OII 2019 Nazionale Teorica

P1 Pietre miliari 1: … dedicato a Leonardo

1. Oggetti misurati, in unità di $r$: indicare nome dell’oggetto e misure Unità di misura (in mm): $r=$ Oggetto 1: $R_1=$ $h_1=$ 2: …

2. Forza minima $F_{\text{min}}=$ $F_{\text{min}}=$ [Espressione] [Valore numerico]

3. Ancora utile? SI NO Spiegare:

4. Componente con maggiore energia: (indicare quale) $K=$ [Espressione]

Topic: Newtonian Mechanics, Rotational Dynamics, Conservation of Energy Metodi: Free-Body Diagram, Torque & Angular Momentum Analysis, Energy Conservation Method Competenze: Measurement & Instrumentation, Mathematical Modeling, Physical Reasoning Objects: Gear, String, Cylinder Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

P1 One billionth … Dedicated to Leonardo

1. Measured objects, in units of $r$: indicate the name of the object and measurements Measurement unit (in mm): $r=$ Subject one: $R_1=$ $h_1=$ 2: …

2. Minimum force $F_{\text{min}}=$ $F_{\text{min}}=$ [Expression] [Number value]

3. Still useful? SI NO Explain:

4. High energy component: (indicate which) $K=$ [Expression]

Topic: Newtonian Mechanics, Rotational Dynamics, Conservation of Energy Metodi: Free-Body Diagram, Torque & Angular Momentum Analysis, Energy Conservation Method Competenze: Measurement & Instrumentation, Mathematical Modeling, Physical Reasoning Objects: Gear, String, Cylinder Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

P2 Pietre miliari 2: Il chilogrammo di Planck

1. Valore di $\eta$ $\eta =$ $\eta =$ [Espressione] [Valore numerico]

2. Equivalenze dimensionali

3. Azione (ridotta) $A=$

4. Livello di azione macroscopico $n=$ $n=$ [Espressione] [Valore numerico]

5. Livello di azione microscopico $n=$ $n=$ [Espressione] [Valore numerico]

6. Livelli energetici $E_n=$ [Espressione]

7. Energia del fotone $\Delta E=$ $\Delta E=$ [Espressione] [Valore numerico]

8. Relazione tra $\lambda$ e $p$

Topic: Modern-Quantum Physics, Kinetic Theory, Oscillations & Waves Metodi: Dimensional Analysis, Bohr Model & Quantization, de Broglie Relation Competenze: Mathematical Modeling, Physical Reasoning, Unit Conversion Objects: Ball, Photon Fonte: Testo (PDF) — p.2 Soluzione: Soluzioni (PDF)

P2 Milestones 2: The Planck kilogram

1. The value of $\eta$ $\eta =$ $\eta =$ [Expression] [Number value]

2. Dimensional equivalencies

3. Action (reduced) $A=$

4. The following shall be added to the list of the following: $n=$ $n=$ [Expression] [Number value]

5. The level of microscopic action $n=$ $n=$ [Expression] [Number value]

6. Energy levels $E_n=$ [Expression]

7. Energy of the photon $\Delta E=$ $\Delta E=$ [Expression] [Number value]

8. The ratio of $\lambda$ to $p$

Topic: Modern-Quantum Physics, Kinetic Theory, Oscillations & Waves Metodi: Dimensional Analysis, Bohr Model & Quantization, de Broglie Relation Competenze: Mathematical Modeling, Physical Reasoning, Unit Conversion Objects: Ball, Photon Fonte: Testo (PDF) — p.2 Soluzione: Soluzioni (PDF)

P3 Là, dove finisce il solenoide

1. Relazioni di simmetria Componenti di $\vec{B}$ parallele all’asse del solenoide: Componenti di $\vec{B}$ perpendicolari all’asse del solenoide:

2. Direzione di $\vec{B}$ sull’asse

3. Relazione tra i campi $\vec{B}$ in $P$ e $P^{\prime }$ ${\vec{B}}^{\prime }(P^{\prime })=$

4. Campo in $P$ $\vec{B}(P)=$

5. Flusso magnetico sul bordo ${\Phi }_{\sigma }$: dimostrazione ${\Phi }_{\sigma }=$

6. Flusso sulla superficie laterale ${\Phi }_{\text{S.Lat}}=$

7. Distanza tra linee di campo $\overline{S_1{S}_2}=$

8. Schema delle linee di campo

Topic: Magnetism, Electromagnetism Metodi: Symmetry Argument, Gauss’s Law, Ampère’s Law Competenze: Diagrammatic Reasoning, Mathematical Modeling, Physical Reasoning Objects: Solenoid Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

P3 There, where the solenoid ends

1. The following shall be added: Components of $\vec{B}$ parallel to the solenoid axis: Components of $\vec{B}$ perpendicular to the axis of the solenoid:

2. Direction of $\vec{B}$ on the axis

3. Relation between fields $\vec{B}$ in $P$ and $P^{\prime }$ ${\vec{B}}^{\prime }(P^{\prime })=$

4. Field in $P$ $\vec{B}(P)=$

5. Magnetic flow on the edge ${\Phi }_{\sigma }$: demonstration ${\Phi }_{\sigma }=$

6. Flow on the lateral surface ${\Phi }_{\text{S.Lat}}=$

7. Distance between field lines $\overline{S_1{S}_2}=$

8. Fieldline scheme

Topic: Magnetism, Electromagnetism Metodi: Symmetry Argument, Gauss’s Law, Ampère’s Law Competenze: Diagrammatic Reasoning, Mathematical Modeling, Physical Reasoning Objects: Solenoid Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

P4 Luce dal ghiaccio: un alone per il Sole

1. minimo valore di $n$ $n_{\text{min}}=$ $n_{\text{min}}=$ [Espressione] [Valore numerico]

2. Frazione $\eta$ di luce che emerge dalla faccia p in funzione di $i$ $\eta (i)=$

3. Minima incidenza per emergere dalla faccia c $i_{\text{min}}=$ $i_{\text{min}}=$ $\delta =$ $\delta =$ [Espressione] [Valore numerico]

4. Massima incidenza per emergere dalla faccia per c $i_{\text{max}}=$ $i_{\text{max}}=$ $r^{\prime }=$ $r^{\prime }=$ $\delta =$ $\delta =$ [Espressione] [Valore numerico]

5. Minima o massima deviazione $i_0=$ $i_0=$ ${\delta }_0=$ ${\delta }_0=$ [Espressione] [Valore numerico] ${\delta }_0$ è… un minimo un massimo

6. Polinomio approssimante $a=$ $a=$ [Espressione] [Valore numerico] Confronto 1: ${ϵ}_1=$ Confronto 2: ${ϵ}_2=$

7. Rapporto di luminosità $\rho$ $\rho =$ $\rho =$ [Espressione] [Valore numerico]

8. Raggio dell’alone Spiegazione:

Topic: Geometric Optics, Wave Optics Metodi: Snell’s Law, Ray Tracing, Approximation & Series Expansion Competenze: Mathematical Modeling, Physical Reasoning, Estimation & Approximation Objects: Prism Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

P4 Light from the ice: a solo for the sun

1. minimo valore di $n$ $n_{\text{min}}=$ $n_{\text{min}}=$ [Expression] [Number value]

2. Fraction of light $\eta$ coming out of face p as a function of $i$ $\eta (i)=$

3. Minimum incidence of emergence from face c $i_{\text{min}}=$ $i_{\text{min}}=$ $\delta =$ $\delta =$ [Expression] [Number value]

4. Maximum incidence of emergence from the face to c $i_{\text{max}}=$ $i_{\text{max}}=$ $r^{\prime }=$ $r^{\prime }=$ $\delta =$ $\delta =$ [Expression] [Number value]

5. Minimum or maximum deviation $i_0=$ $i_0=$ ${\delta }_0=$ ${\delta }_0=$ [Expression] [Number value] ${\delta }_0$ è… a minimum a maximum of

6. Polynomial approximation $a=$ $a=$ [Expression] [Number value] Comparison one: ${ϵ}_1=$ Comparison two: ${ϵ}_2=$

7. The lighting ratio $\rho$ $\rho =$ $\rho =$ [Expression] [Number value]

8. The joy of the harvest Explanation:

Topic: Geometric Optics, Wave Optics Metodi: Snell’s Law, Ray Tracing, Approximation & Series Expansion Competenze: Mathematical Modeling, Physical Reasoning, Estimation & Approximation Objects: Prism Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

P1 Pietre miliari 1: … dedicato a Leonardo

1. Oggetti misurati $N=39$; $r=9.31$ mm; $b=22.15$ mm $=\beta r$ $2.26\le 2.379\le 2.50$ Ruota dent. $R_1=33.10$ mm $={\alpha }_{1}r$; $3.38\le 3.555\le 3.73$ $h_1=R_1/12=5.52$ mm $={\gamma }_{1}r$ $0.563\le 0.5925\le 0.622$ Vite s. fine $R_2=6.42$ mm $={\alpha }_{2}r$; $0.655\le 0.6896\le 0.724$ $h_2=28.76$ mm $={\gamma }_{2}r$ $2.93\le 3.089\le 3.24$ Manovella $R_3=2.38$ mm $={\alpha }_{3}r$; $0.243\le 0.2556\le 0.268$ $h_3=24.42$ mm $={\gamma }_{3}r$ $2.49\le 2.623\le 2.75$

2. Forza minima $F_{\text{min}}=\frac{Mgr}{Nb}=0.01078Mg<Mg$ $\Rightarrow$ Utile ! $0.0102\le 0.0108\le 0.0113$ $[Mg]$

3. Ancora utile? $F=\frac{4Mgr}{Nb}=0.0431Mg<Mg$ $\Rightarrow$ Ancora utile ! $0.0410\le 0.0431\le 0.0453$ $[Mg]$

4. Componente con maggiore energia Ruota dent. $K_1=95$ $\left[\frac{\pi }{4}\rho v^2{r}^3\right]$ $70\le 95\le 120$ Vite s. fine $K_2=1060$ $\left[\frac{\pi }{4}\rho v^2{r}^3\right]$ $790\le 1060\le 1330$ Manovella $K_3=2950$ $\left[\frac{\pi }{4}\rho v^2{r}^3\right]$ $2200\le 2950\le 3700$

Topic: Newtonian Mechanics, Rotational Dynamics, Conservation of Energy Metodi: Free-Body Diagram, Torque & Angular Momentum Analysis, Energy Conservation Method Competenze: Measurement & Instrumentation, Mathematical Modeling, Physical Reasoning Objects: Gear, String, Cylinder Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)

P1 One billionth … Dedicated to Leonardo

1. Measured objects $N=39$; $r=9.31$ mm; $b=22.15$ mm $=\beta r$ $2.26\le 2.379\le 2.50$ It’s a wheel. $R_1=33.10$ mm $={\alpha }_{1}r$; $3.38\le 3.555\le 3.73$ $h_1=R_1/12=5.52$ mm $={\gamma }_{1}r$ $0.563\le 0.5925\le 0.622$

• I ‘m not . fine $R_2=6.42$ mm $={\alpha }_{2}r$; $0.655\le 0.6896\le 0.724$ $h_2=28.76$ mm $={\gamma }_{2}r$ $2.93\le 3.089\le 3.24$ Manovella $R_3=2.38$ mm $={\alpha }_{3}r$; $0.243\le 0.2556\le 0.268$ $h_3=24.42$ mm $={\gamma }_{3}r$ $2.49\le 2.623\le 2.75$

2. Minimum force $F_{\text{min}}=\frac{Mgr}{Nb}=0.01078Mg<Mg$ $\Rightarrow$ It ‘s useful . $0.0102\le 0.0108\le 0.0113$ $[Mg]$

3. Still useful? $F=\frac{4Mgr}{Nb}=0.0431Mg<Mg$ $\Rightarrow$ Still useful ! $0.0410\le 0.0431\le 0.0453$ $[Mg]$

4. High energy component It’s a wheel. $K_1=95$ $\left[\frac{\pi }{4}\rho v^2{r}^3\right]$ $70\le 95\le 120$

• I ‘m not . the end of $K_2=1060$ $\left[\frac{\pi }{4}\rho v^2{r}^3\right]$ $790\le 1060\le 1330$ Manovella $K_3=2950$ $\left[\frac{\pi }{4}\rho v^2{r}^3\right]$ $2200\le 2950\le 3700$

Topic: Newtonian Mechanics, Rotational Dynamics, Conservation of Energy Metodi: Free-Body Diagram, Torque & Angular Momentum Analysis, Energy Conservation Method Competenze: Measurement & Instrumentation, Mathematical Modeling, Physical Reasoning Objects: Gear, String, Cylinder Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)

P2 Pietre miliari 2: Il chilogrammo di Planck

1. Valore di $\eta$ $\eta =\frac{c^2}{h\Delta \nu }=1.47552139973527\dots$

2. Equivalenze dimensionali $\dots =M{L}^2{T}^{-1}$

3. Azione (ridotta) $A=4\ell mv$

4. Livello macroscopico $n=\frac{16m{\ell }^2}{hT}$ $\Rightarrow$ $2.99\le 3.02\le 3.04$ $\times 10^{32}$

5. Livello microscopico $n=\frac{4m\ell v}{h}=3$

6. Livelli energetici $E_n=\frac{h^2}{32{\ell }^{2}m}{n}^2$

7. Energia del fotone $\Delta E=\frac{3h^2}{32{\ell }^{2}m}$ $\Rightarrow$ $2.00\le 2.034\le 2.05$ $[\text{J}\times 10^{-23}]$ $\Rightarrow$ $1.26\le 1.269\le 1.28$ $[\text{eV}\times 10^{-4}]$

8. Relazione di De Broglie $\lambda =h/p$

Topic: Modern-Quantum Physics, Kinetic Theory, Oscillations & Waves Metodi: Dimensional Analysis, Bohr Model & Quantization, de Broglie Relation Competenze: Mathematical Modeling, Physical Reasoning, Unit Conversion Objects: Ball, Photon Fonte: Testo (PDF) — p.6 Soluzione: Soluzioni (PDF)

P2 Milestones 2: The Planck kilogram

1. The value of $\eta$ $\eta =\frac{c^2}{h\Delta \nu }=1.47552139973527\dots$

2. Dimensional equivalencies $\dots =M{L}^2{T}^{-1}$

3. Action (reduced) $A=4\ell mv$

4. The macroscopic level $n=\frac{16m{\ell }^2}{hT}$ $\Rightarrow$ $2.99\le 3.02\le 3.04$ $\times 10^{32}$

5. Microscopic level $n=\frac{4m\ell v}{h}=3$

6. Energy levels $E_n=\frac{h^2}{32{\ell }^{2}m}{n}^2$

7. Energy of the photon $\Delta E=\frac{3h^2}{32{\ell }^{2}m}$ $\Rightarrow$ $2.00\le 2.034\le 2.05$ $[\text{J}\times 10^{-23}]$ $\Rightarrow$ $1.26\le 1.269\le 1.28$ $[\text{eV}\times 10^{-4}]$

8. The report by Mr De Broglie $\lambda =h/p$

Topic: Modern-Quantum Physics, Kinetic Theory, Oscillations & Waves Metodi: Dimensional Analysis, Bohr Model & Quantization, de Broglie Relation Competenze: Mathematical Modeling, Physical Reasoning, Unit Conversion Objects: Ball, Photon Fonte: Testo (PDF) — p.6 Soluzione: Soluzioni (PDF)

P3 Là, dove finisce il solenoide

1. Relazioni di simmetria $B_{\perp }^{\prime }=-B_{\perp }$ $B_{\parallel }^{\prime }=B_{\parallel }$

2. $\vec{B}$ sull’asse Il vettore $\vec{B}$ è diretto lungo l’asse del solenoide.

3. $\vec{B}$ al centro del bordo del solenoide ${\vec{B}}^{\prime }(P^{\prime })=\vec{B}(P)$

4. Campo al centro del bordo $\vec{B}(P)=\frac{1}{2}{\vec{B}}_0$

5. Flusso magnetico sul bordo ${\Phi }_{\sigma }=\frac{\pi }{2}{R}^2{B}_0$

6. Flusso sulla superficie laterale ${\Phi }_{\text{S.Lat}}=\frac{\pi }{2}{R}^2{B}_0$

7. Distanza tra linee di campo $\overline{S_1{S}_2}=\frac{2R}{\sqrt{2}}$

8. Schema delle linee di campo

Topic: Magnetism, Electromagnetism Metodi: Symmetry Argument, Gauss’s Law, Ampère’s Law Competenze: Diagrammatic Reasoning, Mathematical Modeling, Physical Reasoning Objects: Solenoid Fonte: Testo (PDF) — p.7 Soluzione: Soluzioni (PDF)

P3 There, where the solenoid ends

1. The following shall be added: $B_{\perp }^{\prime }=-B_{\perp }$ $B_{\parallel }^{\prime }=B_{\parallel }$

2. $\vec{B}$ on the axis The vector $\vec{B}$ is directed along the axis of the solenoid.

3. $\vec{B}$ at the centre of the edge of the solenoid ${\vec{B}}^{\prime }(P^{\prime })=\vec{B}(P)$

4. Field in the center of the edge $\vec{B}(P)=\frac{1}{2}{\vec{B}}_0$

5. Magnetic flow on the edge ${\Phi }_{\sigma }=\frac{\pi }{2}{R}^2{B}_0$

6. Flow on the lateral surface ${\Phi }_{\text{S.Lat}}=\frac{\pi }{2}{R}^2{B}_0$

7. Distance between field lines $\overline{S_1{S}_2}=\frac{2R}{\sqrt{2}}$

8. Fieldline scheme

Topic: Magnetism, Electromagnetism Metodi: Symmetry Argument, Gauss’s Law, Ampère’s Law Competenze: Diagrammatic Reasoning, Mathematical Modeling, Physical Reasoning Objects: Solenoid Fonte: Testo (PDF) — p.7 Soluzione: Soluzioni (PDF)

P4 Luce dal ghiaccio: un alone per il Sole

1. minimo valore di $n$ $n_{\text{min}}=\frac{1}{\mathrm{sen}60^{\circ }}$ $\Rightarrow$ $1.150\le 1.155\le 1.155$

2. Frazione $\eta$ di luce che emerge dalla faccia p in funzione di $i$ $\eta (i)=1-\frac{\sqrt{3}\mathrm{sen}i}{\sqrt{n^2-{\mathrm{sen}}^{2}i}}$

3. Minima incidenza per per emergere dalla faccia c $i_{\text{min}}=i_1=\mathrm{arcsen}[n\mathrm{sen}(60^{\circ }-\mathrm{arcsen}(1/n)]$ $\Rightarrow$ $13.28\le 13.47\le 13.64$ ${[}^{\circ }]$ $\delta =i_{\text{min}}+30^{\circ }$ $\Rightarrow$ $43.28\le 43.47\le 43.64$ ${[}^{\circ }]$

4. Massima incidenza per per emergere dalla faccia c $i_{\text{max}}=90^{\circ }$ $i_0=r_1^{\prime }=i_{\text{min}}$ $\Rightarrow$ $13.28\le 13.47\le 13.64$ ${[}^{\circ }]$ ${\delta }_1=\delta$ $\Rightarrow$ $43.28\le 43.47\le 43.64$ ${[}^{\circ }]$

5. Minima o massima deviazione $i_0=\mathrm{arcos}(n\mathrm{sen}30^{\circ })$ $\Rightarrow$ $40.82\le 40.92\le 41.02$ ${[}^{\circ }]$ ${\delta }_0=2i_0-60^{\circ }$ $\Rightarrow$ $21.64\le 21.84\le 22.04$ ${[}^{\circ }]$ ${\delta }_0$ è un minimo

6. Polinomio approssimante $a=\frac{y_0-b{x}_0-c}{x_0^2}=\frac{-b}{2x_0}$ $\Rightarrow$ $8.40\le 8.418\le 8.44$ $[10^{-3}{\text{gradi}}^{-1}]$ Confronto 1: ${ϵ}_1=0.7\%$ Confronto 2: ${ϵ}_2=0.4\%$

7. Rapporto di luminosità $\rho =\frac{\Delta i_1}{\Delta i_2^{\prime }+\Delta i_2^{\prime \prime }}$ $\Rightarrow$ $18\le 22\le$ …

8. Raggio dell’alone Spiegazione:

Topic: Geometric Optics, Wave Optics Metodi: Snell’s Law, Ray Tracing, Approximation & Series Expansion Competenze: Mathematical Modeling, Physical Reasoning, Experimental Data Analysis Objects: Prism Fonte: Testo (PDF) — p.8 Soluzione: Soluzioni (PDF)

P4 Light from the ice: a solo for the sun

1. minimo valore di $n$ $n_{\text{min}}=\frac{1}{\mathrm{sen}60^{\circ }}$ $\Rightarrow$ $1.150\le 1.155\le 1.155$

2. Fraction of light $\eta$ coming out of face p as a function of $i$ $\eta (i)=1-\frac{\sqrt{3}\mathrm{sen}i}{\sqrt{n^2-{\mathrm{sen}}^{2}i}}$

3. Minimum incidence of emergence from face c $i_{\text{min}}=i_1=\mathrm{arcsen}[n\mathrm{sen}(60^{\circ }-\mathrm{arcsen}(1/n)]$ $\Rightarrow$ $13.28\le 13.47\le 13.64$ ${[}^{\circ }]$ $\delta =i_{\text{min}}+30^{\circ }$ $\Rightarrow$ $43.28\le 43.47\le 43.64$ ${[}^{\circ }]$

4. Maximum incidence of emergence from face c $i_{\text{max}}=90^{\circ }$ $i_0=r_1^{\prime }=i_{\text{min}}$ $\Rightarrow$ $13.28\le 13.47\le 13.64$ ${[}^{\circ }]$ ${\delta }_1=\delta$ $\Rightarrow$ $43.28\le 43.47\le 43.64$ ${[}^{\circ }]$

5. Minimum or maximum deviation $i_0=\mathrm{arcos}(n\mathrm{sen}30^{\circ })$ $\Rightarrow$ $40.82\le 40.92\le 41.02$ ${[}^{\circ }]$ ${\delta }_0=2i_0-60^{\circ }$ $\Rightarrow$ $21.64\le 21.84\le 22.04$ ${[}^{\circ }]$ ${\delta }_0$ is a minimum

6. Polynomial approximation $a=\frac{y_0-b{x}_0-c}{x_0^2}=\frac{-b}{2x_0}$ $\Rightarrow$ $8.40\le 8.418\le 8.44$ $[10^{-3}{\text{gradi}}^{-1}]$ Comparison one: ${ϵ}_1=0.7\%$ Comparison two: ${ϵ}_2=0.4\%$

7. The luminous ratio $\rho =\frac{\Delta i_1}{\Delta i_2^{\prime }+\Delta i_2^{\prime \prime }}$ $\Rightarrow$ $18\le 22\le$ …

8. The joy of the harvest Explanation:

Topic: Geometric Optics, Wave Optics Metodi: Snell’s Law, Ray Tracing, Approximation & Series Expansion Competenze: Mathematical Modeling, Physical Reasoning, Experimental Data Analysis Objects: Prism Fonte: Testo (PDF) — p.8 Soluzione: Soluzioni (PDF)