GPhO 2016

T1. Stabilizing unstable states (11 points)

Part A. Stabilization via feedback (3.5 points)

Let us study, how an initially unstable equilibrium position can be stabilized. First we consider a reversed pendulum: a thin long rod of length $l$ is fixed at its lowest point to a hinge so that it can freely rotate around the hinge. We describe the position of the rod via the angle $\phi$ between the rod and a vertical line. We shall assume that $\phi \ll 1$ ($\phi$ is much smaller than 1). The free fall acceleration is $g=9.8m/{\mathrm{s}}^2$.

i. (1.5 pts) Express the angular acceleration of the rod $\ddot{\phi }$ in terms of $\phi$, and the parameters $l$ and $g$. Show that the inclination angle $\phi$ as a function of time $t$ is expressed as $\phi (t)=A{e}^{t/\tau }+B{e}^{-t/\tau }$, where $A$ and $B$ are constants which depend on the initial position and initial angular speed of the rod, and $\tau$ is a characteristic time. Express $\tau$ in terms of $l$ and $g$, assuming that you try to keep the rod close to a vertical line.

ii. (0.5 pts) Now, a key tries to keep a long thin rod standing on his palm. For instance, as soon as the rod starts inclining, to the right (in our model — at a positive value of $\phi$), he moves the hinge also to the right, etc.; please raise and lower your palm to make your hand an equation arrives.

iii. (0.5 pts) Use only the front side of the sheets of paper.

iv. (1 pt) Equilibrium on a bike is also kept by displacing the contact point of the wheels with the ground (which is achieved by turning the handlebar while driving forth. Estimate the minimal driving speed $v_m$ of a bicyclist for whom the bicyclist is able to keep balance even without using his hands. The bicycle’s wheel radius is $r=0.3\mathrm{m}$; for this question, assume that the contact point of the wheels with the ground can be moved with a speed not exceeding $v$; the characteristic falling time is the same as for a rod of length $L=2\mathrm{m}$; the distance between the centres of the wheels is $L=1\mathrm{m}$.

Part B. Tightrope walker (3.5 points)

A tightrope walker cannot move the support point in the direction perpendicular to the rope. His equilibrium is kept by displacing the centre of gravity, instead. Let us make a simple model of a man balancing on a rope.

Lower half of the body is modelled by a point mass $m$ at height $H$, and the upper half of the body — by an equal point mass $m$ at height $1.4H$. The mutual position of these point masses can be changed by bowing right or left; for the sake of simplicity, let us assume that the distance of the point masses from the rope will remain unchanged, i.e. these behave as if being fixed to the endpoints of two thin rods of lengths $H$ and $1.4H$ respectively, see figure. Let the rods form angles ${\alpha }_1$ and ${\alpha }_2$ with the vertical line (positive angles correspond to clock-wise rotation), so that the angle between the rods is $\beta ={\alpha }_1-{\alpha }_2$. A tightrope walker can control the value of the angle $\beta$ by bowing.

i. (1 pt) Let us assume that initially, the tightrope walker is standing in an almost perfect equilibrium (${\alpha }_1={\alpha }_2=0$). Due to instability of this equilibrium, he starts slowly falling clockwise, which he notices at $t=t_0$ when ${\alpha }_1={\alpha }_2={\alpha }_0>0$. He hurries rapidly to stop falling: assume that the angle $\beta$ takes almost instantaneously a new value ${\beta }_0$. Express the new values of the angles ${\alpha }_1$ and ${\alpha }_2$ in terms of $\beta$ and ${\alpha }_0$.

ii. (1 pt) So, the tightrope walker is now bowing and keeps that value ($\beta ={\beta }_0$) for the time period $T$, over which he intends to restore his vertical position. He notices that after stopping falling, his body starts moving back, towards the vertical position. Suppose that at $t=T/2$, the pendulum was motionless and inclined by a small angle ${\phi }_0$. Sketch the graph of the inclination angle $\phi$ as a function of time, and determine the angular displacement of the pendulum $\Delta \phi$ for the moment $t=T$, i.e. $\Delta \phi =\phi (T)-\phi (T/2)$. You may assume in your calculations that $\Delta \phi \ll {\phi }_0$ (this is valid because $T\ll l/v_0$).

iii. (1.5 pts) Since we still neglect gravity, only inertial force exerts a torque on the pendulum. Determine the average value of this torque (with respect to the suspension point, averaged over the full period $2T$).

iv. (1 pt) Now, let us take into account that there is also the gravity field of the Earth. Determine, which inequality must be satisfied so the vertical position would remain stable. The vertical position of such a pendulum (some of these parameters may not be needed for your inequality).

Part C. Kapitza’s pendulum (4 points)

In 1908 Andrew Stephenson found that the upper position of a pendulum can be stable, if its suspension point oscillates with a high frequency. The explanation of this phenomenon was provided in 1951 by Russian physicist Pyotr Kapitza. In what follows we’ll find the stability criterion of such a pendulum. Apart from being a fun part of physics, this kind of phenomenon demonstrates the method of separating fast and slow processes which plays an important role in physics. High frequency oscillations can drive a slow motion in various systems, e.g. high frequency electric fields set on charges with an effective average force known as the ponderomotive force.

We consider a pendulum of length $l$, similar to that of Part-A-Question-i, but now the rod is massless, with a point mass at its end, and the suspension point oscillates vertically (see the figure). Let the velocity $v$ of the suspension point depend on time $t$ as shown in the graph below ($v>0$ corresponds to upward motion); the oscillations’ half-period $T\ll l/v_0$. We also assume that $v_0/T\gg g$ so that for questions i–ii you may ignore the free fall acceleration. In order to simplify calculations, you’ll need to study this process in the frame of reference of the suspension point (keep in mind: reference frame’s acceleration $\vec{a}$ gives rise to an inertial force $-M\vec{a}$ acting on a body of mass $M$).

i. (1.5 pts) Suppose that at $t=T/2$, the pendulum was motionless and inclined by a small angle ${\phi }_0$. Sketch the graph of the inclination angle $\phi$ as a function of time, and determine the angular displacement of the pendulum $\Delta \phi$ for the moment $t=T$, i.e. $\Delta \phi =\phi (T)-\phi (T/2)$. You may assume in your calculations that $\Delta \phi \ll {\phi }_0$ (this is valid because $T\ll l/v_0$).

ii. (1.5 pts) Since we still neglect gravity, only inertial force exerts a torque on the pendulum. Determine the average value of this torque (with respect to the suspension point, averaged over the full period $2T$).

iii. (1 pt) Now, let us take into account that there is also the gravity field of the Earth. Determine, which inequality must be satisfied so the vertical position would remain stable. Note: some of these parameters may not be needed for your inequality.

Fonte: Testo (PDF) — p.1

Topic: Oscillations & Waves, Newtonian Mechanics Metodi: Simple Harmonic Motion Analysis, Torque & Angular Momentum Analysis, Differential Equations, Small-Angle Approximation Competenze: Physical Reasoning, Mathematical Modeling, Diagrammatic Reasoning Objects: Pendulum, Rod, Wheel

T1. Stabilizing unstable states (11 points)

Parte A. Stabilizzazione tramite feedback (3.5 punti)

Studiamo come una posizione di equilibrio inizialmente instabile può essere stabilizzata. Prima consideriamo un pendolo invertito: un lungo e sottile bastone di lunghezza $l$ è fissato al suo punto più basso verso un pendolo in modo che possa girare liberamente intorno al pendolo. Descriviamo la posizione della canna attraverso l’angolo $\phi$ tra la canna e una linea verticale. Supponiamo che $\phi \ll 1$ ($\phi$ è molto più piccolo di 1). L’accelerazione della caduta libera è $g=9.8m/{\mathrm{s}}^2$.

**i. (1.5 pts) ** Express the angular acceleration of the rod $\ddot{\phi }$ in terms of $\phi$, and the parameters $l$ and $g$. Mostra che l’angolo di inclination $\phi$ come funzione di tempo $t$ è espresso come $\phi (t)=A{e}^{t/\tau }+B{e}^{-t/\tau }$, dove $A$ e $B$ sono costanti che dipendono dalla posizione iniziale e dalla velocità angolare iniziale della canna, e $\tau$ è un tempo caratteristico. Esprimere $\tau$ in termini di $l$ e $g$, supponendo che si cerchi di mantenere la canna vicino a una linea verticale.

ii. (0.5 pts) Now, a key tries to keep a long thin rod standing on his palm. Per esempio, non appena la canna inizia a inclinare, a destra (nel nostro modello ad un valore positivo di $\phi$), si muove la pendenza anche a destra, ecc.; per favore, sollevare e abbassare la palma per far arrivare la mano all’equazione.

**iii. (0.5 pts) ** Utilizzare solo il lato anteriore dei fogli di carta.

**iv. (1 pt) ** L’equilibrio su una bici è anche mantenuto spostando il punto di contatto delle ruote con il terreno (che è raggiunto girando la manovra mentre si guida avanti. Estimate the minimal driving speed $v_m$ of a bicyclist for whom the bicyclist is able to keep balance even without using his hands. Il radius della ruota della bicicletta è $r=0.3\mathrm{m}$; per questa domanda, supponi che il punto di contatto delle ruote con il terreno possa essere spostato con una velocità non superiore a $v$; il tempo di caduta caratteristico è lo stesso di quello di una canna di lunghezza $L=2\mathrm{m}$; la distanza tra i centri delle ruote è $L=1\mathrm{m}$.

Parte B. Tightrope walker (3,5 punti)

Un stropper non può spostare il punto di supporto in direzione perpendicolare alla corda. Il suo equilibrio viene mantenuto spostando il centro di gravità, invece. Facciamo un modello semplice di un uomo che bilancia su una corda.

La metà inferiore del corpo è modellata da una massa di punto $m$ ad altezza $H$, e la metà superiore del corpo da una massa di punto uguale $m$ ad altezza $1.4H$. La posizione reciproca di queste masse puntistiche può essere cambiata per la destra o la sinistra; per semplicità, supponiamo che la distanza delle masse puntistiche dalla corda rimanga invariata, cioè. Questi comportamenti comportano come se fossero fissati agli endpoints di due thin rods di lunghezza $H$ e $1.4H$ rispettivamente, vedi figura. Lasciate che le barre formino angoli ${\alpha }_1$ e ${\alpha }_2$ con la linea verticale (angoli positivi corrispondono alla rotazione a orologio), in modo che l’angolo tra le barre sia $\beta ={\alpha }_1-{\alpha }_2$. A tightrope walker can control the value of the angle $\beta$ by bowing.

**i. (1 pt) ** Supponiamo che inizialmente, il stroppiatore sia in equilibrio quasi perfetto (${\alpha }_1={\alpha }_2=0$). A causa dell’instabilità di questo equilibrio, ** inizia a scendere lentamente al punto di vista orario **, che egli nota a $t=t_0$ quando ${\alpha }_1={\alpha }_2={\alpha }_0>0$. He hurries rapidly to stop falling: assume che l’angolo $\beta$ prende quasi instantaneously a new value ${\beta }_0$. Esprimere i nuovi valori degli angoli ${\alpha }_1$ e ${\alpha }_2$ in termini di $\beta$ e ${\alpha }_0$.

**ii. (1 pt) ** Quindi, il stroppiatore è ora in movimento e mantiene quel valore ($\beta ={\beta }_0$) per il periodo di tempo $T$, durante il quale intende ripristinare la sua posizione verticale. Si accorge che dopo aver smesso di cadere, il suo corpo inizia a tornare indietro, verso la posizione verticale. Supponiamo che a $t=T/2$, il pendolo fosse motionless e inclinato da un piccolo angolo ${\phi }_0$. Sketch the graph of the inclination angle $\phi$ as a function of time, and determine the angular displacement of the pendulum $\Delta \phi$ for the moment $t=T$, cioè $\Delta \phi =\phi (T)-\phi (T/2)$. Si può assumere nei vostri calcoli che $\Delta \phi \ll {\phi }_0$ (this is valid because $T\ll l/v_0$).

iii. (1.5 pts) Since we still neglect gravity, only inertial force exerts a torque on the pendulum. Determina il valore medio di questa torque (with respect to the suspension point, averaged over the full period $2T$).

iv. (1 pt) Now, let us take into account that there is also the gravity field of the Earth. Determine quale disuguaglianza deve essere soddisfatta in modo che la posizione verticale rimanga stabile. La posizione verticale di un pendolo (alcuni di questi parametri potrebbero non essere necessari per la vostra inequità).

Parte C. Il pendolo di Kapitza (4 punti)

Nel 1908 Andrew Stephenson scoprì che la posizione superiore di un pendolo può essere stabile, se il suo punto di sospensione oscilla con una alta frequenza. L’esposizione di questo fenomeno è stata fornita nel 1951 dal fisico russo Pyotr Kapitza. In quello che segue troveremo il criterio di stabilità di un pendolo simile. Oltre ad essere una parte divertente della fisica, questo tipo di fenomeno dimostra il metodo di separare processi veloci e lenti che svolge un ruolo importante nella fisica. Le oscillazioni ad alta frequenza possono guidare un movimento lento in vari sistemi, ad esempio: campi elettrici ad alta frequenza, a carica di una forza media efficace, nota come forza ponderomotiva.

We consider a pendulum of length $l$, similar to that of Part-A-Question-i, but now the rod is massless, with a point mass at its end, and the suspension point oscillates vertically (see the figure). Let the velocity $v$ of the suspension point depend on time $t$ as shown in the graph below ($v>0$ corresponds to upward motion); the oscillations’ half-period $T\ll l/v_0$. Supponiamo anche che $v_0/T\gg g$ in modo che per le domande iii si possa ignorare l’accelerazione della caduta libera. Per semplificare i calcoli, è necessario studiare questo processo nel frame of reference del punto di sospensione (ricordate: l’accelerazione del frame di riferimento $\vec{a}$ dà luogo a una forza inerziale $-M\vec{a}$ che agisce su un corpo di massa $M$).

i. (1.5 pts) Suppose that at $t=T/2$, the pendulum was motionless and inclined by a small angle ${\phi }_0$. Sketch the graph of the inclination angle $\phi$ as a function of time, and determine the angular displacement of the pendulum $\Delta \phi$ for the moment $t=T$, cioè $\Delta \phi =\phi (T)-\phi (T/2)$. Si può assumere nei vostri calcoli che $\Delta \phi \ll {\phi }_0$ (this is valid because $T\ll l/v_0$).

ii. (1.5 pts) Since we still neglect gravity, only inertial force exerts a torque on the pendulum. Determina il valore medio di questa torque (with respect to the suspension point, averaged over the full period $2T$).

iii. (1 pt) Now, let us take into account that there is also the gravity field of the Earth. Determine quale disuguaglianza deve essere soddisfatta in modo che la posizione verticale rimanga stabile. Nota: alcuni di questi parametri potrebbero non essere necessari per la vostra inequità.

Fonte: Testo (PDF) — p.1

Topic: Oscillations & Waves, Newtonian Mechanics Metodi: Simple Harmonic Motion Analysis, Torque & Angular Momentum Analysis, Differential Equations, Small-Angle Approximation Competenze: Physical Reasoning, Mathematical Modeling, Diagrammatic Reasoning Objects: Pendulum, Rod, Wheel

T1. Stabilizing unstable states (11 points)

Part A. Stabilization by feedback (3.5 points)

Let’s study how an initially unstable equilibrium position can be stabilized. First we consider a reversed pendulum: a thin long rod of length $l$ is fixed at its lowest point to a hinge so that it can freely rotate around the hinge. We describe the position of the rod via the angle $\phi$ between the rod and a vertical line. We shall assume that $\phi \ll 1$ ($\phi$ is much smaller than 1). The free fall acceleration is $g=9.8m/{\mathrm{s}}^2$.

**i. (1.5 pts) ** Express the angular acceleration of the rod $\ddot{\phi }$ in terms of $\phi$, and the parameters $l$ and $g$. Show that the inclination angle $\phi$ as a function of time $t$ is expressed as $\phi (t)=A{e}^{t/\tau }+B{e}^{-t/\tau }$, where $A$ and $B$ are constants which depend on the initial position and initial angular speed of the rod, and $\tau$ is a characteristic time. Express $\tau$ in terms of $l$ and $g$, assuming that you try to keep the rod close to a vertical line.

ii. (0.5 pts) Now, a key tries to keep a long thin rod standing on his palm. For example, as soon as the rod starts inclining, to the right (in our model at a positive value of $\phi$), he moves the hinge also to the right, etc.; please raise and lower your palm to make your hand an equation arrives.

**iii. (0.5 pts) ** Use only the front side of the sheets of paper.

iv. (1 pt) Equilibrium on a bike is also kept by displacing the contact point of the wheels with the ground (which is achieved by turning the handlebar while driving forward. Estimate the minimal driving speed $v_m$ of a bicyclist for whom the bicyclist is able to keep balance even without using his hands. The bicycle’s wheel radius is $r=0.3\mathrm{m}$; for this question, assume that the contact point of the wheels with the ground can be moved with a speed not exceeding $v$; the characteristic falling time is the same as for a rod of length $L=2\mathrm{m}$; the distance between the centres of the wheels is $L=1\mathrm{m}$.

Part B. The following table shows the results of the tests:

A tightrope walker cannot move the support point in the direction perpendicular to the rope. His equilibrium is kept by displacing the center of gravity, instead. Let’s make a simple model of a man balancing on a rope.

The lower half of the body is modelled by a point mass $m$ at height $H$, and the upper half of the body by an equal point mass $m$ at height $1.4H$. The mutual position of these point masses can be changed by bowing right or left; for the sake of simplicity, let’s assume that the distance of the point masses from the rope will remain unchanged, i.e. These behave as if being fixed to the endpoints of two thin rods of lengths $H$ and $1.4H$ respectively, see figure. Let the rods form angles ${\alpha }_1$ and ${\alpha }_2$ with the vertical line (positive angles correspond to clock-wise rotation), so that the angle between the rods is $\beta ={\alpha }_1-{\alpha }_2$. A tightrope walker can control the value of the angle $\beta$ by bowing.

i. (1 pt) ** Let’s assume that initially, the tightrope walker is standing in almost perfect equilibrium (${\alpha }_1={\alpha }_2=0$). Due to instability of this equilibrium, he ** starts slowly falling clockwise, which he notices at $t=t_0$ when ${\alpha }_1={\alpha }_2={\alpha }_0>0$. He hurries rapidly to stop falling: assume that the angle $\beta$ takes almost instantaneously a new value ${\beta }_0$. Express the new values of the angles ${\alpha }_1$ and ${\alpha }_2$ in terms of $\beta$ and ${\alpha }_0$.

ii. (1 pt) So, the tightrope walker is now bowing and keeps that value ($\beta ={\beta }_0$) for the time period $T$, over which he intends to restore his vertical position. He notices that after stopping falling, his body starts moving back, towards the vertical position. Suppose that at $t=T/2$, the pendulum was motionless and inclined by a small angle ${\phi }_0$. Sketch the graph of the inclination angle $\phi$ as a function of time, and determine the angular displacement of the pendulum $\Delta \phi$ for the moment $t=T$, i.e. $\Delta \phi =\phi (T)-\phi (T/2)$. You may assume in your calculations that $\Delta \phi \ll {\phi }_0$ (this is valid because $T\ll l/v_0$).

iii. (1.5 pts) Since we still neglect gravity, only inertial force exerts a torque on the pendulum. Determine the average value of this torque (with respect to the suspension point, averaged over the full period $2T$).

iv. (1 pt) Now, let us take into account that there is also the gravity field of the Earth. Determine which inequality must be satisfied so that the vertical position would remain stable. The vertical position of such a pendulum (some of these parameters may not be needed for your inequality).

Part C. The following table shows the results of the calculations:

In 1908, Andrew Stephenson found that the upper position of a pendulum can be stable, if its suspension point oscillates with a high frequency. The explanation of this phenomenon was provided in 1951 by Russian physicist Pyotr Kapitza. In what follows we’ll find the stability criterion of such a pendulum. Apart from being a fun part of physics, this kind of phenomenon demonstrates the method of separating fast and slow processes which plays an important role in physics. High frequency oscillations can drive a slow motion in various systems, e.g. high frequency electric fields set on charges with an effective average force known as the ponderomotive force.

We consider a pendulum of length $l$, similar to that of Part-A-Question-i, but now the rod is massless, with a point mass at its end, and the suspension point oscillates vertically (see the figure). Let the velocity $v$ of the suspension point depend on time $t$ as shown in the graph below ($v>0$ corresponds to upward motion); the oscillations’ half-period $T\ll l/v_0$. We also assume that $v_0/T\gg g$ so for questions iii you may ignore the free fall acceleration. In order to simplify calculations, you’ll need to study this process in the frame of reference of the suspension point (keep in mind: reference frame’s acceleration $\vec{a}$ gives rise to an inertial force $-M\vec{a}$ acting on a body of mass $M$).

i. (1.5 pts) Suppose that at $t=T/2$, the pendulum was motionless and inclined by a small angle ${\phi }_0$. Sketch the graph of the inclination angle $\phi$ as a function of time, and determine the angular displacement of the pendulum $\Delta \phi$ for the moment $t=T$, i.e. $\Delta \phi =\phi (T)-\phi (T/2)$. You may assume in your calculations that $\Delta \phi \ll {\phi }_0$ (this is valid because $T\ll l/v_0$).

ii. (1.5 pts) Since we still neglect gravity, only inertial force exerts a torque on the pendulum. Determine the average value of this torque (with respect to the suspension point, averaged over the full period $2T$).

iii. (1 pt) Now, let us take into account that there is also the gravity field of the Earth. Determine which inequality must be satisfied so that the vertical position would remain stable. Note: some of these parameters may not be needed for your inequality.

Fonte: Testo (PDF) — p.1

Topic: Oscillations & Waves, Newtonian Mechanics Metodi: Simple Harmonic Motion Analysis, Torque & Angular Momentum Analysis, Differential Equations, Small-Angle Approximation Competenze: Physical Reasoning, Mathematical Modeling, Diagrammatic Reasoning Objects: Pendulum, Rod, Wheel

T2. Gravitational waves (10 points)

Part A. Dipole radiation (2.4 points)

Static electric and gravity fields are described by identical set of equations — as long as we are far from black holes. However, if we add terms describing time variations of the fields, the equations become different. Therefore, expressions for electromagnetic waves cannot be directly carried over to gravitational waves. Still, for expressions given below, the difference will be only in the value of numerical prefactors.

Charges moving with acceleration lose kinetic energy by radiating electromagnetic waves; this radiation is known as the dipole radiation. The total radiation power is expressed as

P_{el} = \frac{\ddot{\vec{d}}^2}{6\pi\varepsilon_0 c^3},\tag{1}

where $\vec{d}$ is the second time derivative of the dipole moment, $c$ is the speed of light, and ${\epsilon }_0$ — vacuum permittivity. Dipole moment for a system of charges $q_i$ is defined as $\vec{d}={\sum }_i{\vec{r}}_i{q}_i$, where ${\vec{r}}_i$ is vector pointing from the origin to the position of $i$-th charge. For harmonically oscillating dipoles, the radiated wave frequency equals to the frequency of oscillations.

i. (1.4 pts) Consider an electron of charge $-e$ and mass $m$, circulating around an atomic nucleus of charge $+Ze$ at distance $r$; neglect quantum mechanical effects. Express the total radiated power, and the wavelength $\lambda$ of the radiated waves in terms of $r$, $Z$, $m$, $e$, and physical constants.

ii. (1 pt) Let us try to carry over Eq. (1) to gravitational waves. Then, the total radiation power $P_{gd}$ would be proportional to ${\ddot{\vec{d}}}^2$, where ${\vec{d}}_g$ is the gravitational dipole moment, and two dots denote the second time-derivative. Analogously to the electrical dipole, gravitational dipole moment for a system of point masses $m_i$ is defined as ${\vec{d}}_g={\sum }_i{\vec{r}}_i{m}_i$. Show that always $P_{gd}=0$.

Part B. Quadrupole radiation (7.6 points)

Let us consider a binary star consisting of two stars of equal mass $M$ which rotate around a circular orbit of radius $R$ with angular speed $\omega$.

i. (1 pt) Express $\omega$ in terms of $M$, $R$, and constants.

ii. (0.8 pts) While there is no gravitational dipole radiation, there is a quadrupole one. In analogy with the dipole radiation power, the quadrupole radiated power can be assumed to be proportional to the square of the third time-derivative of the quadrupole moment. For this problem, it is enough to know that for a binary star, the quadrupole moment $Q$ can be estimated as $Q\approx M{R}^2$, where the factor before $M{R}^2$ is of the order of unity. The dimension of $Q$ is $[Q]=\mathrm{kg}{\mathrm{m}}^2$. So, the power of the quadrupole radiation $P_q$ depends on the orbital radius $R$, the angular speed $\omega$, the mass $M$, and physical constants. Express $P_q$ using dimensional analysis.

iii. (0.8 pts) The effect of gravitational waves is measured by strain $h=\Delta l/l$; here $l$ is a distance between two points in space, and $\Delta l$ is the change of that distance due to the wave. As usual for waves, the energy flux density $S$ (radiation energy per unit time and unit area) is proportional to the squared wave amplitude: $S=K{h}_0^2$ ($h_0$ denotes the wave amplitude). Based on dimensional arguments, express the factor $K$ in terms of constants and the angular frequency of the wave $\omega$.

iv. (1 pt) The dipole radiation is distributed over propagation directions anisotropically, but let us ignore this: for the sake of simplicity, assume isotropic radiation. Express the amplitude $h_0$ of gravitational waves at distance $L$ in terms of $M$, $R$, and physical constants.

The energy of the binary star decreases in time due to the emission of gravitational waves. So, the distance $R$ between the two stars decreases. This process will continue until the stars collide and merge ($R$ becomes still smaller in the case of cylindrically symmetric stars, but the maximum value of the cylinder height — at the moment of the original star). Let the initial magnetic field at the outer shell be $B_0$. Express the magnetic field $B$ as a function of time $t$ in the region where field lines are still circular (valid for $t\gg 1/{\omega }_n$) in terms of $B_0$ and ${\omega }_n$. [N.B. — see note below regarding the LIGO question text.]

v. (1 pt) Express $R$ in terms of time $t$ until the distance $R$ is reduced to a certain value $R_{\ast }$. Determine analytically the time ${\tau }_{\ast }$ when the stars approach the distance $R_{\ast }$.

vi. (2 pts) Very strong magnetic fields affect chemical properties of matter by changing the shape of electron orbits. This happens when the Lorentz force acting on an orbital electron becomes comparable than the Coulomb force due to the atomic core size. Estimate the maximal possible distance of two black holes when LIGO experiment (reported on 11th February 2016), gravitational waves emitted right before a merger of two black holes were detected. The strain $h$ (see question iii) was measured as a function of time; the result is given in the graph below. Using this graph and assuming that the masses of the two black holes were equal, estimate the mass of each of them numerically. Gravitational constant $G=6.67\times 10^{-11}{\mathrm{m}}^3{\mathrm{s}}^{-2}\mathrm{k}{\mathrm{g}}^{-1}$; $c=3.00\times 10^{8}m/s$.

vii. (1.5 pts) Using the same data as for question vi, estimate the distance to these black holes.

Fonte: Testo (PDF) — p.1

Topic: Astrophysics, Gravitation Metodi: Dimensional Analysis, Newton’s Law of Gravitation, Order-of-Magnitude Estimation, Conservation of Energy Competenze: Estimation & Approximation, Mathematical Modeling, Physical Reasoning Objects: Star, Black Hole, Electron, Nucleus

T2. Gravitational waves (10 points)

Parte A. Radiamento dipole (2.4 punti)

I campi elettrici e gravitazionali statici sono descritti da un insieme identico di equazioni finché siamo lontani dai buchi neri. Tuttavia, se aggiungiamo termini che descrivono le variazioni di tempo dei campi, le equazioni diventano diverse. Pertanto, le espressioni per le onde elettromagnetiche non possono essere direttamente portate su onde gravitazionali. Tuttavia, per le espressioni di seguito, la differenza sarà solo nel valore dei prefatori numerici.

Le cariche che si muovono con accelerazione perdono energia cinetica mediante l’irradiamento di onde elettromagnetiche; questa radiazione è conosciuta come la dipole radiation. Il potere totale di radiazione è espresso come

P_{el} = \frac{\ddot{\vec{d}}^2}{6\pi\varepsilon_0 c^3},\tag{1}

dove $\vec{d}$ è la seconda derivata del momento di dipolo, $c$ è la velocità della luce e ${\epsilon }_0$ la permissività di vuoto. Dipole moment for a system of charges $q_i$ is defined as $\vec{d}={\sum }_i{\vec{r}}_i{q}_i$, where ${\vec{r}}_i$ is vector pointing from the origin to the position of $i$-th charge. Per i dipoli armonicamente oscillanti, la frequenza delle onde irradiate è pari alla frequenza delle oscillazioni.

i. (1.4 pts) Consider an electron of charge $-e$ and mass $m$, circulating around an atomic nucleus of charge $+Ze$ at distance $r$; neglect quantum mechanical effects. Express the total radiated power, and the wavelength $\lambda$ of the radiated waves in terms of $r$, $Z$, $m$, $e$, and physical constants.

**ii. (1 pt) ** Provamo a portare sopra l’Eq. (1) ad onde gravitazionali. Quindi, la potenza di radiazione totale $P_{gd}$ sarebbe proporzionale a ${\ddot{\vec{d}}}^2$, dove ${\vec{d}}_g$ è il momento di dipolo gravitazionale, e due punti denotano la seconda derivazione temporale. Analogamente al dipolo elettrico, il momento dipolo gravitazionale per un sistema di masse puntate $m_i$ è definito come ${\vec{d}}_g={\sum }_i{\vec{r}}_i{m}_i$. Show that always $P_{gd}=0$.

Parte B. Quadruple di radiazione (7,6 punti)

Consideriamo una stella binaria costituita da due stelle di massa uguale $M$ che ruotano attorno ad un’orbita circolare di raggio $R$ con velocità angolare $\omega$.

**i. (1 pt) ** Express $\omega$ in termini di $M$, $R$, e costanti.

ii. (0.8 pts) While there is no gravitational dipole radiation, there is a quadrupole one. In analogia con la potenza di radiazione di dipole, la potenza di radiazione quadrupola può essere presunta come proporzionale al quadrato del terzo derivato temporale del momento quadrupolare. Per questo problema, è sufficiente sapere che per una stella binaria, il momento quadrupolo $Q$ può essere stimato come $Q\approx M{R}^2$, dove il fattore prima $M{R}^2$ è dell’ordine di unità. La dimensione di $Q$ è $[Q]=\mathrm{kg}{\mathrm{m}}^2$. So, the power of the quadrupole radiation $P_q$ depends on the orbital radius $R$, the angular speed $\omega$, the mass $M$, and physical constants. Espresso $P_q$ con analisi dimensionale.

iii. (0.8 pts) The effect of gravitational waves is measured by strain $h=\Delta l/l$; here $l$ is a distance between two points in space, and $\Delta l$ is the change of that distance due to the wave. Come sempre per le onde, la densità di flusso di energia $S$ (radiation energy per unit time and unit area) è proporzionale all’ampiezza di onda squared: $S=K{h}_0^2$ ($h_0$ denota l’ampiezza d’onda). Basato su argomenti dimensionali, esprime il fattore $K$ in termini di costanti e la frequenza angolare dell’onda $\omega$.

iv. (1 pt) The dipole radiation is distributed over propagation directions anisotropically, but let us ignore this: for the sake of simplicity, assume isotropic radiation. Esprimere l’ampiezza $h_0$ di onde gravitazionali a distanza $L$ in termini di $M$, $R$, e costanti fisici.

L’energia della stella binaria diminuisce nel tempo a causa dell’emissione di onde gravitazionali. So, the distance $R$ between the two stars decreases. Questo processo continuerà fino a quando le stelle non collidono e non si fondono ($R$ diventa ancora più piccolo nel caso di stelle cilindricamente simmetriche, ma il valore massimo della altezza del cilindro al momento della stella originale). Lasciate che il campo magnetico iniziale alla conchiglia esterna sia $B_0$. Esprimere il campo magnetico $B$ come funzione di tempo $t$ nella regione dove le linee di campo sono ancora circolari (valid per $t\gg 1/{\omega }_n$) in termini di $B_0$ e ${\omega }_n$. [N.B. vedere nota sotto riguardo al testo della domanda LIGO.]

**v. (1 pt) ** Express $R$ in terms of time $t$ until the distance $R$ is reduced to a certain value $R_{\ast }$. Determine analiticamente il tempo ${\tau }_{\ast }$ quando le stelle si avvicinano alla distanza $R_{\ast }$.

**vi. (2 pts) ** I campi magnetici molto forti influenzano le proprietà chimiche della materia cambiando la forma delle orbite degli elettroni. Questo accade quando la forza di Lorentz che agisce su un elettrone orbitale diventa comparabile a quella di Coulomb a causa della dimensione del nucleo atomico. Estimare la massima distanza possibile di due buchi neri quando LIGO esperimento (reportato l’11 febbraio 2016), onde gravitazionali emesse proprio prima di una fusione di due buchi neri sono stati rilevati. Il strain $h$ (vedi domanda iii) è stato misurato come funzione di tempo; il risultato è dato nel grafico qui sotto. Usando questo grafico e supponendo che le masse dei due buchi neri fossero uguali, stimare la massa di ciascuno di loro numericamente. Costante gravitazionale $G=6.67\times 10^{-11}{\mathrm{m}}^3{\mathrm{s}}^{-2}\mathrm{k}{\mathrm{g}}^{-1}$; $c=3.00\times 10^{8}m/s$.

vii. (1.5 pts) Using the same data as for question vi, estimate the distance to these black holes.

Fonte: Testo (PDF) — p.1

Topic: Astrophysics, Gravitation Metodi: Dimensional Analysis, Newton’s Law of Gravitation, Order-of-Magnitude Estimation, Conservation of Energy Competenze: Estimation & Approximation, Mathematical Modeling, Physical Reasoning Objects: Star, Black Hole, Electron, Nucleus

T2. Gravitational waves (10 points)

Part A. Dipole radiation (2.4 points)

Static electric and gravity fields are described by identical set of equations as long as we are far from black holes. However, if we add terms describing time variations of the fields, the equations become different. Therefore, expressions for electromagnetic waves cannot be directly carried over to gravitational waves. Still, for expressions given below, the difference will be only in the value of numerical prefactors.

Charges moving with acceleration lose kinetic energy by radiating electromagnetic waves; this radiation is known as the dipole radiation. The total radiation power is expressed as

P_{el} = \frac{\ddot{\vec{d}}^2}{6\pi\varepsilon_0 c^3},\tag{1}

where $\vec{d}$ is the second time derivative of the dipole moment, $c$ is the speed of light, and ${\epsilon }_0$ vacuum permittivity. Dipole moment for a system of charges $q_i$ is defined as $\vec{d}={\sum }_i{\vec{r}}_i{q}_i$, where ${\vec{r}}_i$ is vector pointing from the origin to the position of $i$-th charge. For harmonically oscillating dipoles, the radiated wave frequency is equal to the frequency of oscillations.

i. (1.4 pts) Consider an electron of charge $-e$ and mass $m$, circulating around an atomic nucleus of charge $+Ze$ at distance $r$; neglect quantum mechanical effects. Express the total radiated power, and the wavelength $\lambda$ of the radiated waves in terms of $r$, $Z$, $m$, $e$, and physical constants.

ii. (1 pt) Let us try to carry over Eq. (1) to gravitational waves. Then, the total radiation power $P_{gd}$ would be proportional to ${\ddot{\vec{d}}}^2$, where ${\vec{d}}_g$ is the gravitational dipole moment, and two dots denote the second time-derivative. Analogously to the electrical dipole, gravitational dipole moment for a system of point masses $m_i$ is defined as ${\vec{d}}_g={\sum }_i{\vec{r}}_i{m}_i$. Show that always $P_{gd}=0$.

Part B. Quadruple radiation (7.6 points)

Let us consider a binary star consisting of two stars of equal mass $M$ which rotate around a circular orbit of radius $R$ with angular speed $\omega$.

i. (1 pt) Express $\omega$ in terms of $M$, $R$, and constants.

ii. (0.8 pts) While there is no gravitational dipole radiation, there is a quadrupole one. In analogy with the dipole radiation power, the quadrupole radiated power can be assumed to be proportional to the square of the third time-derivative of the quadrupole moment. For this problem, it’s enough to know that for a binary star, the quadrupole moment $Q$ can be estimated as $Q\approx M{R}^2$, where the factor before $M{R}^2$ is of the order of unity. The dimension of $Q$ is $[Q]=\mathrm{kg}{\mathrm{m}}^2$. So, the power of the quadrupole radiation $P_q$ depends on the orbital radius $R$, the angular speed $\omega$, the mass $M$, and physical constants. Express $P_q$ using dimensional analysis.

iii. (0.8 pts) The effect of gravitational waves is measured by strain $h=\Delta l/l$; here $l$ is a distance between two points in space, and $\Delta l$ is the change of that distance due to the wave. As usual for waves, the energy flux density $S$ (radiation energy per unit time and unit area) is proportional to the square wave amplitude: $S=K{h}_0^2$ ($h_0$ denotes the wave amplitude). Based on dimensional arguments, express the factor $K$ in terms of constants and the angular frequency of the wave $\omega$.

iv. (1 pt) The dipole radiation is distributed over propagation directions anisotropically, but let us ignore this: for the sake of simplicity, assume isotropic radiation. Express the amplitude $h_0$ of gravitational waves at distance $L$ in terms of $M$, $R$, and physical constants.

The energy of the binary star decreases in time due to the emission of gravitational waves. So, the distance $R$ between the two stars decreases. This process will continue until the stars collide and merge ($R$ becomes still smaller in the case of cylindrically symmetrical stars, but the maximum value of the cylinder height at the moment of the original star). Let the initial magnetic field at the outer shell be $B_0$. Express the magnetic field $B$ as a function of time $t$ in the region where field lines are still circular (valid for $t\gg 1/{\omega }_n$) in terms of $B_0$ and ${\omega }_n$. [N.B. see note below regarding the LIGO question text.]

**v. (pt) ** Express $R$ in terms of time $t$ until the distance $R$ is reduced to a certain value $R_{\ast }$. Determine analytically the time ${\tau }_{\ast }$ when the stars approach the distance $R_{\ast }$.

vi. (2 pts) Very strong magnetic fields affect chemical properties of matter by changing the shape of electron orbits. This happens when the Lorentz force acting on an orbital electron becomes comparable to the Coulomb force due to the atomic core size. Estimate the maximum possible distance of two black holes when LIGO experiment (reported on 11th February 2016), gravitational waves emitted right before a merger of two black holes were detected. The strain $h$ (see question iii) was measured as a function of time; the result is given in the graph below. Using this graph and assuming that the masses of the two black holes were equal, estimate the mass of each of them numerically. Gravitational constant $G=6.67\times 10^{-11}{\mathrm{m}}^3{\mathrm{s}}^{-2}\mathrm{k}{\mathrm{g}}^{-1}$; $c=3.00\times 10^{8}m/s$.

vii. (1.5 pts) Using the same data as for question vi, estimate the distance to these black holes.

Fonte: Testo (PDF) — p.1

Topic: Astrophysics, Gravitation Metodi: Dimensional Analysis, Newton’s Law of Gravitation, Order-of-Magnitude Estimation, Conservation of Energy Competenze: Estimation & Approximation, Mathematical Modeling, Physical Reasoning Objects: Star, Black Hole, Electron, Nucleus

T3. Magnetars (9 points)

Magnetic fields are everywhere around us. Some typical magnetic B-field values: Earth’s magnetic field $\approx 25$–$60\mu \mathrm{T}$, at Sunspots: $0.3\mathrm{T}$; strong permanent magnets: around $1\mathrm{T}$; continuously maintained magnetic fields in laboratory: up to $45\mathrm{T}$; neutron stars and magnetars: up to $10^{11}\mathrm{T}$. In what follows we study few aspects of strong magnetic fields.

Magnetic field energy density $w=B^2\frac{1}{2{\mu }_0}$, where ${\mu }_0=1.3\times 10^{-6}N/{\mathrm{A}}^2$ is the vacuum permeability, and $\mu$ — the relative permeability of the medium. Systems tries to move towards a lower energy state and so ferromagnetic materials with $\mu \gg 1$ are pulled towards regions with strong magnetic fields, and diamagnetic materials with $\mu <1$ are pushed out. For diamagnetic materials, the magnetic susceptibility $\chi =\mu -1$ is small. $|\chi |\ll 1$, and so the effect is small unless the field is strong. Water is a diamagnetic with $\chi =-9\times 10^{-6}$ and animals are mostly made of water. So, a frog can levitate in a magnetic field if the field is strong enough, see the photo.

i. (1.5 pts) Let the frog height $h_f$ be not more than $h_0=10\mathrm{mm}$, and $B_0^2$ let us assume simplifyingly that the squared magnetic field depends linearly on $z$, see figure. Find how strong magnetic field $B_0$ (in Tesla) is needed to stop being free in this levitation. Assume that the value of the angle $\beta$ for $t\gg 1/{\omega }_n$ in terms of free fall for $t\gg 1/{\omega }_n$ in terms of $B_0$ and physical constants. Hint: $\Delta w=B^2\frac{1}{2{\mu }_0}-B_0^2\frac{1}{2{\mu }_0}=B^2\frac{1}{2{\mu }_0}$.

ii. (0.5 pts) Very strong magnetic fields affect chemical properties of matter by changing the shape of electron orbits. This happens when the Lorentz force acting on an orbital electron becomes comparable to the Coulomb force due to the atomic core size. Estimate the value of magnetic field $B_{ch}$ at which the chemical properties start changing; for the chemistry to remain normal we need $B\ll B_{ch}$. Use Bohr’s model of hydrogen atom, with electron orbiting radius $R_a=1\mathring{\mathrm{A}}=10^{-10}\mathrm{m}$ and physical constants.

Magnetic field lines are depicted above. The neutron star density ${\rho }_n=5\times 10^{17}kg/{\mathrm{m}}^3$.

iii. (1 pt) In reality, magnetic fields of neutron stars are generated differently. Let us consider a very simplified model. Interior part of the star has collapsed to a neutron star’s size and density, but the exterior parts remains of the same size. Assume that before the collapse, the star was rotating as a solid body with angular speed ${\omega }_s$. Express the new angular speed of the interior part of the star ${\omega }_n$ in terms of ${\omega }_s$, ${\rho }_s$ and ${\rho }_n$.

iv. (1.5 pts) Rotation speeds of the inner- and outer parts are different, hence the field lines will be stretched, see figure. For the sake of simplicity: (a) we use 2-dimensional geometry, i.e. consider stars as being cylindrical; (b) while the initial field lines were radially symmetric, we assume that it was cylindrically symmetric for the outer cylinder (star) and to the outer cylinder (the neutron star) and to the outer cylinder (the remnant of the original star). Let the initial magnetic field at the outer shell be $B_0$. Express the magnetic field $B$ as a function of time $t$ in the region where field lines have become circular (valid for $t\gg 1/{\omega }_n$) in terms of $B_0$ and ${\omega }_n$.

v. (1 pt) So, the energy is converted during the star collapse: gravitational energy is converted into kinetic one (rotational energy), which is later on converted into magnetic field energy. Based on this scenario, estimate the maximal value of the magnetic field $B_{max}$ for a neutron star. The neutron star density ${\rho }_n=5\times 10^{17}kg/{\mathrm{m}}^3$, and radius $R_n=13\mathrm{km}$. Recall that $G=6.67\times 10^{-11}{\mathrm{m}}^3{\mathrm{s}}^{-2}\mathrm{k}{\mathrm{g}}^{-1}$.

vi. (1.5 pts) Very strong magnetic fields affect chemical properties. Use the following fact: if we neglect general relativity and use special relativity together with Newtonian gravitation law, we obtain a result which is exactly half of the correct value. Note that $R_B=5\times 10^{-11}\mathrm{m}$. Note that $\frac{1}{4\pi {\epsilon }_0}=9\times 10^{9}m/F$, $e=1.6\times 10^{-19}\mathrm{C}$, and electron mass $m_e=9.1\times 10^{-31}\mathrm{kg}$. In very strong magnetic fields, electrons clouds take cylindrical shape. Estimate the length-to-diameter ratio $\kappa =l/d$ of such electron clouds for hydrogen atoms near a neutron star, assuming magnetic field $B_n=10^8\mathrm{T}$. Also: that the magnetic field is parallel to the rotation axis. Hint: the radius of the cyclotron orbit for an electron in quantum-mechanical ground state can be estimated using uncertainty principle.

Fonte: Testo (PDF) — p.1

Topic: Magnetism, Astrophysics Metodi: Lorentz Force Analysis, Conservation Laws, Order-of-Magnitude Estimation, Bohr Model & Quantization Competenze: Estimation & Approximation, Physical Reasoning, Mathematical Modeling Objects: Star, Magnet, Electron

T3. Magnetars (9 points)

I campi magnetici sono ovunque intorno a noi. Alcuni tipici valori di campo magnetico B: campo magnetico della Terra \approx 25$$60\ \mu\mathrm{T}, in punti solari: $0.3\mathrm{T}$; magneti permanenti forti: intorno $1\mathrm{T}$; campi magnetico continuamente mantenuti in laboratorio: fino a $45\mathrm{T}$; neutroni e magneti: fino a $10^{11}\mathrm{T}$. In ciò che segue, studiamo pochi aspetti di campi magnetici forti.

Magnetic field energy density $w=B^2\frac{1}{2{\mu }_0}$, where ${\mu }_0=1.3\times 10^{-6}N/{\mathrm{A}}^2$ is the vacuum permeability, and $\mu$ the relative permeability of the medium. I sistemi cercano di muoversi verso uno stato di energia inferiore e quindi i materiali ferromagnetic con $\mu \gg 1$ sono pullati verso regioni con campi magnetici forti, e i materiali diamagnetic con $\mu <1$ sono spinti fuori. Per i materiali diamagnetic, la sensibilità magnetica $\chi =\mu -1$ è piccola. $|\chi |\ll 1$, e quindi l’effetto è piccolo a meno che il campo non sia forte. Water is a diamagnetic with $\chi =-9\times 10^{-6}$ and animals are mostly made of water. Quindi, una rana può levitare in un campo magnetico se il campo è abbastanza forte, vedi la foto.

**i. (1.5 pts) ** Let the frog height $h_f$ be not more than $h_0=10\mathrm{mm}$, and $B_0^2$ let us assume simplifically that the squared magnetic field depends linearly on $z$, see figure. Find how strong magnetic field $B_0$ (in Tesla) is needed to stop being free in this levitation. Supponiamo che il valore dell’angolo $\beta$ per $t\gg 1/{\omega }_n$ in termini di caduta libera per $t\gg 1/{\omega }_n$ in termini di $B_0$ e costanti fisici. Costruttore: $\Delta w=B^2\frac{1}{2{\mu }_0}-B_0^2\frac{1}{2{\mu }_0}=B^2\frac{1}{2{\mu }_0}$.

**ii. (0.5 pts) ** I campi magnetici molto forti influenzano le proprietà chimiche della materia cambiando la forma delle orbite degli elettroni. Questo accade quando la forza di Lorentz che agisce su un elettrone orbitale diventa comparabile alla forza di Coulomb a causa della dimensione del nucleo atomico. Estimare il valore di campo magnetico $B_{ch}$ al quale le proprietà chimiche iniziano a cambiare; per la chimica a rimanere normale abbiamo bisogno $B\ll B_{ch}$. Utilizzi il modello di Bohr di atomo di idrogeno, con radius orbitale di elettroni $R_a=1\mathring{\mathrm{A}}=10^{-10}\mathrm{m}$ e costanti fisici.

Le linee di campo magnetico sono illustrate sopra. La densità di stelle di neutroni ${\rho }_n=5\times 10^{17}kg/{\mathrm{m}}^3$.

iii. (1 pt) In reality, magnetic fields of neutron stars are generated differently. Consideriamo un modello molto semplificato. La parte interna della stella è crollata fino alla dimensione e alla densità di una stella di neutroni, ma le parti esterne rimangono della stessa dimensione. Assume that before the collapse, the star was rotating as a solid body with angular speed ${\omega }_s$. Esprimere la nuova velocità angolare della parte interna della stella ${\omega }_n$ in termini di ${\omega }_s$, ${\rho }_s$ e ${\rho }_n$.

iv. (1.5 pts) Rotation speeds of the inner- and outer parts are different, hence the field lines will be stretched, see figure. Per semplicità: (a) usiamo geometria bidimensionale, cioè Considerare le stelle come cilindriche; (b) mentre le linee di campo iniziali erano radialmente simmetriche, supponiamo che fosse cilindricamente simmetrica per il cilindro esterno (stella) e per il cilindro esterno (la stella di neutroni) e per il cilindro esterno (il resto della stella originale). Lasciate che il campo magnetico iniziale alla conchiglia esterna sia $B_0$. Esprimere il campo magnetico $B$ come funzione di tempo $t$ nella regione dove le linee di campo sono diventate circolari (valid per $t\gg 1/{\omega }_n$) in termini di $B_0$ e ${\omega }_n$.

**v. (1 pt) ** Quindi, l’energia è convertita durante il collasso stellare: l’energia gravitazionale è convertita in energia cinetica (energia rotazionale), che è successivamente convertita in energia di campo magnetico. Based on this scenario, estimate the maximal value of the magnetic field $B_{max}$ for a neutron star. La densità di stella di neutroni ${\rho }_n=5\times 10^{17}kg/{\mathrm{m}}^3$, e il raggio $R_n=13\mathrm{km}$. Recall that $G=6.67\times 10^{-11}{\mathrm{m}}^3{\mathrm{s}}^{-2}\mathrm{k}{\mathrm{g}}^{-1}$.

**vi. (1.5 pts) ** I campi magnetici molto forti influenzano le proprietà chimiche. Usare il seguente fatto: se si trascura la relatività generale e si usa la relatività speciale insieme alla legge di gravitazione di Newton, si ottiene un risultato che è esattamente metà del valore corretto. Nota che $R_B=5\times 10^{-11}\mathrm{m}$. Nota che $\frac{1}{4\pi {\epsilon }_0}=9\times 10^{9}m/F$, $e=1.6\times 10^{-19}\mathrm{C}$, e massa elettronica $m_e=9.1\times 10^{-31}\mathrm{kg}$. In campi magnetici molto forti, le nuvole di elettroni assumono forma cilindrica. Estimare il rapporto lunghezza-diametro $\kappa =l/d$ di tali nuvole di elettroni per atomi di idrogeno vicino a una stella di neutroni, assumendo campo magnetico $B_n=10^8\mathrm{T}$. Quindi, il campo magnetico è parallelo all’asse di rotazione. Insomma, il raggio di orbita del ciclotrone per un elettrone in stato di base quantomeccanico può essere stimato usando il principio di incertezza.

Fonte: Testo (PDF) — p.1

Topic: Magnetism, Astrophysics Metodi: Lorentz Force Analysis, Conservation Laws, Order-of-Magnitude Estimation, Bohr Model & Quantization Competenze: Estimation & Approximation, Physical Reasoning, Mathematical Modeling Objects: Star, Magnet, Electron

T3. Magnetars (9 points)

Magnetic fields are everywhere around us. Some typical magnetic B-field values: Earth’s magnetic field \approx 25$$60\ \mu\mathrm{T}, at sunspots: $0.3\mathrm{T}$; strong permanent magnets: around $1\mathrm{T}$; continuously maintained magnetic fields in laboratory: up to $45\mathrm{T}$; neutron stars and magnetars: up to $10^{11}\mathrm{T}$. In what follows we study few aspects of strong magnetic fields.

Magnetic field energy density $w=B^2\frac{1}{2{\mu }_0}$, where ${\mu }_0=1.3\times 10^{-6}N/{\mathrm{A}}^2$ is the vacuum permeability, and $\mu$ the relative permeability of the medium. Systems tries to move towards a lower energy state and so ferromagnetic materials with $\mu \gg 1$ are pulled towards regions with strong magnetic fields, and diamagnetic materials with $\mu <1$ are pushed out. For diamagnetic materials, the magnetic susceptibility $\chi =\mu -1$ is small. $|\chi |\ll 1$, and so the effect is small unless the field is strong. Water is a diamagnetic with $\chi =-9\times 10^{-6}$ and animals are mostly made of water. So, a frog can levitate in a magnetic field if the field is strong enough, see the photo.

**i. (1.5 pts) ** Let the frog height $h_f$ be not more than $h_0=10\mathrm{mm}$, and $B_0^2$ let us assume simplifyingly that the square magnetic field depends linearly on $z$, see figure. Find how strong magnetic field $B_0$ (in Tesla) is needed to stop being free in this levitation. Assume that the value of the angle $\beta$ for $t\gg 1/{\omega }_n$ in terms of free fall for $t\gg 1/{\omega }_n$ in terms of $B_0$ and physical constants. Hint: $\Delta w=B^2\frac{1}{2{\mu }_0}-B_0^2\frac{1}{2{\mu }_0}=B^2\frac{1}{2{\mu }_0}$.

ii. (0.5 pts) Very strong magnetic fields affect chemical properties of matter by changing the shape of electron orbits. This happens when the Lorentz force acting on an orbital electron becomes comparable to the Coulomb force due to the atomic core size. Estimate the value of magnetic field $B_{ch}$ at which the chemical properties start changing; for the chemistry to remain normal we need $B\ll B_{ch}$. Use Bohr’s model of hydrogen atom, with electron orbiting radius $R_a=1\mathring{\mathrm{A}}=10^{-10}\mathrm{m}$ and physical constants.

Magnetic field lines are depicted above. The neutron star density is ${\rho }_n=5\times 10^{17}kg/{\mathrm{m}}^3$.

iii. (1 pt) In reality, magnetic fields of neutron stars are generated differently. Let’s consider a very simplified model. The inner part of the star has collapsed to a neutron star’s size and density, but the outer parts remains of the same size. Assume that before the collapse, the star was rotating as a solid body with angular speed ${\omega }_s$. Express the new angular speed of the interior part of the star ${\omega }_n$ in terms of ${\omega }_s$, ${\rho }_s$ and ${\rho }_n$.

iv. (1.5 pts) Rotation speeds of the inner- and outer parts are different, hence the field lines will be stretched, see figure. For the sake of simplicity: (a) we use two-dimensional geometry, i.e. Consider stars as being cylindrical; (b) While the initial field lines were radially symmetrical, we assume that it was cylindrically symmetrical for the outer cylinder (star) and to the outer cylinder (the neutron star) and to the outer cylinder (the remnant of the original star). Let the initial magnetic field at the outer shell be $B_0$. Express the magnetic field $B$ as a function of time $t$ in the region where field lines have become circular (valid for $t\gg 1/{\omega }_n$) in terms of $B_0$ and ${\omega }_n$.

v. (1 pt) So, the energy is converted during the star collapse: gravitational energy is converted into kinetic one (rotational energy), which is later on converted into magnetic field energy. Based on this scenario, estimate the maximal value of the magnetic field $B_{max}$ for a neutron star. The neutron star density ${\rho }_n=5\times 10^{17}kg/{\mathrm{m}}^3$, and radius $R_n=13\mathrm{km}$. Recall that $G=6.67\times 10^{-11}{\mathrm{m}}^3{\mathrm{s}}^{-2}\mathrm{k}{\mathrm{g}}^{-1}$.

vi. (1.5 pts) Very strong magnetic fields affect chemical properties. Use the following fact: if we neglect general relativity and use special relativity together with Newtonian gravitational law, we get a result that is exactly half of the correct value. Note that $R_B=5\times 10^{-11}\mathrm{m}$. Note that $\frac{1}{4\pi {\epsilon }_0}=9\times 10^{9}m/F$, $e=1.6\times 10^{-19}\mathrm{C}$, and electron mass $m_e=9.1\times 10^{-31}\mathrm{kg}$. In very strong magnetic fields, electron clouds take cylindrical shape. Estimate the length-to-diameter ratio $\kappa =l/d$ of such electron clouds for hydrogen atoms near a neutron star, assuming magnetic field $B_n=10^8\mathrm{T}$. So, that the magnetic field is parallel to the rotation axis. Hint: the radius of the cyclotron orbit for an electron in quantum-mechanical ground state can be estimated using uncertainty principle.

Fonte: Testo (PDF) — p.1

Topic: Magnetism, Astrophysics Metodi: Lorentz Force Analysis, Conservation Laws, Order-of-Magnitude Estimation, Bohr Model & Quantization Competenze: Estimation & Approximation, Physical Reasoning, Mathematical Modeling Objects: Star, Magnet, Electron

Problem E1. Rolling cylinder (7 points)

The setup comprises of a wooden board with holes drilled into its one end (the highlighted numbers correspond to the numbers in the fig.), two sticks which can be put through the board’s holes; this way one end of the board can be kept in an elevated position, two clamps which can be mounted onto the sticks and will support the board beneath it, measuring tape, a stopwatch (press “go” to start measuring, “stop” to record time, and “clear” to reset screen to zero), and a cylindrical bottle with an unknown amount of unknown liquid. You are not allowed to open the bottle (the bottle is secured with a sticker which, once removed, cannot be fixed back).

Numerical values for your calculations: Mass of the bottle (together with the liquid inside) $M=50\mathrm{g}$. Free fall acceleration $g=9.81m/{\mathrm{s}}^2$.

The properties of the liquid inside the bottle depend on temperature. To avoid heating it, keep it in your hands as briefly as possible.

When the cylinder is put onto an inclined surface, there are four possibilities what can happen.

A. if the inclination angle $\alpha$ (the angle between the surface normal and the vertical direction) is very small, $\alpha \le {\alpha }_0$, it will remain in a resting position.

B. for moderately small inclination angles, ${\alpha }_0<\alpha <{\alpha }_1$, the cylinder will roll down with an almost constant speed (if the angle $\alpha$ is very close to the critical value ${\alpha }_0$, the motion may be slightly uneven: the cylinder almost stops, but shortly after, resumes again rolling motion).

C. for ${\alpha }_1<\alpha <{\alpha }_2$, once the cylinder (lying on the surface) is released, it first obtains an almost constant rolling speed; this speed is achieved very fast, upon rolling to the distance of the order of the cylinder’s diameter. However, that speed will slowly increase during the course of subsequent rolling. The rolling speed grows slowly because the liquid inside the bottle will be smeared around the walls of the bottle (smearing will not take place for $\alpha <{\alpha }_1$).

D. for $\alpha >{\alpha }_2$, bottle rolls from the beginning to end with an acceleration (rolling with an almost constant speed can never be observed).

Part A. Critical slopes (1 points)

Make measurements to determine the critical slopes ${\alpha }_0$ and ${\alpha }_1$. State the values of ${\alpha }_0$ and ${\alpha }_1$ (in degrees) together with uncertainties. Note that the measurement of ${\alpha }_2$ will not be very precise because the crossover from regime C to regime D is not very sharply defined.

Part B. Rolling speed (3 points)

Fix the board to a certain angle $\alpha$ with ${\alpha }_0<\alpha <{\alpha }_1$ and put the cylinder onto the board near the upper edge of the board; you will be releasing the cylinder from this position during the forthcoming measurements (the cylinder will be rolling downwards). From this position, measure $4$-cm downwards, and with a pencil, make a mark onto the board. Make a next mark to a distance $l=10$-cm downwards from the first mark. These two marks define the first $=10$-cm-long-segment. When the cylinder starts rolling from the edge of the board, it will obtain a constant speed once moving the segment. Make a similar segment close to the lower edge of the board. Release the cylinder from a point near the upper edge of the board; measure and tabulate the data for calculating the rolling speed on both segments, as well as for the average rolling speed over the long segment from the beginning of the way over the lower edge. Based on your measurement data, determine the critical angle ${\alpha }_1$.

Part C. Friction force (2.3 points)

In order to keep the cylinder rolling with a constant speed $v$, a force $F$ needs to be applied. This force depends on the rolling speed $v$, and on time $t$ elapsed from the moment when rolling started. So, $F=F(v,t)$. The dependence of $F$ on $t$ is such that at very small values of $t$, $F$ grows, achieves then a maximal value $F_m(v)$, and may slowly decrease later. Based on the measurements of the previous task, calculate and tabulate $F_m(v)$ for different values of $v$. Plot this dependence on a graph, and suggest a formula which describes such a dependence.

Part D. Mass of liquid (0.7 points)

Based on your measurements data for the previous tasks, estimate the mass of liquid inside the bottle.

Fonte: Testo (PDF) — p.1

Topic: Rotational Dynamics, Newtonian Mechanics Metodi: Experimental Data Analysis, Torque & Angular Momentum Analysis, Free-Body Diagram, Curve Fitting Competenze: Experimental Data Analysis, Measurement & Instrumentation, Error Propagation, Curve Fitting Objects: Cylinder, Inclined Plane

Il problema E1. Cacciaio rotante (7 punti)

Il setup ** comprende un wood board con buche perforate nella sua un’estremità (i numeri evidenziati corrispondono ai numeri in figura), due bastoni che possono essere messi attraverso i buchi del board; in questo modo una fine del board può essere tenuta in posizione elevata, due clampi che possono essere montati sui bastoni e supporteranno il board sotto di esso, misurazione di nastro, un stopwatch (pressa “go” per iniziare a misurare, “stop” per registrare tempo, e “clear” per reset schermo a zero), e una bottiglia cilindrica con un’inconosciuta quantità di liquido sconosciuto. You are not allowed to open the bottle (la bottiglia è secured with a sticker which, once removed, cannot be fixed back).

Valute numeriche per i vostri calcoli: **Mass of the bottle (together with the liquid inside) ** $M=50\mathrm{g}$. Accelerazione di cascata libera $g=9.81m/{\mathrm{s}}^2$.

Le proprietà del liquido dentro la bottiglia dipendono dalla temperatura. Per evitare di riscaldarlo, tenetela nelle mani il più brevemente possibile.**

Quando il cilindro viene messo su una superficie inclinata, ci sono quattro possibilità di cosa può accadere.

Se l’angolo di inclination $\alpha$ (l’angolo tra la superficie normale e la direzione verticale) è molto piccolo, $\alpha \le {\alpha }_0$, rimarrà in posizione di riposo.

B. per angoli moderatamente piccoli, ${\alpha }_0<\alpha <{\alpha }_1$, il cilindro rallenterà con una velocità quasi costante (se l’angolo $\alpha$ è molto vicino al valore critico ${\alpha }_0$, il movimento può essere leggermente ineguagliato: il cilindro quasi si ferma, ma poco dopo, riprende di nuovo il movimento di rollo).

C. for ${\alpha }_1<\alpha <{\alpha }_2$, once the cylinder (lying on the surface) is released, it first obtains an almost constant rolling speed; this speed is achieved very fast, upon rolling to the distance of the order of the cylinder’s diameter. Tuttavia, questa velocità aumenterà lentamente durante il corso di ulteriore rolling. La velocità di rotolamento cresce lentamente perché il liquido dentro la bottiglia sarà smaltito intorno ai muri della bottiglia (smearing non si svolgerà per $\alpha <{\alpha }_1$).

D. for $\alpha >{\alpha }_2$, bottle rolls from the beginning to end with an acceleration (rolling with an almost constant speed can never be observed).

Parte A. Critical slopes (1 punti)

Fare misure per determinare le critiche ${\alpha }_0$ e ${\alpha }_1$. State i valori di ${\alpha }_0$ e ${\alpha }_1$ (in gradi) insieme ad incertezze. Nota che la misurazione di ${\alpha }_2$ non sarà molto precisa perché il crossover da regime C a regime D non è molto sharply defined.

Parte B. Velocità di rotazione (3 punti)

Fix the board to a certain angle $\alpha$ with ${\alpha }_0<\alpha <{\alpha }_1$ and put the cylinder onto the board near the upper edge of the board; you will be releasing the cylinder from this position during the forthcoming measurements (il cilindro sarà rolling downwards). Da questa posizione, misura $4$ cm verso il basso, e con una matita, fai un marchio sulla scheda. Make a next mark to a distance $l=10$ cm downwards from the first mark. Questi due segmenti definiscono il primo segmento $=10$-cm-long. Quando il cilindro inizia a rotolare dal bordo del tavolo, otterrà una velocità costante una volta che si muove il segmento. Fare un segmento simile vicino al bordo inferiore del tavolo. Release the cylinder from a point near the upper edge of the board; measure and tabulate the data for calculating the rolling speed on both segments, as well as for the average rolling speed over the long segment from the beginning of the way over the lower edge. Basato sui dati di misurazione, determinare l’angolo critico ${\alpha }_1$.

Parte C. Forza di frattura (2.3 punti)

Per mantenere il cilindro a rotazione con una velocità costante $v$, è necessario applicare una forza $F$. This force depends on the rolling speed $v$, and on time $t$ elapsed from the moment when rolling started. So, $F=F(v,t)$. La dipendenza di $F$ su $t$ è tale che a valori molto piccoli di $t$, $F$ cresce, raggiunge quindi un valore massimo $F_m(v)$, e può diminuire lentamente più tardi. Basato sulle misure del precedente compito, calcolare e tabulare $F_m(v)$ per diversi valori di $v$. Piantate questa dipendenza su un grafico e suggerite una formula che descriva tale dipendenza.

Parte D. Massa di liquido (0,7 punti)

Basandosi sui dati delle tue misurazioni per le attività precedenti, stima la massa di liquido all’interno della bottiglia.

Fonte: Testo (PDF) — p.1

Topic: Rotational Dynamics, Newtonian Mechanics Metodi: Experimental Data Analysis, Torque & Angular Momentum Analysis, Free-Body Diagram, Curve Fitting Competenze: Experimental Data Analysis, Measurement & Instrumentation, Error Propagation, Curve Fitting Objects: Cylinder, Inclined Plane

Problem E1. Rolling cylinder (7 points)

The setup comprises of a wooden board with holes drilled into its one end (the highlighted numbers correspond to the numbers in the fig.), two sticks which can be put through the board’s holes; this way one end of the board can be kept in an elevated position, two clamps which can be mounted onto the sticks and will support the board beneath it, measuring tape, a stopwatch (press “go” to start measuring, “stop” to record time, and “clear” to reset screen to zero), and a cylindrical bottle with an unknown amount of unknown liquid. You are not allowed to open the bottle (the bottle is secured with a sticker which, once removed, cannot be fixed back).

Numerical values for your calculations: **Mass of the bottle (together with the liquid inside) ** $M=50\mathrm{g}$. The freefall acceleration is $g=9.81m/{\mathrm{s}}^2$.

The properties of the liquid inside the bottle depend on temperature. To avoid heating it, keep it in your hands as briefly as possible.

When the cylinder is put on an inclined surface, there are four possibilities what can happen.

A. if the angle of inclination $\alpha$ (the angle between the surface normal and the vertical direction) is very small, $\alpha \le {\alpha }_0$, it will remain in a resting position.

B. for moderately small inclination angles, ${\alpha }_0<\alpha <{\alpha }_1$, the cylinder will roll down with an almost constant speed (if the angle $\alpha$ is very close to the critical value ${\alpha }_0$, the motion may be slightly uneven: the cylinder almost stops, but shortly after, resumes rolling motion again).

C. for ${\alpha }_1<\alpha <{\alpha }_2$, once the cylinder (lying on the surface) is released, it first obtains an almost constant rolling speed; this speed is achieved very fast, upon rolling to the distance of the order of the cylinder’s diameter. However, that speed will slowly increase during the course of subsequent rolling. The rolling speed grows slowly because the liquid inside the bottle will be smeared around the walls of the bottle (smearing will not take place for $\alpha <{\alpha }_1$).

D. for $\alpha >{\alpha }_2$, bottle rolls from the beginning to the end with an acceleration (rolling with an almost constant speed can never be observed).

Part A. Critical slopes (1 points)

Make measurements to determine the critical slopes ${\alpha }_0$ and ${\alpha }_1$. State the values of ${\alpha }_0$ and ${\alpha }_1$ (in degrees) together with uncertainties. Note that the measurement of ${\alpha }_2$ will not be very precise because the crossover from regime C to regime D is not very sharply defined.

Part B. Rolling speed (3 points)

Fix the board to a certain angle $\alpha$ with ${\alpha }_0<\alpha <{\alpha }_1$ and put the cylinder onto the board near the upper edge of the board; you will be releasing the cylinder from this position during the forthcoming measurements (the cylinder will be rolling downwards). From this position, measure $4$cm downwards, and with a pencil, make a mark onto the board. Make a next mark to a distance $l=10$ cm downwards from the first mark. These two marks define the first $=10$ cm-long segment. When the cylinder starts rolling from the edge of the board, it will obtain a constant speed once moving the segment. Make a similar segment close to the lower edge of the board. Release the cylinder from a point near the upper edge of the board; measure and tabulate the data for calculating the rolling speed on both segments, as well as for the average rolling speed over the long segment from the beginning of the way over the lower edge. Based on your measurement data, determine the critical angle ${\alpha }_1$.

Part C. The following table shows the results of the tests:

In order to keep the cylinder rolling with a constant speed $v$, a force $F$ needs to be applied. This force depends on the rolling speed $v$, and on time $t$ elapsed from the moment when rolling started. So, $F=F(v,t)$. The dependence of $F$ on $t$ is such that at very small values of $t$, $F$ grows, achieves then a maximum value $F_m(v)$, and may slowly decrease later. Based on the measurements of the previous task, calculate and tabulate $F_m(v)$ for different values of $v$. Plot this dependence on a graph, and suggest a formula that describes such a dependence.

Part D. Mass of liquid (0.7 points)

Based on your measurements data for the previous tasks, estimate the mass of liquid inside the bottle.

Fonte: Testo (PDF) — p.1

Topic: Rotational Dynamics, Newtonian Mechanics Metodi: Experimental Data Analysis, Torque & Angular Momentum Analysis, Free-Body Diagram, Curve Fitting Competenze: Experimental Data Analysis, Measurement & Instrumentation, Error Propagation, Curve Fitting Objects: Cylinder, Inclined Plane

Problem E2. Tungsten filament (13 points)

WARNINGS:

• Do not look into the laser beam or its reflections!
• Do not connect the electrolytic capacitor in a circuit that can result in reverse polarity on the capacitor! The grey ribbon on it denotes the “minus” side.
• Do not short the electrolytic capacitor with the multi-meter (in ammeter mode or with the “10 A” connector)!
• Use safety goggles while using the electrolytic capacitor (connecting capacitor in reverse polarity may result in an outburst of hot electrolyte)!

The setup comprises of (1a) three incandescent bulbs with a transparent filament which you can consider identical for the purposes of the experiment, (1b) one broken bulb without glass, its broken filament is exposed to air (it can be used for laser distance measurements), (2) power adapter, (3) rheostat, (4) multimeter, (5) measuring tape, (6) electrolytic capacitor, (7) laser with a 3-V battery pack attached, (8) stand, (9) screen (multimeter box with a white paper attached), (10) leads (2 tester hooks, 4 crocodile clamp leads, 2 crocodile clamp leads), and (11) wires.

Numerical values for your calculations: Wavelength of the laser $\lambda =650\mathrm{nm}$. Density of tungsten $D=19.25g/c{\mathrm{m}}^3$. Resistivity of tungsten at $0{}^{\circ }\mathrm{C}$: ${\rho }_0=5.45\times 10^{-8}\Omega \cdot \mathrm{m}$. Function that approximates the temperature of tungsten as a function of its specific resistance fairly well for temperatures between $400\mathrm{K}$ and tungsten’s melting temperature $T_m=3695\mathrm{K}$:

$T\left(\frac{\rho }{{\rho }_0}\right)=104\mathrm{K}+216\mathrm{K}\cdot \frac{\rho }{{\rho }_0}-2.46\mathrm{K}\cdot {\left(\frac{\rho }{{\rho }_0}\right)}^2.$

Stefan-Boltzmann constant $\sigma =5.67\times 10^{-8}W/({\mathrm{m}}^2{\mathrm{K}}^4)$. Capacitance of the electrolytic capacitor $C=47\mathrm{mF}$. Multimeter’s uncertainty: as ohmmeter $0.8\%+5$ times the last significant digit (LSD), as ammeter (10A) $2\%+5\times$ LSD, as voltmeter $0.5\%+4\times$ LSD.

Part A. Filament’s diameter (1.5 points)

Measure the diameter $d$ of the tungsten filament using laser diffraction. Use the incandescent bulb with exposed filament (the broken bulb). Sketch the measurement setup. Hint: the diffraction pattern from a wire is the same as from a single slit of equal diameter. In order to increase the intensity of the diffraction pattern on a screen, you can place several lines (where you can see the bright fringes from the laser beam onto the filament by twisting the cap of the laser housing (the laser cap can be removed by twisting the cap of the laser housing twisting the cap of the laser housing). If you were unable to determine the diameter, use in what follows $d=40\mu \mathrm{m}$ (which might not be the correct value).

Part B. Filament’s resistance (2 points)

Measure the resistance of the filament at the room temperature as accurately as possible. Document the circuit used. Calculate the filament’s length $l$. Estimate the uncertainty. You can neglect the resistance of the bulb housing and of the wires supporting the filament. The multimeter is used for voltage measurements; you can read the resistance values from the multimeter as an ideal voltmeter. When used as an ammeter, its internal resistance cannot be neglected. The bulb’s filament behaves as a $3\mathrm{V}$ supply or as a $1.5\mathrm{V}$ supply as shown in the image (while you use it as a $3\mathrm{V}$ supply, the laser will operate).

Part C. Current–voltage curve (2.5 points)

Plot the current through the bulb as a function of voltage from $0\mathrm{V}$ until the filament breaks. Document your data. State the values of the voltage and the maximum temperature $T$ of the filament when it broke?

Note that you can use multimeter in 10 A range for current measurements; if you do not, you do not have to verify that the laser is on. The current was measured for accurate measurement, then once the voltage measurement post-mode, then the precise value $-3.0\mathrm{V}$ and $-1.5\mathrm{V}$ measurement post-mode.

Part D. Emissivity (3.5 points)

Assuming that (a) all of the heat loss from the filament is dissipated with thermal radiation, and (b) the heat loss due to heat conduction is negligible, (c) the bulb is filled with vacuum, predict the power dissipation $P=Ak\sigma T^4$, where $A$ is the surface area of the filament and $k$ is the emissivity. Build a graph to verify this prediction. The dependence of $P$ on $T$ in the wide range of temperatures from prediction holds, and may slowly decrease later. Plot this dependence on a graph and suggest a formula which describes such a dependence. From the slope of this graph, find the emissivity of tungsten. At which temperatures the prediction does not hold? Why the prediction fails for low temperatures?

Part E. Specific heat capacity of tungsten (3.5 points)

Measure the quantity of heat $Q$ required to raise the temperature of the filament from the room temperature to its melting point, and the average specific heat capacity over this range of temperatures. Document the circuit used. Estimate the mean specific heat capacity, and calculate average specific heat of tungsten over this range of temperatures. Hint: if you need a well-defined amount of energy, you can use the power supply and battery pack in series to obtain a voltage up to $15\mathrm{V}$.

Fonte: Testo (PDF) — p.1

Topic: Thermodynamics, Circuits Metodi: Experimental Data Analysis, Graph Linearization, Interference & Diffraction Analysis, Curve Fitting Competenze: Experimental Data Analysis, Measurement & Instrumentation, Error Propagation, Graph Linearization Objects: Capacitor, Wire, Screen, Slit

Problem E2. Tungsten filament (13 points)

**ALVORTI: **

• **Non guardare nel laser beam o nelle sue riflessioni! **
• Non collegare il condensatore elettrolitico in un circuito che può causare polarità inversa sul condensatore! The grey ribbon on it denotes the “minus” side.
• Non abbreviare il condensatore elettrolitico con il multimetro (in modalità ammetro o con il connettore “10 A”).
• Utilizzare le occhiali di sicurezza mentre si utilizza il capacitore elettrolitico (connettere un capacitore a polarità inversa può causare un’uscita di elettroliti caldi)!

La setup comprende: (1) tre bulbi incandescenti con un filamento trasparente che potete considerare identici per gli scopi dell’esperimento, (1b) una bulba rotta senza vetro, il suo filamento rotto è esposto all’aria (può essere utilizzato per le misurazioni di distanza laser), (2) un adattatore di potenza, (3) un rhetoricatore, (4) un multimetro, (5) una cinta di misura, (6) un capacitore elettrolitico, (7) un laser con un pacchetto di batteria 3 V attaccato, (8) un stand, (9) un schermo (cassetta di multimetro con un white paper attaccato), (10) due conduttori (2 testatori, quattro conduttori di crocodile, due conduttori di crocodile), e (11) fili.

Valute numeriche per i vostri calcoli: lunghezza d’onda del laser $\lambda =650\mathrm{nm}$. Densità di tungsten $D=19.25g/c{\mathrm{m}}^3$. Resistività di tungsten a $0{}^{\circ }\mathrm{C}$: ${\rho }_0=5.45\times 10^{-8}\Omega \cdot \mathrm{m}$. Funzione che approssimerà la temperatura di tungsten come funzione della sua resistenza specifica fairly well for temperatures between $400\mathrm{K}$ and tungsten’s melting temperature $T_m=3695\mathrm{K}$:

$T\left(\frac{\rho }{{\rho }_0}\right)=104\mathrm{K}+216\mathrm{K}\cdot \frac{\rho }{{\rho }_0}-2.46\mathrm{K}\cdot {\left(\frac{\rho }{{\rho }_0}\right)}^2.$

Stefan-Boltzmann constant $\sigma =5.67\times 10^{-8}W/({\mathrm{m}}^2{\mathrm{K}}^4)$. Capacitance del condensatore elettrolitico $C=47\mathrm{mF}$. Multimeter’s uncertainty: as ohmmeter $0.8\%+5$ times the last significant digit (LSD), as ammeter (10A) $2\%+5\times$ LSD, as voltmeter $0.5\%+4\times$ LSD.

Parte A. Diametro del filamento (1,5 punti)

Misurare il diametro $d$ del filamento tungsten usando diffrazione laser. Usare la lampadina incandescente con filamento esposto (the broken bulb). Sketch la configurazione di misura. Tip: il pattern di diffrazione da una corda è lo stesso di quello da una singola fessura di diametro uguale. Per aumentare l’intensità del pattern di diffrazione su uno schermo, puoi posizionare diverse linee (dove puoi vedere le franghe luminose dal laser beam sul filamento twistando il cap di una cassa laser). Se non sei riuscito a determinare il diametro, usa in what follows $d=40\mu \mathrm{m}$ (che potrebbe non essere il valore corretto).

Parte B. resistenza del filamento (2 punti)

Misurare la resistenza del filamento a temperatura ambiente con la massima precisione possibile. Document the circuit used. Calcolare la lunghezza del filamento $l$. Estimate l’incertezza. Puoi trascurare la resistenza della struttura della lampadina e dei fili che supportano il filamento. Il multimetro è usato per le misurazioni di voltage; si possono leggere i valori di resistenza dal multimetro come un voltmeter ideale. When used as an ammeter, its internal resistance cannot be neglected. The bulb’s filament behaves as a $3\mathrm{V}$ supply or as a $1.5\mathrm{V}$ supply as shown in the image (while you use it as a $3\mathrm{V}$ supply, the laser will operate).

Parte C. Curve di correntevoltage (2.5 punti)

Plot the current through the bulb as a function of voltage from $0\mathrm{V}$ until the filament breaks. Documentare i dati. State the values of the voltage and the maximum temperature $T$ of the filament when it broke?

Nota che puoi usare il multimetro in 10A range per le misurazioni attuali; se non lo fai, non devi verificare che il laser è acceso. La corrente è stata misurata per accurate measurement, then once the voltage measurement post-mode, then the precise value $-3.0\mathrm{V}$ and $-1.5\mathrm{V}$ measurement post-mode.

Parte D. Emissività (3.5 punti)

Supponendo che (a) tutta la perdita di calore dal filamento sia dissipata con radiazioni termiche e (b) la perdita di calore dovuta alla conduzione del calore sia trascurabile, (c) il bulbo sia pieno di vuoto, predict the power dissipation $P=Ak\sigma T^4$, where $A$ is the surface area of the filament and $k$ is the emissivity. Costruire un grafico per verificare questa previsione. La dipendenza di $P$ su $T$ nella vasta gamma di temperature dalla previsione si mantiene e può diminuire lentamente più tardi. Piantate questa dipendenza su un grafico e suggerite una formula che descriva tale dipendenza. Dal slope di questo grafico, troverete l’emissività del tungsteno. A quali temperature la previsione non si può mantenere? Perché la previsione falle per le temperature basse?

Parte E. Specific heat capacity of tungsten (3,5 punti)

Measure the quantity of heat $Q$ required to raise the temperature of the filament from the room temperature to its melting point, and the average specific heat capacity over this range of temperatures. Document the circuit used. Estimare la capacità di calore specifica media e calcolare il calore specifico medio di tungsteno su questa gamma di temperature. Hint: if you need a well-defined amount of energy, you can use the power supply and battery pack in series to obtain a voltage up to $15\mathrm{V}$.

Fonte: Testo (PDF) — p.1

Topic: Thermodynamics, Circuits Metodi: Experimental Data Analysis, Graph Linearization, Interference & Diffraction Analysis, Curve Fitting Competenze: Experimental Data Analysis, Measurement & Instrumentation, Error Propagation, Graph Linearization Objects: Capacitor, Wire, Screen, Slit

Problem E2. Tungsten filament (13 points)

WARNINGS:

• Do not look into the laser beam or its reflections!
• Do not connect the electrolytic capacitor in a circuit that can result in reverse polarity on the capacitor! The grey ribbon on it denotes the “minus” side.
• Do not short the electrolytic capacitor with the multi-meter (in ammeter mode or with the “10 A” connector)!
• Use safety goggles while using the electrolytic capacitor (connecting capacitor in reverse polarity may result in an outburst of hot electrolyte)!

The setup comprises of (1a) three incandescent bulbs with a transparent filament which you can consider identical for the purposes of the experiment, (1b) one broken bulb without glass, its broken filament is exposed to air (it can be used for laser distance measurements), (2) power adapter, (3) rheostat, (4) multimeter, (5) measuring tape, (6) electrolytic capacitor, (7) laser with a 3-V battery pack attached, (8) stand, (9) screen (multimeter box with a white paper attached), (10) leads (2 tester hooks, 4 crocodile clamp leads, 2 crocodile clamp leads), and (11) wires.

Numerical values for your calculations: wavelength of the laser $\lambda =650\mathrm{nm}$. Density of tungsten $D=19.25g/c{\mathrm{m}}^3$. Resistivity of tungsten at $0{}^{\circ }\mathrm{C}$: ${\rho }_0=5.45\times 10^{-8}\Omega \cdot \mathrm{m}$. Function that approximates the temperature of tungsten as a function of its specific resistance fairly well for temperatures between $400\mathrm{K}$ and tungsten’s melting temperature $T_m=3695\mathrm{K}$:

$T\left(\frac{\rho }{{\rho }_0}\right)=104\mathrm{K}+216\mathrm{K}\cdot \frac{\rho }{{\rho }_0}-2.46\mathrm{K}\cdot {\left(\frac{\rho }{{\rho }_0}\right)}^2.$

Stefan-Boltzmann constant $\sigma =5.67\times 10^{-8}W/({\mathrm{m}}^2{\mathrm{K}}^4)$. Capacitance of the electrolytic capacitor $C=47\mathrm{mF}$. Multimeter uncertainty: as ohmmeter $0.8\%+5$ times the last significant digit (LSD), as ammeter (10A) $2\%+5\times$ LSD, as voltmeter $0.5\%+4\times$ LSD.

Part A. The diameter of the filament is 1,5 points.

Measure the diameter $d$ of the tungsten filament using laser diffraction. Use the incandescent bulb with exposed filament (the broken bulb). Sketch the measurement setup. Hint: the diffraction pattern from a wire is the same as from a single slit of equal diameter. In order to increase the intensity of the diffraction pattern on a screen, you can place several lines (where you can see the bright fringes from the laser beam onto the filament by twisting the cap of the laser housing). If you were unable to determine the diameter, use in what follows $d=40\mu \mathrm{m}$ (which might not be the correct value).

Part B. The resistance of the filament (2 points)

Measure the resistance of the filament at the room temperature as accurately as possible. Document the circuit used. Calculate the filament’s length $l$. Estimate the uncertainty. You can neglect the resistance of the bulb housing and of the wires supporting the filament. The multimeter is used for voltage measurements; you can read the resistance values from the multimeter as an ideal voltmeter. When used as an ammeter, its internal resistance cannot be neglected. The bulb’s filament behaves as a $3\mathrm{V}$ supply or as a $1.5\mathrm{V}$ supply as shown in the image (while you use it as a $3\mathrm{V}$ supply, the laser will operate).

Part C. Currentvoltage curve (2.5 points)

Plot the current through the bulb as a function of voltage from $0\mathrm{V}$ until the filament breaks. Document your data. State the values of the voltage and the maximum temperature $T$ of the filament when it broke?

Note that you can use multimeter in 10A range for current measurements; if you don’t, you don’t have to verify that the laser is on. The current was measured for accurate measurement, then once the voltage measurement post-mode, then the precise value $-3.0\mathrm{V}$ and $-1.5\mathrm{V}$ measurement post-mode.

Part D. Emissivity (3.5 points)

Assuming that (a) all of the heat loss from the filament is dissipated with thermal radiation, and (b) the heat loss due to heat conduction is negligible, (c) the bulb is filled with vacuum, predicts the power dissipation $P=Ak\sigma T^4$, where $A$ is the surface area of the filament and $k$ is the emissivity. Build a graph to verify this prediction. The dependence of $P$ on $T$ in the wide range of temperatures from prediction holds, and may slowly decrease later. Plot this dependence on a graph and suggest a formula which describes such a dependence. From the slope of this graph, find the emissivity of tungsten. At which temperatures does the prediction not hold? Why does the prediction fail for low temperatures?

Part E. Specific heat capacity of tungsten (3.5 points)

Measure the quantity of heat $Q$ required to raise the temperature of the filament from the room temperature to its melting point, and the average specific heat capacity over this range of temperatures. Document the circuit used. Estimate the mean specific heat capacity, and calculate average specific heat of tungsten over this range of temperatures. Hint: if you need a well-defined amount of energy, you can use the power supply and battery pack in series to obtain a voltage up to $15\mathrm{V}$.

Fonte: Testo (PDF) — p.1

Topic: Thermodynamics, Circuits Metodi: Experimental Data Analysis, Graph Linearization, Interference & Diffraction Analysis, Curve Fitting Competenze: Experimental Data Analysis, Measurement & Instrumentation, Error Propagation, Graph Linearization Objects: Capacitor, Wire, Screen, Slit