OII 2026 Teorica

T1: Jumper (10 pts)

Two uniform rods, each of mass $m$ and length $\ell$, are connected at one end by a hinge equipped with a torsional spring. The spring exerts equal and opposite restoring torques $\tau$ on the rods about the hinge, $\tau =2k\theta ,$ where $2\theta$ is the angle between the rods (in radians), and $k$ ($k\gg mg\ell$) is the torsional spring constant.

The system is placed on a flat horizontal surface with the free ends of the rods resting on the floor and the hinge above them. The hinge is then pushed downward until the rods lie flat on the floor in opposite directions, corresponding to $\theta =\pi /2$. Friction between the rods and the floor is negligible.

Jumper jumps after the hinge is released. Determine the maximum height $h$ reached by the center of mass of the system to a precision better than $1\%$.

Topic: Newtonian Mechanics, Rotational Dynamics Metodi: Conservation of Energy, Torque & Angular Momentum Analysis Competenze: Physical Reasoning, Mathematical Modeling Objects: Rod (object) Fonte: Testo (PDF) Soluzione: Soluzioni (PDF)

T1: Jumper (10 pts)

Two uniform rods, each of mass $m$ and length $\ell$, are connected at one end by a hinge equipped with a torsional spring. The spring exerts equal and opposite restoring torques $\tau$ on the rods about the hinge, $\tau =2k\theta ,$ where $2\theta$ is the angle between the rods (in radians), and $k$ ($k\gg mg\ell$) is the torsional spring constant.

The system is placed on a flat horizontal surface with the free ends of the rods resting on the floor and the hinge above them. The hinge is then pushed downward until the rods lie flat on the floor in opposite directions, corresponding to $\theta =\pi /2$. Friction between the rods and the floor is negligible.

Jumper jumps after the hinge is released. Determine the maximum height $h$ reached by the center of mass of the system to a precision better than $1\%$.

Topic: Newtonian Mechanics, Rotational Dynamics Metodi: Conservation of Energy, Torque & Angular Momentum Analysis Competenze: Physical Reasoning, Mathematical Modeling Objects: Rod (object) Fonte: Testo (PDF) Soluzione: Soluzioni (PDF)

T2: Hysteresis (10 pts)

Consider a solenoid with $N\gg 1$ turns, radius $R$ and length $\ell \gg R$. The solenoid has a ferromagnetic core with hysteresis curve as shown in the picture. $M$ denotes the magnetization of the core and would be zero without core material. It is related to $H$, the magnetic field strength generated by the current in the solenoid coil, and $B$ by $B={\mu }_0(H+M)$. $M_0$ and $H_0$ are of the same order of magnitude. Assume that $M$ can only change if $|H|\ge H_0$.

An ideal capacitor with capacitance $C$ is now connected to the solenoid forming a closed circuit. Assume that all wires have negligible resistance.

a) (3.0 pts) Initially there is a current $I_i$ flowing through the circuit and the capacitor is uncharged. $I_i$ is large enough that $H(I_i)\gg H_0$. After one oscillation the current again reaches a maximum value. Determine the difference between $I_i$ and this value.

b) (2.3 pts) Find the maximum current that can be attained after many oscillations.

c) (4.7 pts) The current $I(t)$ exhibits two qualitatively distinct phases of behaviour, A and B. The system can remain in phase A indefinitely, whereas any single interval spent in phase B has bounded duration. Find the maximal possible duration of a phase-B interval, if $I_i$ is chosen optimally.

Topic: Magnetism, Electromagnetic Induction Metodi: Faraday’s Law of Induction, Differential Equations Competenze: Physical Reasoning, Mathematical Modeling Objects: Solenoid (object) Fonte: Testo (PDF) Soluzione: Soluzioni (PDF)

T2: Hysteresis (10 pts)

Consider a solenoid with $N\gg 1$ turns, radius $R$ and length $\ell \gg R$. The solenoid has a ferromagnetic core with hysteresis curve as shown in the picture. $M$ denotes the magnetization of the core and would be zero without core material. It is related to $H$, the magnetic field strength generated by the current in the solenoid coil, and $B$ by $B={\mu }_0(H+M)$. $M_0$ and $H_0$ are of the same order of magnitude. Assume that $M$ can only change if $|H|\ge H_0$.

An ideal capacitor with capacitance $C$ is now connected to the solenoid forming a closed circuit. Assume that all wires have negligible resistance.

a) (3.0 pts) Initially there is a current $I_i$ flowing through the circuit and the capacitor is uncharged. $I_i$ is large enough that $H(I_i)\gg H_0$. After one oscillation the current again reaches a maximum value. Determine the difference between $I_i$ and this value.

b) (2.3 pts) Find the maximum current that can be attained after many oscillations.

c) (4.7 pts) The current $I(t)$ exhibits two qualitatively distinct phases of behaviour, A and B. The system can remain in phase A indefinitely, whereas any single interval spent in phase B has bounded duration. Find the maximal possible duration of a phase-B interval, if $I_i$ is chosen optimally.

Topic: Magnetism, Electromagnetic Induction Metodi: Faraday’s Law of Induction, Differential Equations Competenze: Physical Reasoning, Mathematical Modeling Objects: Solenoid (object) Fonte: Testo (PDF) Soluzione: Soluzioni (PDF)

T3: Dry ice hockey (10 pts)

At pressure $p_0=100\mathrm{kPa}$, solid ${\mathrm{CO}}_2$ (dry ice) sublimates (goes from solid to gaseous state) at $T_s=-78.5{}^{\circ }\mathrm{C}$. Its saturated vapour pressure follows the Clausius—Clapeyron relation: $\frac{\mathrm{d}{p}_{\mathrm{sat}}}{\mathrm{d}T}=\frac{\mu \lambda p_{\mathrm{sat}}}{R{T}^2},$ where the latent heat of sublimation is $\lambda =600kJ/kg$, the molar mass is $\mu =0.044kg/mol$, and the gas constant is $R=8.3J/(\mathrm{mol}\cdot \mathrm{K})$. The thermal conductivity of the ${\mathrm{CO}}_2$ gas is $\kappa =10mW/(\mathrm{m}\cdot \mathrm{K})$ and its dynamic viscosity is $\eta =10\mu \mathrm{Pa}\cdot \mathrm{s}$. The density of dry ice is $\rho =1500{kg/m}^3$ and the gravitational acceleration is $g=10{m/s}^2$.

A puck of radius $r=10\mathrm{mm}$ consists of a disc of dry ice of thickness $h=1\mathrm{mm}$, and a metal disc of mass $M=0.01\mathrm{kg}$ on top of it. The initial temperature of the puck is $T_s$ and the ambient pressure is $p_0$.

The puck is placed on a horizontal metal plate which is held at a constant temperature $T=T_s+\Delta T$, and given an initial horizontal velocity $v=10mm/s$. After a very long time, the horizontal displacement $L$ of the metal disc is measured.

a) (2 pts) When $\Delta T$ is sufficiently small, $L$ is negligible and independent of $\Delta T$. However, when $\Delta T$ reaches a critical value $\Delta T_c$, the function $L(\Delta T)$ starts to grow. Estimate $\Delta T_c$.

b) (8 pts) Estimate the maximal value $L_{\max }$ of the function $L(\Delta T)$.

Treat the ${\mathrm{CO}}_2$ gas as ideal. Assume no tilting of the puck at any moment, all surfaces are perfectly smooth, and the thermal conductivities of the metal, dry ice, and gas satisfy ${\kappa }_{\mathrm{metal}}\gg {\kappa }_{\mathrm{ice}}\gg \kappa$.

Topic: Thermodynamics, Fluid Mechanics Metodi: Ideal Gas Law, Order-of-Magnitude Estimation Competenze: Physical Reasoning, Mathematical Modeling Objects: Puck (object) Fonte: Testo (PDF) Soluzione: Soluzioni (PDF)

T3: Dry ice hockey (10 pts)

At pressure $p_0=100\mathrm{kPa}$, solid ${\mathrm{CO}}_2$ (dry ice) sublimates (goes from solid to gaseous state) at $T_s=-78.5{}^{\circ }\mathrm{C}$. Its saturated vapour pressure follows the Clausius—Clapeyron relation: $\frac{\mathrm{d}{p}_{\mathrm{sat}}}{\mathrm{d}T}=\frac{\mu \lambda p_{\mathrm{sat}}}{R{T}^2},$ where the latent heat of sublimation is $\lambda =600kJ/kg$, the molar mass is $\mu =0.044kg/mol$, and the gas constant is $R=8.3J/(\mathrm{mol}\cdot \mathrm{K})$. The thermal conductivity of the ${\mathrm{CO}}_2$ gas is $\kappa =10mW/(\mathrm{m}\cdot \mathrm{K})$ and its dynamic viscosity is $\eta =10\mu \mathrm{Pa}\cdot \mathrm{s}$. The density of dry ice is $\rho =1500{kg/m}^3$ and the gravitational acceleration is $g=10{m/s}^2$.

A puck of radius $r=10\mathrm{mm}$ consists of a disc of dry ice of thickness $h=1\mathrm{mm}$, and a metal disc of mass $M=0.01\mathrm{kg}$ on top of it. The initial temperature of the puck is $T_s$ and the ambient pressure is $p_0$.

The puck is placed on a horizontal metal plate which is held at a constant temperature $T=T_s+\Delta T$, and given an initial horizontal velocity $v=10mm/s$. After a very long time, the horizontal displacement $L$ of the metal disc is measured.

a) (2 pts) When $\Delta T$ is sufficiently small, $L$ is negligible and independent of $\Delta T$. However, when $\Delta T$ reaches a critical value $\Delta T_c$, the function $L(\Delta T)$ starts to grow. Estimate $\Delta T_c$.

b) (8 pts) Estimate the maximal value $L_{\max }$ of the function $L(\Delta T)$.

Treat the ${\mathrm{CO}}_2$ gas as ideal. Assume no tilting of the puck at any moment, all surfaces are perfectly smooth, and the thermal conductivities of the metal, dry ice, and gas satisfy ${\kappa }_{\mathrm{metal}}\gg {\kappa }_{\mathrm{ice}}\gg \kappa$.

Topic: Thermodynamics, Fluid Mechanics Metodi: Ideal Gas Law, Order-of-Magnitude Estimation Competenze: Physical Reasoning, Mathematical Modeling Objects: Puck (object) Fonte: Testo (PDF) Soluzione: Soluzioni (PDF)