From dc3bae6b1ef1bb94c566dbf3ee0683736a61db6f Mon Sep 17 00:00:00 2001 From: Lugaricci Date: Wed, 27 May 2026 17:32:14 +0200 Subject: [PATCH] pridej eintoph vseho mozneho --- Frenetův rámec.md | 62 +++ Inerciální a neinerciální soustava.md | 43 ++ Mechanické kmity.md | 0 Mechanika tuhého tělesa.md | 0 Otázky test.md | 413 ++++++++++++++++++ Pohyb po kružnici 2.md | 24 + Speciální teorie relativity.md | 47 ++ Srážka těles.md | 39 ++ Zákony zachování.md | 40 ++ dokumenty/Fyzika1_skripta.pdf | Bin 0 -> 14802564 bytes ...rka prikladu - sbirka_prikladu_Fyzika1.pdf | Bin 0 -> 1086678 bytes 11 files changed, 668 insertions(+) create mode 100644 Frenetův rámec.md create mode 100644 Inerciální a neinerciální soustava.md create mode 100644 Mechanické kmity.md create mode 100644 Mechanika tuhého tělesa.md create mode 100644 Otázky test.md create mode 100644 Pohyb po kružnici 2.md create mode 100644 Speciální teorie relativity.md create mode 100644 Srážka těles.md create mode 100644 Zákony zachování.md create mode 100644 dokumenty/Fyzika1_skripta.pdf create mode 100644 dokumenty/Sbirka prikladu - sbirka_prikladu_Fyzika1.pdf diff --git a/Frenetův rámec.md b/Frenetův rámec.md new file mode 100644 index 0000000..a36a9e1 --- /dev/null +++ b/Frenetův rámec.md @@ -0,0 +1,62 @@ +Odbočka. + +Pohyb po křivce se dá zapsat jako: +$$ +r(t) = \begin{pmatrix} x(t)\\y(t)\\ z(t)\end{pmatrix} +$$ + +Pohyb křivce můžeme ale také popsat pomocí tří navzájem kolmých. +1) tečný vektor -> ukazuje ve směru pohybu +2) normálový vektor -> ukazuje do středu pomyslné rovnice +3) binominální vektor -> je kolmý na plochu otáčení. +**Všechny jsou jednotkové.** + +Na to, abychom to pochopili, musíme nejprve nějak vstřebat fakt, že si můžeme na křivce v každém bodě představit, že se vlastně pohybujeme po kružnici (směrově). Tahle kružnice má jakýsi poloměr $R$. Místo přímo poloměru používáme tzv. **zakřivení** $\kappa = \frac{1}{R}$. + +Tečný vektor: +$$ +\vec{T}(t) = \frac{\vec{\dot{r}}(t)}{||\vec{\dot{r}}(t)||} +$$ +Normálový vektor: +$$ +\vec{N}(t) = \frac{\vec{\dot{T}}(t)}{||\vec{\dot{T}}(t)||} +$$ +Binominální vektor: +$$ +\vec{B}(t) = \vec{T}(t) \vec{\times}N(t) +$$ + +Pro popis pohybu po křivce existuje sada Frenet–Serret vztahů: + +$$ +\frac{d\vec{T}}{ds} = \kappa \vec{N} +$$ +Doslova rozdíl vektorů dT - tedy když se nepatrně posunu a koukám se, kam ukazuje tečný směr křivky - lépe řečeno [^1]rozdíl špiček vektorů - tak je ve směru normálového vektoru (co ukazuje na střed) v posunutém bodě a velikost tohoto rozdílu je **zakřivení**. + +$$ +\frac{d\vec{N}}{ds} = -\kappa \vec{T} + \tau \vec{B} +$$ + +Rozdíl normálového vektoru je roven složce je roven složce $-\kappa T$, která popisuje o pohybu na ploše. Změna mezi normálovými vektoru (od starému k novému) je rovná opačnému směru tečného vektoru v druhém bodě (v tom kam se pohnu) násobeném zakřivením. +Zde ale také počítáme s možností pohybu mimo plochu (ve 3D). Takže tam je i složka $\tau B$ s tzv. **torzí**, která popisuje právě pohyb, kdy se vektor kolmý na plochu mění. + +$$ +\frac{d\vec{B}}{ds} = -\tau \vec{N} +$$ + + +**Zakřivení a Torze**: + + +$$ +\kappa = \frac{\dot{r}(t)\times \ddot{r}(t)}{||\dot{r}(t)||^3} +$$ +$$ +\tau = \frac{(\dot{r}(t)\times \ddot{r}(t))\cdot \dddot{r}(t)}{||\dot{r}(t)\times \ddot{r}(t)||^2} +$$ +kde: +$$ +\dot{r}(t)\times \ddot{r}(t) +$$ +je normála na oskulační rovinu - co to je? Nepředstavitelný koncept roviny, která nejlépe doléha na nějakou jinou rovinu - nevím :') +[^1]: za tenhle termín mi pls neutrhněte hlavu diff --git a/Inerciální a neinerciální soustava.md b/Inerciální a neinerciální soustava.md new file mode 100644 index 0000000..aee5ca0 --- /dev/null +++ b/Inerciální a neinerciální soustava.md @@ -0,0 +1,43 @@ +Celé toto téma se točí kolem problému popisu pohybu ve dvou různých systémech. Například mám senzor pohybu v místnosti, který měří pohyb ve vzdálenosti vůči sobě samému, v domě je ale systém, který snímá pohyb v celém domě a snaží si ho přepočítat do vlastní soustavy. Pak dokonce můžume mít systém s pohybem a zrychlením. Například počítat pohyb na zemi vůči středu sluneční soustavy, etc. + +# Inerciální vztažná soustava +Toto je vlastně speciální případ dvou a více soustav, které se vůči sobě **pohybují konstantní rychlostí** (která může být i nulová). +Pokud máme nějaké bod v prostoru, tak bude platit **ve všech soustavách, že:** +- $\vec{r_{a}} - \vec{r_{b}} = \vec{R}$ a tedy: $\vec{r_{a}} = \vec{R} - \vec{r_{b}}$ +kde $\vec{_{r_{a}}}$ je polohový vektor bodu vůči středu soustavy A, $\vec{_{r_{b}}}$ je polohový vektor bodu vůči středu soustavy B a $\vec{R}$ je polohový vektor, který značí vzdálenost dvou soustav. +**Představa je intuitivní**: Když jdu ze svého domu na náměstí a pak se vrátím z náměstí ke kamarádova (teda opačná trasa než od kamaráda na náměstí), tak skončím u kamaráda a skončil bych tam stejně tak, kdybych šel od mého domu rovnou ke kamarádovi. + +Pak je třeba ještě řešit otázku rychlosti (což jde třeba derivací): +platí pro rychlost bodu vůči soustavě A $\vec{v_{a}}$ a rychlost bodu vůči soustavě $\vec{v_{b}}$ a vzájemnou rychlost $V$: +- $\vec{v_{a}} = V + v_{b}$ +**Úplně stejně intuitivně**: Když se pohybuje vlak vůči mě 50km/h a člověk ve vlaku pochoduje zpět rychlostí 5km/h, tak můj pohled rychlosti člověka je 45 = 50 - 5. + +Poslední krok je zjistit zrychlení bodu vůči mojí soustavě: +derivací: $a_{a} = 0 + a_{b}$ tedy $a_{a} = a_{b}$. Protože jsme si na začátku řekli, že rychlost soustav je konstantní, tak musí mít nulovou rychlost. +**Takže každý zrychlující bod v prostoru zrychluje vůči oběma soustavám stejným způsobem.** Svět je krásný, poníci skáčou po lučinách, a my žijeme šťastní až do... + +# Neinerciální vztažná soustava +Hrome... cože... +Když se věci hýbou jinak, než konstantní rychlostí, tak to vypadá, že vznikají v obou soustavách jiné síly. Newtonovi zákony neplatí. Svět je v plamenech. Kde jsou poníci?!? + +Počkat, ono to ale jde napravit. +Za prvé si uvědomíme, že **co se týče polohy, tak se doslova nic nezměnilo**. Poloha je věc geometrie a tedy pořád platí: $r_{a} = R + r_{b}$ + +**Dále na řadě je úprava rychlosti.** Zde je úprava vlastně malá, ale ne na první pohled zřejmá. +$$ +v_{a} = V + v_{b} + \omega \times r_{b} +$$ +Přibylo: $\omega \times r_{b}$ Tedy... Pokud se bod nachází v mojí neinerciální soustavě, třeba na kolotoči, pak nám tam vznikne nedostatek. Stále platí, že potřebuji sečíst rychlost dvou soustav a rychlost bodu vůči soustavě. To ale nestačí. Řekněme, že já se vůči kolotoči nepohybuju a kolotoč se otáčí. Pak sedadlo na kolotoči se vůči kolotoči také nepohybuje (protože se otáčí stejnou rychlostí jako kolotoč.). Vůči mě se ale sedadlo zřejmě pohybuje. +Ten člen nám tedy toto řeší, protože vektorový součin úhlové rychlosti a vektoru polohy nám dají vektor rychlosti bodu. + +**A teď přejdeme k tomu nejdůležitějšímu a to ke zrychlení a silám.** Všechny ty nestandardní síly, co jsou v jedné soustavě, ale v druhé jsou vlastně důsledek úpravy rychlosti, co jsme udělali před chvílí. (vyplyne to z derivace) +$$ +a_{a} = A + a_{b} + \epsilon \times r' + 2 \omega \times v' + \omega \times (\omega \times r') +$$ +Derivace $\frac{d}{dt}(V) = A$, to vektor zrychlení mezí soustavami a nazývá se **setrvačné zrychlení**. +Derivace $\frac{d}{dt}(a_{b})$ se musí dělat podle triku. + +Unavenej.. dodělat potom... + + + diff --git a/Mechanické kmity.md b/Mechanické kmity.md new file mode 100644 index 0000000..e69de29 diff --git a/Mechanika tuhého tělesa.md b/Mechanika tuhého tělesa.md new file mode 100644 index 0000000..e69de29 diff --git a/Otázky test.md b/Otázky test.md new file mode 100644 index 0000000..3d8141e --- /dev/null +++ b/Otázky test.md @@ -0,0 +1,413 @@ +# Fundamentální +### Napište základní jednotky SI, stručně popište způsob jejich zavedení a rozložte jednotku tesla na součin základních jednotek SI. + +Jednotky SI: +**Je jich 7** + +| Veličina | Jednotka | Symbol | Definice | +| --------- | ------------ | ------ | ------------------------------------------------------------------------------------------------------------------------------- | +| Délka | **metr** | *m* | vzdálenost, kterou světlo ve vakuu urazí za přesně 1/299 792 458 sekundy | +| Čas | **sekunda** | *s* | čas, během kterého atom Cs-133 vykoná přesně 9 192 631 770 kmitů | +| Hmotnost | **kilogram** | *kg* | taková hmotnost, že Planckova konstanta h = přesně 6,626 07015 × 10⁻³⁴ J·s | +| Proud | **ampér** | *A* | | +| Teplota | **kelvin** | *K* | 1 K je taková teplota, že Boltzmannova konstanta k = přesně 1,380 649 × 10⁻²³ J/K | +| Množství | **mol** | *mol* | 1 mol obsahuje přesně 6,022 14076 × 10²³ elementárních entit | +| Svítivost | **kandela** | *cd* | 1 cd je svítivost zdroje, který vyzařuje monochromatické záření o frekvenci 540 × 10¹² Hz s výkonem 1/683 W ve stanoveném směru | + +--- +Zavedení: + +Máme více způsobů zavedení: + +1) **Přes fyzikální konstanty (od roku 2019)** - to znamená, že se vybera nějaká fyzikální konstanta a přiřadí se jí nějaká určená hodnota. Násobek tou hodnotou nebo podíl je pak nová jednotka SI - např: rychlost světla je c = 299 792 458 m/s +2) **Zavedení přes etalony** - hodnota zavedena dle nějakých předmětů, které tu hodnotu určují - tzv. etalony. Např. do roku 1983 byl metr definovaný platinovou tyčí v Paříži + +--- +Tesla: +Je jednotka magnetické indukce + +**Z Lorenzovy síly:** +$$ +F = q(v \times B) +$$ +přepsáno: +$$ +B = \frac{F}{qv} +$$ +Jednotky: +$$ +T = \frac{N\cdot s}{C\cdot m} = \frac{(kg \cdot m)\cdot s}{C \cdot m \cdot (s^2)} = \frac{kg}{C\cdot s} = \frac{kg}{(A\cdot s)\cdot s} = kg\cdot A^{-1}\cdot s^{-2} +$$ +### Uveďte znění Newtonových pohybových zákonů a jejich odpovídající matematické vyjádření. + +1) Těleso zůstává v klidu nebo v rovnoměrném přímočarém pohybu, pokud není nuceno svůj pohybový stav změnit působením vnějších sil. + +$$ +\vec{F} = \vec{o} \to \vec{v} = konst. +$$ + nebo asi také: +$$ +\vec{F} \sim \frac{\Delta \vec{v}}{\Delta t} +$$ + +2) Časová změna hybnosti tělesa je rovna síle, která na něj působí. +$$ +\frac{\delta \vec{p}}{\delta t} = \vec{F} +$$ +3) Jestliže dvě tělesa na sebe vzájemně působí silami, pak jsou tyto síly stejně velké, ale opačně orientované, leží na společné silové přímce a v každém okamžiku platí: +$$ +F_{12} = -F_{21} +$$ +--- +shrnutí: +1) **klid a pohyb** +2) **hybnost a síla** +3) **opačná síla** +### Napište první a druhou větu impulzovou pro soustavu hmotných bodů. +1) věta mluví o bilanci sil v soustavě -> výslednice sil v soustavě hmotných bodů je rovna derivace celkové hybnosti soustavy. + +$$ +\frac{d}{dt} \sum^n_{i=1} p_{i} = \frac{d}{dt} \sum^n_{i=1} m_{i}v_{i} = F_{vysl.} +$$ + +2) druhá věta impulsová mluví o celkovém momentu sil celé soustavy, víme, že moment síly je derivace momentu hybnosti za čas +$$ +\frac{d}{dt}\sum^n_{i=1} L_{i} = M_{výsl.} +$$ +### Uveďte znění Newtonova gravitačního zákona a Keplerových zákonů; doplňte o matematická vyjádření nebo náčrtky. + + + +> [!Znění gravitačního zákona:] +> Dva hmotné body se přitahují silou, jejíž velikost je přímo úměrná součinu jejich hmotností a nepřímo úměrná kvadrátu jejich vzdálenosti. + +Vyjádřeno **skalárně**: +$$ +F = G \frac{m_{1}m_{2}}{r^2} +$$ + +Vyjádřeno **vektorově** (na těleso 2 působením tělesa 1): +$$ +F_{21} = -G \frac{m_{1}m_{2}}{||\vec{r_{2}}-\vec{r_{1}}||^3} (\vec{r_{2}} -\vec{r_{1}}) +$$ +--- + +> [!První keplerův zákon:] +> Planety obíhají kolem Slunce po [eliptických](https://cs.wikipedia.org/wiki/Elipsa "Elipsa") [drahách](https://cs.wikipedia.org/wiki/Oběžná_dráha "Oběžná dráha") (přesněji trajektoriích), v jejichž jednom společném ohnisku je Slunce. + +> [!Druhý keplerův zákon:] +> Obsahy ploch opsaných průvodičem planety (spojnice planety a Slunce) za stejný čas jsou stejně velké. + +$$ +w = \frac{\delta S}{\delta t} = \frac{L}{2m} = konst. +$$ +kde L je moment hybnosti a m je hmotnost obíhajícího tělesa (která je obsažena i v L, takže se vyruší a nezáleží na ni) + +Jak spočítat L obecně: +$$ +L = m \sqrt{ GMa(1-e^2) } +$$ +m - hmostnost obíhajícího objektu, G - gravitační konstanta a - velká poloosa, e - excentricita +$$ +L = m r_{min}v_{max} = mr_{max}v_{min} +$$ +tedy moment hybnosti v periheliu (nejmenší vzdálenost, největší rychlost) je rovna rychlosti v afeliu (největší vzdálenost, nejmenší hybnost). + +kde $L$ je moment hybnosti soustavy a +> [!Třetí keplerův zákon:] +> Poměr druhých mocnin oběžných dob dvou planet je stejný jako poměr třetích mocnin délek jejich hlavních poloos (středních vzdáleností těchto planet od Slunce). + +$$ +\frac{T^2}{a^3} = \frac{4\pi^2}{GM} = konst. +$$ + +kde $T$ je perioda oběhu a $a$ je excentricita. $M$ je hmotnost obíhaného objektu. +### Napište Lagrangeovy rovnice 2. druhu (pro případ konzervativních sil). +$$ +\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 +$$ + +Kde $L = T - U$ je lagrangián, $T$ je kinetická energie, $U$ je zobecněná energie a $q$ je zobecněná souřadnice. +### Napište tvary rovnice kontinuity jak pro tok tekutiny, tak pro elektrický náboj v diferenciálním i integrálním tvaru. +**Rovnice kontinuity pro tekutiny:** +$$ +\frac{\partial \rho}{\partial t} + \nabla \cdot(\rho v) +$$ +$$ +\iiint_{V} \frac{\partial p}{\partial t}\space dV = - \iint_{\partial V} (\rho \vec{v})\cdot dS +$$ +--- +**Rovnice kontinuity pro elektrický náboj:** +$$ +\frac{\partial \rho}{\partial t} + \nabla \cdot j = 0 +$$ +kde $j = \rho \vec{v}$ je, $\rho$ je nábojová hustota +$$ +I = -\frac{\partial q}{\partial t} = \iint_{\partial V}j\cdot dS = - \iiint_{V} \frac{\partial \rho}{\partial t} \space dV +$$ + +Kde $J$ je hustota elektrického proudu. +### Uveďte postuláty speciální teorie relativity. + +> [!První postulát] +> Všechny fyzikální zákony mají ve všech inerciálních vztažných soustavách stejný tvar. + +- de facto zobecnění Galileův zákon + +> [!Druhý postulát] +> Rychlost světa ve vakuu je všech inerciálních vztažných soustavách stejná. + +### Napište Coulombův zákon pro dva bodové náboje a pro silové působení náboje o hustotě ρ rozprostřeného v objemu V′ na bodový náboj q; doplňte vhodným náčrtkem. +**Pro dva bodové náboje:** +Coulombův zákon ve skalárním tvaru: +$$ +F_{C} = \frac{1}{4\pi \epsilon_{0}} \frac{q_{1}q_{2}}{r^2} +$$ + +Coulombův zákon ve vektorovém tvaru: +$$ +F_{21} = \frac{1}{4\pi \epsilon_{0}} \frac{q_{1}q_{2}}{||\vec{r_{2}}-\vec{r_{1}}||^3}(\vec{r_{2}}-\vec{r_{1}}) +$$ +--- +**Pro distribuovaný náboj a bod:** +$$ +F = \frac{q}{4\pi \epsilon_{0}} \int_{V} \rho(\vec{r})dV +$$ +### Napište Maxwellovy rovnice v diferenciálním a integrálním tvaru pro materiálové prostředí a vakuum. +**Legenda**: + +E - > elektrická intenzita +D -> elektrická indukce +H -> magnetická intenzita +B -> magnetická indukce + +--- +**Diferenciální tvar:** +Gaussův zákon elektrostatiky: +$$ +\nabla \cdot \vec{E} = 0 +$$ +Gaussův zákon magnetického pole (neexistence magnetických monopólů: +$$ +\nabla \cdot \vec{B} = 0 +$$ +Faradayův zákon elektromagnetické indukce: +$$ +\nabla \times E = - \frac{\partial B}{\partial t} +$$ +Ampérův-Maxwellův zákon celkového proudu: +$$ +\nabla \times B = \epsilon_{0}\micro_{0} \space \frac{\partial E}{\partial t} +$$ +--- +**Integrální tvar**: +Gaussův zákon elektrostatiky: +$$ +\iint_{\partial V} \vec{E}\cdot dS = 0 +$$ +Gaussův zákon magnetického pole (neexistence magnetických monopólů: +$$ +\iint_{\partial V}\vec{B}\cdot dS = 0 +$$ +Faradayův zákon elektromagnetické indukce: +$$ +\oint_{\partial S} \vec{E}\cdot dr = -\frac{d}{dt} \iint_{S}B\cdot dS +$$ +Ampérův-Maxwellův zákon celkového proudu: +$$ +\oint_{\partial S} \vec{B}\cdot dr = \iint_{S} \epsilon_{0}\micro_{0} \space\frac{\partial E}{\partial t}\cdot dS +$$ + +# Obecné + +### Měření délky matematického kyvadla bylo zatíženo standardní nejistotou u(ℓ) a měření periody kmitů standardní nejistotou u(T). Odpovídající průměrné hodnoty veličina označíme jako ¯ℓ a ¯T . Určete vzorec pro kombinovanou standardní nejistotu gravitačního zrychlení g, jestliže všechny ostatní zdroje nejistot lze považovat za zanedbatelné. + +## Pohyb po kružnici + (inerc. neinerc.) +### Definujte velikost úhlu v radiánech, doplňte vhodným náčrtkem. + +Úhel je délka oblouku dělena poloměrem kružnice. +$$ +\varphi [rad] = \frac{s}{r} +$$, kde s je délka oblouku a r je poloměr kružnice. +### Rozepište vektor zrychlení a pomocí jeho tečné a normálové složky. Pro obě uveďte vztahy. +$$ +\vec{a} = \frac{d\vec{v}}{dt} = \frac{d}{dt}(v\vec{\tau}) = \frac{dv}{dt}\vec{\tau} + \frac{d\vec{\tau}}{dt}v +$$ +kde složka $$\frac{dv}{dt}\vec{\tau} = \vec{a_{t}}$$ vyjadřuje změnu rychlosti ve směru tečného vektoru (**je to tečné zrychlení** - mění rychlost) +a složka $$\frac{d\vec{\tau}}{dt}v = \vec{a_{n}}$$ +vyjadřuje změnu tečny se stejnou rychlostí (**je to normálové zrychlení** - mění směr) + +vzorec nahoře je rozepsání: +$$ +\vec{v} = v\vec{\tau} +$$ +kde v je velikost rychlosti a tau je tečný jednotkový vektor. Vzorec se dál rozepíše dle pravidla násobení v derivaci. + +Lze to rozepsat ještě pomocí Frenetova rámce a vzorců do lepší podoby: +Protože +$$ +\frac{ds}{dt} = v +$$ +a +$$ +\frac{d\vec{\tau}}{ds} = \kappa \vec{N} = \frac{1}{R}\vec{N} +$$ +$$ +\frac{d\vec{\tau}}{dt}v = \left( \frac{d\vec{\tau}}{ds} \right) \left( \frac{ds}{dt} \right) v = (\kappa \vec{N})(v)v = \vec{N} \frac{v^2}{R} +$$ +Tedy: +$$ +\vec{a} = \vec{a_{t}} + \vec{a_{n}} = \frac{dv}{dt}\vec{\tau} + \frac{v^2}{R} \vec{N} +$$ +### Pro případ rovnoměrného pohybu po kružnici napište vztahy mezi úhlovou rychlostí, periodou a frekvencí. +$$ +f =\frac{1}{T} +$$ +$$ +\omega = 2\pi f = \frac{2\pi (rad)}{T} +$$ +proč? úhlová rychlost říká, kolikrát za čas oběhne pohyb celý kruh. Protože celý kruh je $2\pi rad$ a $\frac{1}{T}$ říká "jedno otočení, za T sekund", tak pro úhlovou rychlost v radiánech (a ne jednotkách otočení) musíme vynásobit $\frac{1}{T}$ počtem radiánů v otočení (což je $2\pi$). + +Damn, do toho jsem se nějak zamotal. + +### Za jakých podmínek působí na hmotný bod v neinerciální vztažné soustavě Coriolisova síla? + +Corlissova síla vzniká, když se těleso pohybuje nějakou rychlostí od středu vlastní soustavy a soustava rotuje. + +Závisí na rychlosti bodu vůči ose otáčení a na úhlové rychlosti soustavy. + +$$ +F_{cor} = 2\omega \times v' +$$ + +## Síla práce dynamika? +### Definujte práci síly F vykonanou po křivce C nejobecnějším způsobem. +$$ +W = \int_{c}F\cdot d\vec{r} +$$ +### Definujte veličinu okamžitý výkon a uveďte její souvislost se silou působící na pohybující se hmotný bod. +$$ +P = \frac{dW}{dt} = \vec{F} \cdot \vec{v} +$$ +### Pomocí jaké operace se z potenciální energie obdrží konzervativní síla? +$$ +\vec{F} = -\nabla U +$$ +### Uveďte alespoň jednu z vlastností nekonzervativní síly a nějaký její příklad. +**Vykoná práce závisí na dráze.** (a ne pouze na počáteční a konečné poloze jako u konzervativních sil). +Třecí síla je třeba nekonzervativní síla. + +## Analytická mechanika +### Definujte pojem cyklická souřadnice, definujte zobecněnou hybnost a uveďte, co tyto dva pojmy spojuje. +### Napište Hamiltony kanonické rovnice. +## Oscilace +### Uveďte vzorce pro vratnou sílu pružnosti, její potenciální energii a vztah, který je spojuje. +$$ +\vec{F} = -kx +$$ +$$ +U = \frac{1}{2}kx^2 +$$ +$$ +\vec{F} = -\nabla U +$$ +### Napište pohybovou rovnici netlumeného lineárního harmonického oscilátoru a některý z možných tvarů jejího řešení + +$$ +F = -kx +$$ +$$ +m\ddot{x} + kx = 0 +$$ +$$ +\ddot{x} + \frac{k}{m}x = 0 +$$ +**Ta poslední je to co je pohybová rovnice:** +$$ +\ddot{x} + \omega^2x = 0 +$$ +--- +Tvary: +$$ +A\cos(\omega t) + B\sin(\omega t) = x +$$ +$$ +A\sin(\omega t + \varphi_{0}) = 0 +$$ +## Mechanika tuhého tělesa +### Uveďte definiční vztahy polohy hmotného středu tuhého tělesa a momentu setrvačnosti tuhého tělesa. +Moment setrvačnosti: +$$ +I = \int_{V}r_{\perp}^2\rho(\vec{r})\space dV +$$ +Střed hmotného tělesa: +$$ +\vec{r_{T}} = \frac{\int_{V}\vec{r}\rho(\vec{r})\space dV}{\int_{V}\rho(\vec{r})\space dV} = \frac{1}{M} \int_{V}\vec{r} \space dm +$$ +### Napište Konigovu větu pro pohyb tuhého tělesa (rotaci uvažujte pouze vzhledem jedné ose). +Konigova věta mluví o kinetické energii tělesa, které je jak pohybu přímočarém, tak v pohybu rotačním. + +$$ +T = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 +$$ + +$m$ je celkové hmotnost tělesa, $v$ je rychlost těžiště, $I$ je moment setrvačnosti, $\omega$ je úhlová rychlost +### Napište Steinerovu větu a doplňte vhodným obrázkem. + +$$ +I_{celk} = I_{T} + md^2 +$$ +kde $I_{celk}$ je celkový moment setrvačnosti tělesa, $I_{T}$ je moment setrvačnosti tělese osou, která prochází těžištěm, $d$ je vzdálenost od těžiště +## STR +### Uveďte vztahy Lorentzovy transformace. +$$ +x' = \gamma(x - Vt) +$$ +$$ +t' = \gamma\left( t - \frac{Vx}{c^2} \right) +$$ +$$ +y = y' +$$ +$$ +\gamma = \sqrt{ \frac{1}{1-\frac{v^2}{c^2}} } +$$ + +Kde S je klidová inerciální soustava. A S' je pohybující se inerciální soustava. +### Napište vztah pro Lorentzův faktor a nakreslete graf jeho vývoje jako funkci vzájemné rychlosti dvou inerciálních soustav. +$$ +\gamma = \frac{1}{\sqrt{ 1 - \frac{v^2}{c^2} }} +$$ +Pro v = 0 graf začíná na hodnotě 1. V hodnotě v = c se jmenovatel rovná nule a graf se blíží zleva k nekonečnu. +### Slovně popište jevy kontrakce délek a dilatace času. Doplňte příslušnými matematickými vztahy. + +Pro pozorovatele se zdá, že pohybující se objekt má menší rozměry ve směru osy pohybu. Tomu se říká kontrakce délek. +$$ +L = \frac{L_{0}}{\gamma} +$$ + +Pozorovateli se zdá, že pohybující se události utíkají pomaleji. Tedy na zemi uběhne víc času, než v pozorované pohybující se soustavě. + +$$ +t = \gamma t_{0} +$$ + + +## Mechanika tekutin +### Napište Gaussův-Ostrogradského teorém a vztah pro výpočet divergence v kartézských souřadnicích. +**Gauss-Ostrgradsky:** +$$ +\iiint_{V} (\nabla \cdot \vec{F})\space dV = \iint_{\partial V} \vec{F}\cdot d\vec{S} +$$ +**Divergence**: +$$ +div(\vec{F}) = \nabla \cdot \vec{F} = \frac{\partial\vec{F_{x}}}{\partial x} + \frac{\partial\vec{F_{y}}}{\partial y} + \frac{\partial\vec{F_{z}}}{\partial z} +$$ +### Uveďte tvar Bernoulliho rovnice pro (kvazi)-jednodimenzionální stacionární proudění nestlačitelné tekutiny +$$ +\frac{1}{2}\rho v^2 + \rho gh + p = konst. +$$ +kde $\rho$ je hustota kapaliny (nestlačitelná - nemění se), $v$ je rychlost prouděnií, $g$ je tíhové zrychlení, $h$ je výška, $p$ je tlak + +## Elekto peklo + \ No newline at end of file diff --git a/Pohyb po kružnici 2.md b/Pohyb po kružnici 2.md new file mode 100644 index 0000000..a96b78a --- /dev/null +++ b/Pohyb po kružnici 2.md @@ -0,0 +1,24 @@ +# Základ +**Pohyb po kružnici je takový pohyb, že vzdálenost od nějakého bodu je konstantní.** +Neplatí, že $r = konst$, protože směr vektoru se mění, ale platí $||r_{v}|| = \sqrt{} = r = konst$ +(kde $r_{v}$ je vektor a $r$ je číselný poloměr). + +Z toho také platí $r^2 = konst = $. + +Rychlost je **derivace vektoru polohy podle času**, tedy: $v = \frac{dr}{dt}$. +... + +Teď víme, že **rychlost a polohový vektor jsou kolmé.** + +## Zavádíme úhel +Úhel můžeme definovat jako +$$ +\varphi = \frac{s}{r} +$$, kde $r$ je obvodová délka a $r$ je poloměr. Celý kruh má úhel rovný $2\pi$. Z toho zavádíme radiány, kde do jednoho obvodu kruhu se vejde $2\pi$ radiánů. + + +# Dostředivé zrychlení + +# Ta poslední věc + +# Shrnutí: diff --git a/Speciální teorie relativity.md b/Speciální teorie relativity.md new file mode 100644 index 0000000..32c5a75 --- /dev/null +++ b/Speciální teorie relativity.md @@ -0,0 +1,47 @@ +Hlavní téma v teorii relativity je... nečekaně relativity. SSR mluví o tom, že každý pozorovatel pohybující se odlišnou rychlostí v prostoru zažívá čas a prostor jinak. Specificky dochází k jevům jako **dilatace času** (když pozorujume druhou soustavu pohybující se blízko rychlosti světla, vidíme, že pro ní utíká čas pomaleji) a **kontrakce délek** (když pozorujume druhou soustavu pohybující se blízko rychlosti světla, vidíme, že je smrštěná). + +Taky se to dá představit takto. Řekněme, že vystřelíme foton (neboli světlo) a kulku rychlostí jedné poloviny rychlosti světla. Kdybychom byly v Newtonovské fyzice, stalo by se to, že kulka by zažívala rychlost světla jako poloviční, než my ze země. To se ale stát nemůže, na rychlosti světla jsou založeny počty třeba v elektromagnetismu, které by vycházely jinak a my víme, že fyzikální zákony se nemění v závislosti na soustavě (a jak rychle se pohybuje vůči jiným soustavám). Jediné možné řešení je teda to, že rychlost toho fotonu je stejná jak vůči zemi, tak vůči kulce. To nám ale nedává smysl, jak může být rychlost fotonu stejná jak pro zemi tak pro kulku, která se vůči kulce pohybuje? +Einstein (Lorentz) vyřešil ten problém asi takto. Rychlost je změna délky za změnu času $\frac{dx}{dt}$. Pokud by řekněme kulka zažívala čas vůči zemi o něco pomalejší rychlostí a také by se kulka vůči zemi smrštila (a stejně tak její nástroje na měření délky), tak by mohlo naměřit stejnou rychlost fotonu. +Je dobré asi ale uvědomit, že změna času ani délky není pro žádnou ze soustav opravdový, každá soustava zažívá svoji realitu jako "normální", akorát si všímá změny vůči druhé soustavě. + +# Jak na pohybové úlohy +Pro výpočet problému blízky rychlosti světla máme na pomoc **lorentzův faktor** a 2 vzorce. +Lorentzův faktor:$$ +\gamma = \frac{1}{\sqrt{ 1-\frac{v^2}{c^2} }} +$$ +Dále máme 2 vzorce pro přepočet polohy a času soustavy S a S': +$$ +x = \gamma(x' - Vt') +$$ +$$ +t = \gamma\left( t' - \frac{Vx'}{c^2} \right) +$$ + +## Speciální případy (dilatace času a kontrakce délek) +Řekněme, že chceme zjistit utíkání času hmotného bodu, co se od nás pohybuje. V tom případě využijeme vzorec: +$$ +t = \gamma\left( t' - \frac{Vx'}{c^2} \right) +$$ protože $x' = 0$, vzorec se zjednoduší +na: +$$ +t = \gamma t' +$$ Čas t tedy zažívá za svůj čas $\gamma$ krát víc času, než čas $t'$. My vidíme hmotný bod, jak je zpomalený (jestli tedy vůbec můžeme vnímat čas hmotného bodu.). Hmotný bod naopak vidí, jak jsme my zrychlení. + +Pokud chceme zjistit pouze, jak moc se objekt zdánlivě smrštil, tak můžeme počítat dle vzorce: +$$ + L = \frac{L'}{\gamma} +$$ +## Hlubší pochopení vzorců: +Pro vzorec +$$ +t = \gamma\left( t' - \frac{Vx'}{c^2} \right) +$$ +jsme si řekli jak spočítat jeho změnu času ze země, ale realita je ještě kapánek složitější. Tento vzorec ještě obsahuje člen $\frac{Vx'}{v^2}$, který říká, že pojem času se mění v závislosti na poloze ve druhé soustavě. + +To si můžeme představit asi takto: +Řekněme, že vyšleme kilometr dlouhou tyč ze země na Saturn blízko rychlosti světla. To že se tyč a země neshodnou za jak dlouho tam doletí, již očekáváme. Nyní si ale představme, že tak kilometr dlouhá tyč má na každém konci maják a oba dva ty majáky blikají současně. Z pohledu země vzhledem k vzorci $-\frac{Vx'}{c^2}$ majáky neblikají ve stejný moment, ale vzdálenější maják blikne malinko dřív, než druhý. To znamená, že vzdálenější události nastávající pro pozorovatele z druhé soustavy dřív, i když jsou v soustavě S' ve stejnou dobu. + + +# Odvození Lorentzova faktoru a vzorců + +# Myšlenky \ No newline at end of file diff --git a/Srážka těles.md b/Srážka těles.md new file mode 100644 index 0000000..a793b64 --- /dev/null +++ b/Srážka těles.md @@ -0,0 +1,39 @@ +**Tento dokument je pouze přepis Bednaříkových skript dle mého omezeného chápáni, pokud. Původní zdroj bude velice pravděpodobně lepší zdroj informací na toto téma.** + +Srážka těles je (docela zjevně) jev, kdy se 2 tělesa přiblíží na tolik, že spolu dokáží vzájemně interagovat. Musíme si ale uvědomit, že srážka není nutně střet dvou tuhých objektů, může třeba dojít, ke střetu atomů, které se nedotknou pevnými obaly, ale navzájem se odpudí svými silami. Je potřeba ten termín vnímat i trošku abstraktnějším pojetí. + +Srážky rozdělujeme do jejich typů: +- Pokud při srážce netrácíme energii deformaci nebo teplem a tedy platí kompletně zákon zachování mechanické energii, jde o **srážku (dokonale) pružnou**. +- Pokud tomu tak neplatí, jde o **srážku nepružnou.** -> Pokud se částice po srážce spojí do jedné částice, jde o srážku **dokonale nepružnou**, pokud se částice odrazí částečně, jde o srážku **nedokonale pružnou**. + +Typicky budeme počítat dokonale pružné a nedokonale pružné příklady (zatím hádám). +Dále pak srážky můžou mít tělesa na různých vektorech rychlosti, to pak je **srážka šikmá**, pokud jsou vektory obou částic na jedné přímce **před i po srážce** jde o srážku **přímou (čelní)**. + +Celé rázy těles jsou o zákonech zachování. +**Vždy bude platit zákon zachování hybnosti.** Tedy součet hybností těles před a po srážce bude stejný. V případě pružné srážky bude platit **zákon zachování mechanické energii (typicky kinetické)** + +# Dokonale pružná čelní srážka +V této situaci se pohybujeme po přímce. Platí zákon zachování hybnosti a protože to je dokonale pružná srážka, platí zákon zachování kinetické energie. + +V této situaci máme tyto veličiny:$v_{1}, v_{2}, u_{1}, u_{2}, m_{1}, m_{2}$. +Kde $v_{1}, u_{1}, m_{1}$ je původní rychlost tělesa 1, jeho rychlost po srážce a jeho hmotnost. Ostatní veličiny analogicky. + +Správně bychom měli pracovat vektorově, kdy $v_{1} = |v_{1}|e$, $v_{2} = |v_{2}|e\dots$ kde $e$ je směrový vektor rychlostí, který bude vždy stejný a nevektorové rychlosti budou pouze jeho násobky. Protože ale pracujeme na přímce, můžeme si to zjednodušit a o směrech nepřemýšlet. + +**Platí zákon zachování hybnosti (ZZH):** +$$ +p_{1} + p_{2} = m_{1}v_{1} + m_{2}v_{2} = m_{1}u_{1} + m_{2}u_{2} = p'_{1} + p'_{2} = P +$$ +tedy hybnosti před srážkou se rovnají hybnosti po srážce, kde hybnost počítáme jako $mv$. $P$ je celková hybnost soustavy. + +**Z pružnosti platí i zákon zachování energie (ZZE):** +$$ +T_{1} + T_{2} = \frac{1}{2}m_{1}v_{1}^2 + \frac{1}{2}m_{2}v_{2}^2 = \frac{1}{2}m_{1}u_{1}^2 + \frac{1}{2}m_{2}u_{2}^2 = T'_{1} + T'_{2} = E_{s} +$$ +Kde $T = \frac{1}{2}mv^2$ je značení pro kinetickou energii a $E_{S}$ je celková kinetické energie soustavy. + +**Nyní máme 2 rovnice a několik neznámých.** To co nevíme umíme obvykle z úprav dopočítat. + +Zajímavé je asi teď jenom pozorování chování takové soustavy. +- dodělat kam se pohybují tělesa dle jejich vzájemných hmotností a hybností +# Dokonale nepružná čelní srážka \ No newline at end of file diff --git a/Zákony zachování.md b/Zákony zachování.md new file mode 100644 index 0000000..72b0846 --- /dev/null +++ b/Zákony zachování.md @@ -0,0 +1,40 @@ +U zákonů zachování máme 2 typy. Zákony co platí vždy (zatím nebylo dokázáno, že je něco porušuje). Pak máme zákony zachování, které ale platí pouze v některých situacích. + +Důležité je dodat, že tyto zákony platí v **uzavřených systémech**. Musíme se koukat na systém jako celek a nesmíme připustit únik nějaké vlastnosti. + +Hruška myslím zmínil: + +> [!NOTE] Zákony zachování +> - **zákon zachování energie** +> - **zákon zachování hybnosti** +> - **zákon zachování momentu hybnosti** +> - **zákon zachování elektrického náboje** - rovnice kontinuity (proud) +> - **zákon zachování mechanické energie** - T + U = konst +> - **zákon zachování klidové energie hmotnosti** - rovnice kontinuity kapalin + +# Exaktní zákony +platí vždy + +- **zákon zachování energie** +- **zákon zachování hybnosti** +- **zákon zachování momentu hybnosti** +- **zákon zachování elektrického náboje** + +- zákon zachování barevného náboje +- zákon zachování slabého isospinu +- zachování rozdílu mezi baryonovým a leptonovým číslem + +# Zákony lokální +Následující zákony platí za některých podmínek, ale ne vždy. Můžou je narušit relativistické rychlosti (tedy platí v Newtonovském modelu), nebo třeba nějaká specifická interakce. + +- **zákon zachování mechanické energie** - platí bez disipativních sil jako tření +- **zákon zachování klidové energie hmotnosti** + platí v nerelativistickách rychlostech + vyrušením antičástic (které obě mají kladnou hmotnost) obě zaniknou (a s nimi i jejich hmotnost), ale vznikne ekvivalentní energie. Proto se dá říct, že to je vlastně způsob jak interpretovat zákon zachování energie. - dále také platí, že teplota, či jiné energie navyšují hmotnost částic (což ale lze obvykle zanedbat) - **v mechanice tekutin z toho přímo plynou rovnice kontinuity** + +Pak máme kvantové vlastnosti (či vlastnosti velmi malých částic), jako jsou baryonová a leptonová čísla, barva, divnost, etc. obvykle narušené nějakou interakcí, jako je slabá jaderná např. + +Pak máme symetrie CPT - obrácený náboj, čas a prostorové převrácení (pravo/levo točivost), která je částěčne narušena také slabou jadernou silou. +pravo/levo točivost - narušena slabou jadernou, časová je některými částicemi a kvantovými jevy + +Mluvím o té CPT trošku nepřesně - samotné třeba jsou narušené, ale to neznamená, že jejich kombinace neplatí, ale i jejich kombinace byly prokázány, že ne vždy platí (jako třeba CP). **Myslím, že je zatím obecně předpokládáno, že CPT (všechny) dohromady platí.** + diff --git 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