a=\pm\SI{0,0013508856}{\meter\per\second\squared}
\]
\end_inset
\begin_inset Float figure
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{tikzpicture}
\end_layout
\begin_layout Plain Layout
\backslash
begin{axis}[width=0.75
\backslash
textwidth, scale only axis, ylabel=$
\backslash
color{red}t
\backslash
color{black}$, domain=0:1, ytick pos=left, samples=256]
\end_layout
\begin_layout Plain Layout
\backslash
addplot[red, only marks] table [x=d, y=t] {podatki.tsv};
\end_layout
\begin_layout Plain Layout
\backslash
addplot[red] (x, {sqrt(x*12.9273)});
\end_layout
\begin_layout Plain Layout
\backslash
addplot[green] (x, {sqrt(x*(12.9273+0.6204))});
\end_layout
\begin_layout Plain Layout
\backslash
addplot[green] (x, {sqrt(x*(12.9273-0.6204))});
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\backslash
end{axis}
\end_layout
\begin_layout Plain Layout
\backslash
begin{axis}[width=0.75
\backslash
textwidth, scale only axis, xlabel=$d$, ylabel=$t^2$, ylabel near ticks,
domain=0:1, xtick pos=bottom, ytick pos=right, axis y line*=right]
\end_layout
\begin_layout Plain Layout
\backslash
addplot[only marks] table [x=d, y=t2] {podatki.tsv};
\end_layout
\begin_layout Plain Layout
\backslash
addplot[black] (x, {x*13.3468});
\end_layout
\begin_layout Plain Layout
\backslash
addplot[green] (x, {x*(13.3468+0.2407)});
\end_layout
\begin_layout Plain Layout
\backslash
addplot[green] (x, {x*(13.3468-0.2407)});
\end_layout
\begin_layout Plain Layout
\backslash
end{axis}
\end_layout
\begin_layout Plain Layout
\backslash
end{tikzpicture}
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:graf"
\end_inset
Grafa
\begin_inset Formula $t$
\end_inset
in
\begin_inset Formula $t^{2}$
\end_inset
v odvisnosti od
\begin_inset Formula $d$
\end_inset
.
Čas se tukaj sicer neobičajno pojavi na ordinatni osi, kar je smiselno,
saj je to izmerjena količina.
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Čas, potreben za kotaljenje kroglice po celotni dolžini klanca
\begin_inset CommandInset label
LatexCommand label
name "subsec:Čas,-potreben-za"
\end_inset
\end_layout
\begin_layout Enumerate
Kroglico spusti po rahlo nagnjenem klancu ter meri čas gibanja.
Pazi, da ta čas ni prekratek.
Meritev ponovi vsaj osemkrat, izračunaj povprečni čas ter zapiši rezultat
meritve z absolutno in relativno napako.
\begin_inset Newline newline
\end_inset
Ne pozabi izmeriti naklonskega kota klanca!
\begin_inset Newline newline
\end_inset
\begin_inset Formula
\[
\phi=\sin\frac{{\SI{0,02}{\meter}}}{\SI{1,00}{\meter}}=\ang{1,14699199839}
\]
\end_inset
\begin_inset Float table
placement h
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $t_{1}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\series bold
\begin_inset Formula $t_{2}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $t_{3}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $t_{4}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $t_{5}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $t_{6}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $t_{7}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $t_{8}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\overline{{t}}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\sigma_{\overline{{t}}}\left[\si{\second}\right]$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\sigma_{\overline{{t}}}:\overline{{t}}\left[\si{\second\per\second}\right]$
\end_inset
\end_layout
\end_inset
|
|
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4,3
\end_layout
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|
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4,2
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|
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\begin_layout Plain Layout
3,8
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|
\begin_inset Text
\begin_layout Plain Layout
3,7
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|
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\begin_layout Plain Layout
3,8
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|
\begin_inset Text
\begin_layout Plain Layout
3,9125
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0,20
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $5,18\%$
\end_inset
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Čas, potreben za spust kroglice po klancu nizdol.
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Enumerate
Izmeri dolžino klanca in oceni napako meritve!
\begin_inset Newline newline
\end_inset
\begin_inset Formula
\[
l=\SI{1}{\meter}\pm\left(<\SI{1}{\milli\meter}\right)
\]
\end_inset
\end_layout
\begin_layout Enumerate
Izračunaj povprečno hitrost ter zapiši rezultat z absolutno in relativno
napako.
\begin_inset Formula
\[
\overline{v}=\overline{{t}}^{-1}=\SI{0,255591}{\meter\per\second}\left(1\pm0,0518\right)=\SI{0,255591}{\meter\per\second}\pm\SI{0,013}{\meter\per\second}
\]
\end_inset
\end_layout
\begin_layout Enumerate
Izračunaj pospešek ter zapiši rezultat z absolutno in relativno napako.
\begin_inset Formula
\[
v_{\text{{max}}}=2\overline{{v}}
\]
\end_inset
\begin_inset Formula
\[
\overline{{a}}=\frac{v_{\text{{max}}}}{\overline{{t}}}=\SI{0,065326}{\meter\per\second\squared}1\pm2\cdot0,0518=\SI{0,065326}{\meter\per\second\squared}\pm\SI{0,003384}{\meter\per\second\squared}
\]
\end_inset
Ta pospešek se ujema z izmerjenim pospeškom iz naloge
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Spreminjanje-lege-po"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Subsection
Izračun pospeška po klancu brez trenja drsečega telesa
\end_layout
\begin_layout Standard
Izračunaj pospešek za telo, ki bi brez trenja drselo po klancu iz naloge
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Čas,-potreben-za"
plural "false"
caps "false"
noprefix "false"
\end_inset
, primerjaj rezultata ter zapiši ugotovitve.
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
F_{g}\tan\phi=F_{d}\rightarrow\cancel{{m}}g(\sin\phi=0,02)=\cancel{m}a=\SI{0,1962}{\meter\per\second\squared}
\]
\end_inset
\end_layout
\begin_layout Standard
Kroglica med premikanjem nekaj energije porablja tudi za vrtenje — to je
moja hipoteza, zakaj je pospešek drsečega telesa večji od izmerjenega pospeška
kotalegeča se telesa.
\end_layout
\end_body
\end_document