OII na Teorica

Astana, Kazakhstan

45th International Physics Olympiad Astana, Kazakhstan Theoretical Competition, Tuesday, 15 July 2014 Lista delle costanti fondamentali Velocità della luce nel vuoto c=299792458m $\cdots$ $-1$ Costante di gravitazione universale $G=6.67\cdot 10$ $-11m$ $3\text{cdotkg}$ $-1\cdots$ $-2$ Accelerazione di gravità $g=9.81m\cdots$ $-2$ Numero di Avogadro N $A=6.02\cdot 10$ 23mol $-1$ Costante universale dei gas R=8.31J $\text{cdotK}$ $-1\text{cdotmol}$ $-1$ Costante di Boltzmann $k=1.38\cdot 10$ $-23J\text{cdotK}$ $-1$ Carica elementare $e=1.60\cdot 10$ $-19C$ Massa a riposo dell’elettrone $me=9.11\cdot 10$ $-31kg$ Massa a riposo del protone m $p=1.67\cdot 10$ $-27kg$ Costante di Planck ridotta $ћ¿1.05\cdot 10$ $-34$ J $\cdots$ Costante dielettrica del vuoto $ϵ0=8.85\cdot 10$ $-12F\text{cdotm}$ $-1$ Permeabilità magnetica del vuoto $\mu 0=1.26\cdot 10$ $-6$ H $\text{cdotm}$ $-1$ Formule matematiche utili (1+x ) $\alpha \approx 1+\text{alphax}+$ 1 2 $\alpha (\alpha -1$) x 2, dove ¿ $x\vee \ll 1$ e $\alpha$ è una costante arbitraria sin $x\text{approxx}-x$ 3 3 , dove ¿ $x\vee \ll 1$ cos x $\approx 1-1$ 2 x 2, dove ¿ $x\vee \ll 1$ $\text{intx}$ ndx= x n+1 n+1+C, $n\ne -1$, con C costante arbitraria $\text{intdx}$ $x-a=log|x-a|+C$, con C costante arbitraria cosh x=e x+e $-x$ 2 sinh x= e $x-e$ $-x$ 2 cosh $2x-sinh$ 2x=1 tanh x= sinh x cosh x (e x)‘=e x (log x ) ‘=1 x (x n) ‘=n x $n-1$

ut (x (t ))=ux(x (t)) xt(t ) (u (x )v (x )) ‘=u (x ) ’ v (x )+u (x )v(x)’ ( u(x) v(x)) ’ =u (x ) ’ v $(x)-u$ (x )v(x)’ v(x) 2

Topic: Newtonian Mechanics, Oscillations & Waves, Thermodynamics Metodi: Approximation & Series Expansion, Calculus-Integration Competenze: Mathematical Modeling, Unit Conversion Objects: — Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

Astana, Kazakhstan

The 45th International Physics Olympiad is held in London. The following is the list of official languages of the Republic of Kazakhstan: Theoretical Competition, Tuesday, 15 July 2014 List of fundamental constants Speed of light in vacuum c=299792458m $\cdots$ $-1$ The universal gravitational constant is the $G=6.67\cdot 10$ $-11m$ $3\text{cdotkg}$ $-1\cdots$ $-2$ Acceleration by gravity $g=9.81m\cdots$ $-2$ Avogadro number N $A=6.02\cdot 10$ 23 mol $-1$ Universal gas constant R=8.31J $\text{cdotK}$ $-1\text{cdotmol}$ $-1$ The Boltzmann constant is the constant $k=1.38\cdot 10$ $-23J\text{cdotK}$ $-1$ Basic load $e=1.60\cdot 10$ $-19C$ The mass at rest of the electron $me=9.11\cdot 10$ $-31kg$ The mass at rest of the proton m $p=1.67\cdot 10$ $-27kg$ Reduced Planck constant $ћ¿1.05\cdot 10$ $-34$ J $\cdots$ The value of the product shall be determined by the following formula: $ϵ0=8.85\cdot 10$ $-12F\text{cdotm}$ $-1$ The magnetic permeability of the vacuum $\mu 0=1.26\cdot 10$ $-6$ H $\text{cdotm}$ $-1$ Useful mathematical formulas (1+x ) $\alpha \approx 1+\text{alphax}+$ 1 2 $\alpha (\alpha -1$) x 2, where ¿ $x\vee \ll 1$ and $\alpha$ is an arbitrary constant sin $x\text{approxx}-x$ 3 3 , dove ¿ $x\vee \ll 1$ where x $\approx 1-1$ 2 x 2, where $x\vee \ll 1$ $\text{intx}$ where x is equal to x n+1 n+1+C, $n\ne -1$, with C as an arbitrary constant $\text{intdx}$ $x-a=log|x-a|+C$, with C arbitrary constant cosh x is equal to e x+e $-x$ 2 X = e $x-e$ $-x$ 2 What ? $2x-sinh$ 2x=1 The value of the product shall be the sum of the values of the product ♪ I ‘m not gonna let you down ♪ (e x)‘=e x (log x) ‘=1 x (x n) ‘=n x $n-1$

ut (x (t ))=ux(x (t)) xt(t ) (u (x )v (x )) ‘=u (x ) ’ v (x )+u (x )v(x)’ ( u(x) v(x)) ’ =u (x ) ’ v $(x)-u$ (x )v(x)’ v(x) 2

Topic: Newtonian Mechanics, Oscillations & Waves, Thermodynamics Metodi: Approximation & Series Expansion, Calculus-Integration Competenze: Mathematical Modeling, Unit Conversion Objects: — Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)