PROBLEMA n. 1 { Viaggio in galleria 90 Pun ti Quesito n. 1. Nel aso elettrostati o, dal T eorema di Gauss, si ottiene

he il amp o in terno alla distribuzione uniforme di ari a negativ a v ale ~ E (r )

E 0 R ~ r do v e E 0

e il mo dulo del amp o p er r

R : In analogia a vremo ~ g (r )

g 0 R ~ r do v e R

e il raggio della T erra e g 0 l’a elerazione di gra vit a standard (a liv ello del mare).

Topic: Gravitation, Electrostatics Metodi: Gauss’s Law, Physical Modeling, Symmetry Argument Competenze: Physical Reasoning, Mathematical Modeling Objects: Planet Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

The problem n. 1 { Travel in Gallery 90 The Commission ti Question No n. 1. In the Other electrical equipment or, from T Other di Gauss, what are you doing? si The result is

he il amp o in Other to the distribution uniform di the following points are inserted: Negative a v Other ~ E (r )

E 0 R ~ r do v e E 0

e il mo Other of the amp o p er r

R : In The following is the list of the products concerned: a We will. ~ g (r )

g 0 R ~ r do v e R

e il Radius of the T error e g 0 The following table shows the results of the evaluation: di The following is the list of the The Commission a The following standard (a Other This is of the The Commission has not yet adopted a proposal.

Quesito n. 2. Detta x la proiezione del v ettore ~ r lungo la galleria, l’equazione di moto

e quella di un moto armoni o: m

x

m g 0 R x )

x + ! 2 x

0 on ! 2

g 0 R ) T o

2 s R g 0 essendo T o il p erio do di un’os illazione ompleta. La durata del viaggio (T )

e met a del p erio do: T

p R =g 0

2:53

10 3 s

42 min (indip enden te dalla latitudine ’ !)

Topic: Newtonian Mechanics, Oscillations & Waves Metodi: Simple Harmonic Motion Analysis, Kinematic Equations, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Planet Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

Question No n. 2. This x la projection of the v Hector ~ r long la the gallery, The equation di Motorcycles

e That one. di un Motorcycles harmonies or: m

x

m g 0 R x )

x + ! 2 x

0 on ! 2

g 0 R ) T o

2 s R g 0 being T o il p the air do di a bone of insulation

It’s full. La Duration of the operation of the Travel (T )

e The Commission a of the p the air do: T

p R =g 0

2:53

10 3 s

42 Minimum number of days (indip) The Commission shall adopt implementing acts. te from latitude ’ !)

Quesito n. 3. La v elo it a massima si tro v a on la onserv azione dell’energia o on la legge oraria: 1 2 mv 2 max

U max U min do v e, p er simmetria, l’energia p otenziale U

e funzione solo di r : U (r )

U (0) Z r 0 m ~ g (r )

~ dr

Z r 0 m g (r ) dr

m(g 0 =2R )r 2 a v endo p osto ~ g (r )

g (r ) ^ r

(g 0 =R ) ~ r e U (0)

0 : Essendo p oi r max

R ed r min

R sen ’ si tro v a U

U max U min

m g 0 2R R 2 R 2 sen 2 ’

= 1 2 mg 0 R os 2 ’ e quindi v max

p 2 U =m

p g 0 R os ‘

3:95 km=s : Oppure x(t)

A os ! t on A

R os ’ ) v (t)

R ! os ’ sen ! t ) ) v max

R ! os ’

p g 0 R os ’ P ag. 1 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 9 Ap rile 2010

Topic: Conservation of Energy, Oscillations & Waves, Newtonian Mechanics Metodi: Energy Conservation Method, Simple Harmonic Motion Analysis, Conservation Laws Competenze: Mathematical Modeling, Physical Reasoning Objects: Planet Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

Question No n. 3. La v I’m going to it a maximum si The Commission v a on la onserv Action energy o on la Law time: 1 2 mv 2 Max

U Max U Minimum number of days do v e, p er the symmetry, energy p The following information shall be provided: U

e function only di r : U (r )

U (0) Z r 0 m ~ g (r )

~ dr

Z r 0 m g (r ) dr

m(g 0 =2R )r 2 a v Other p Other ~ g (r )

g (r ) ^ r

(g 0 =R ) ~ r e U (0)

0 : Being p oi r Max

R ed r Minimum number of days

R Other ’ si The Commission v a U

U Max U Minimum number of days

m g 0 2R R 2 R 2 Other 2 ’

= 1 2 mg 0 R os 2 ’ e So, what do you mean? v Max

p 2 U =m

p g 0 R os ‘

3:95 km=s : Or x(t)

A os ! t on A

R os ’ ) v (t)

R ! os ’ Other ! t ) ) v Max

R ! os ’

p g 0 R os ’ P ag. 1 The following information is provided: { The Olympics di The following is a list of Ga ra National team: The Commission of the Try it . T eo ri a { The following is the list of countries: { 9 Ap Reels 2010

Quesito n. 4. La rotazione terrestre nel riferimen to (non inerziale) della T erra determina una forza en trifuga opp osta alla omp onen te del amp o parallela alla galleria: F

= m

2 x. Questa dunque non altera l’equazione ma ne mo di a solo un o eÆ ien te. m

x

m[(g 0 =R )

2 )℄x ) ! 0 2

(g 0 =R )

2 ) T 0 o

2 p (g 0 =R )

2 Numeri amen te g 0 =R

1:54

10 6 s 2 e

2 4 2 T 2 g

(7:29

10 5 ) 2

5:32

10 9 s 2 essendo T g la durata del giorno. Si pu o quindi approssimare la soluzione p onendo g 0 R

2

1 R

2 g 0

g 0 R )

g 0 R

1

2

1 + R

2 2g 0

g 0 R

1 2 T 0

1 2 T 0 o

g 0 R

1

1 + R

2 2g 0

r g 0 R

1 + R

2 2g 0

T La v ariazione (p ositiv a)

e quindi Æ T

2 2g 0 T

4 s Si osservi

he nel riferimen to non inerziale si do vrebb e onsiderare an he la forza di Coriolis (F Cor

2m ~ !

~ v ); questa p er o risulta sempre p erp endi olare alla direzione del moto e in questo aso

e ann ullata dalla reazione vin olare.

Topic: Newtonian Mechanics, Rotational Dynamics Metodi: Free-Body Diagram, Vector Decomposition, Approximation & Series Expansion Competenze: Mathematical Modeling, Physical Reasoning Objects: Planet Fonte: Testo (PDF) — p.2 Soluzione: Soluzioni (PDF)

Question No n. 4. La rotation of the land In the References to (not (Inert) of the T error determines the One of them.

What ? en the trifuge Op. Other to the The Commission shall adopt implementing acts. Other te of the amp o Other to the Gallery: F

= m

2 x. This one So, what? Not change The equation ma ne mo di a only un o It is te. m

x

m[(g 0 =R )

2 )℄x ) ! 0 2

(g 0 =R )

2 ) T 0 o

2 p (g 0 =R )

2 Number of the te g 0 =R

1:54

10 6 s 2 e

2 4 2 T 2 g

(7:29

10 5 ) 2

5:32

10 9 s 2 being T g la Duration of the operation of the I’m going to be a day. Si pu o So, what do you mean? Approaching la The solution p by the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight g 0 R

2

1 R

2 g 0

g 0 R )

g 0 R

1

2

1 + R

2 2g 0

g 0 R

1 2 T 0

1 2 T 0 o

g 0 R

1

1 + R

2 2g 0

r g 0 R

1 + R

2 2g 0

T La v Air conditioning (p Other a)

e So, what do you mean? Æ T

2 2g 0 T

4 s Si Watch out

he In the References to Not Inertial si do The Commission shall adopt implementing acts. e to be considered an he la

What ? di Coriolis (F The Cor = 2m ~ !

~ v ); This one p er o The following is the list of the Always p Other Other, of a kind used for the manufacture of goods to the Direction of the Motorcycles e in This one. Other

e The Commission Oiled from reaction I’m going to have a drink.

Quesito n. 5. Sia

la frazione di energia p ersa in un viaggio (semi-os illazione ompleta). La ondizione

he Æ U

U on

= 0:01 Ma dall’espressione U (r ) vista al pun to 3. si ri a v a Æ U U (r )

2 Æ r r P er r

R si ha Æ r

R 2 Æ U U (R )

R 2

U U (R ) Æ r

R 2 mg 0 R os 2 ’ 2 2 mg 0 R

1 2

R os 2 ’ La distanza dalla stazione di arriv o si ottiene in ne ome D

Æ r os ’

= 1 2

R os ‘

15:9 km An he qui, in alternativ a, si p otev a onsiderare ome energia p otenziale quella legata al moto armoni o ( ome nel aso di una molla) data da U 0

1 2 k x 2 on k

m g 0 R Si v ede subito

he questa di eris e da quella gra vitazionale { al olata sopra { solo p er un termine ostan te

he quindi risulta inin uen te: U (r )

1 2 m g 0 R r 2

1 2 m g 0 R x 2 + 1 2 m g 0 R y 2

U 0 (x) + ost do v e y

e la distanza della galleria dal en tro della T erra. P ag. 2 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 9 Ap rile 2010 La ondizione si esprime ora ome Æ U 0

U 0 e Æ U 0 U 0 (x)

2 Æ x x ) Æ x

x 2 U 0 U 0 (x) P er x

R os ’ e U 0

U 0 (R os ’) si ritro v a D

jÆ xj

1 2

R os ’

Topic: Conservation of Energy, Oscillations & Waves Metodi: Energy Conservation Method, Simple Harmonic Motion Analysis, Conservation Laws Competenze: Mathematical Modeling, Physical Reasoning Objects: Planet Fonte: Testo (PDF) — p.2 Soluzione: Soluzioni (PDF)

Question No n. 5. Both

la fraction di energy p Other in un Travel (semi-oiled (Full and complete) La the condition

he Æ U

U on

= 0:01 Ma from the expression U (r ) view al the following points are added: to 3. si ri a v a Æ U U (r )

2 Æ r r P er r

R si ha Æ r

R 2 Æ U U (R )

R 2

U U (R ) Æ r

R 2 mg 0 R os 2 ’ 2 2 mg 0 R

1 2

R os 2 ’ La distance from station di I ‘m coming o si The result is in ne human beings D

Æ r os ’

= 1 2

R os ‘

15:9 km An he Here, this is the in Other a, si p Other a to be considered human beings energy p The following information shall be provided: That one. of which: al Motorcycles harmonies or (man) In the Other di One of them. (From the bottom of the page) Date of the date da U 0

1 2 k x 2 on k

m g 0 R Si v Other Right now.

he This one di eris and da That one. The following is the list of the The Commission { Other, of a kind used for the manufacture of goods of heading 8106 above { only p er un term Other te

he So, what do you mean? The following is the list of the in Other te: U (r )

1 2 m g 0 R r 2

1 2 m g 0 R x 2 + 1 2 m g 0 R y 2

U 0 (x) + Other do v e y

e la distance of the Gallery from en The Commission of the T He’s wrong. P ag. 2 The following information is provided: { The Olympics di The following is a list of Ga ra National team: The Commission of the Try it . T eo ri a { The following is the list of countries: { 9 Ap Reels 2010 La the condition si the expression Now human beings Æ U 0

U 0 e Æ U 0 U 0 (x)

2 Æ x x ) Æ x

x 2 U 0 U 0 (x) P er x

R os ’ e U 0

U 0 (R os ’) si withdrawal v a D

jÆ xj

1 2

R os ’

Quesito n. 6. Se l’os illazione si dimezza in ampiezza, l’energia totale div en ta 1/4 di quella iniziale, men tre il p erio do di os illazione non am bia. Si pu o quindi s riv ere n uo v amen te Æ U

U ) Æ U T

U T Se la v ariazione di energia

e len ta si pu o onfondere il rapp orto in remen tale on la deriv ata (v edi suggerimen to in termini di p otenza media e istan tanea) e s riv ere dU dt

T U ) U (t)

U 0 e t=

= T

e U 0

U (R ) La ondizione da imp orre

e U (N T )

1 4 U 0 ) e N T =(T

)

1 4 )

N

ln 4 L’ampiezza si sar a dimezzata dop o N

ln 4

139 viaggi.

Topic: Oscillations & Waves, Conservation of Energy Metodi: Energy Conservation Method, Differential Equations, Simple Harmonic Motion Analysis Competenze: Mathematical Modeling, Physical Reasoning Objects: Planet Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

Question No n. 6. Se The bone is insulated si half in the width, energy Total Div en ta 1/4 di That one. the initial, Men and women three il p the air do di Other Not am I’m not going to lie. Si pu o So, what do you mean? The following is the list of the The Commission n uo v I will not let you go. te Æ U

U ) Æ U T

U T Se la v Air conditioning di energy

e Other ta si pu o Flooding il REP vegetable in remen such on la derivatives Other (v The Commission shall adopt implementing acts. Suggestions to in the terms di p the authorisation average e Other (including e The following is the list of the The Commission dU dt

T U ) U (t)

U 0 e t=

= T

e U 0

U (R ) La the condition da Employment Horrors

e U (N T )

1 4 U 0 ) e N T =(T

)

1 4 )

N

ln 4 The width si will be a half after o N

ln 4

139 travel.

Quesito n. 7. Nel mo dello della T erra a due strati, on n u leo p esan te, il amp o gra vitazione nel n u leo res e linearmen te on r ed

e o vviamen te maggiore

he nel aso visto prima. Questo

e v ero an he nel man tello; infatti la densit a del man tello

e ne essariamen te minore di quella della T erra omogenea e di onseguenza,

ssata una sup er ie sferi a di raggio r nella regione del man tello, la massa esterna

e ertamen te minore risp etto al aso della T erra omogenea. Di onseguenza la massa della parte in terna alla sup er ie

e sempre maggiore ed

e quindi maggiore an he il amp o gra vitazionale; dunque il viaggio durer a meno. Il moto p er o non

e pi

u armoni o dato

he, solo p er una distribuzione omogenea, v ale la dip endenza lineare tra amp o e distanza dal en tro. PROBLEMA n. 2 { Ci lo termo dinami o 90 Pun ti

Topic: Gravitation, Newtonian Mechanics Metodi: Gauss’s Law, Physical Modeling, Symmetry Argument Competenze: Physical Reasoning Objects: Planet Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

Question No n. 7. In the mo of the of the T error a two layers, on n The European Union p Other te, il amp o The following is the list of the living In the n The European Union Res and Linearmen te on r ed

e o I’m going to go. te major

he In the Other See also I’m not going to. This one.

e v I was an he In the Man the textile; In fact, la density a of the Man Other textile materials

e The Commission shall adopt the following measures: te Minor di That one. of the T error of a kind used for the manufacture of goods e di the presence of a

Other One of them. Supplementary er ie The spheres a di Radius r In the Region of the Man the textile, la Other Other

e Other te Minor Responsibility Other al Other of the T error It’s the same. Di the frequency la Other of the Part in Other to the Supplementary er ie

e Always major ed

e So, what do you mean? major an he il amp o The following is the list of the the life cycle; So, what? il Travel The duration a

I’m not. Il Motorcycles p er o Not

e pi

u harmonies or given

he, only p er One of them. distribution the same, v Other la Dip Other, of a kind used for the manufacture of goods Linear between amp o e distance from en I’m going to try. The problem n. 2 { Ci lo term Other, of a kind used for the manufacture of goods 90 The Commission ti

Quesito n. 1. P er la trasformazione isobara: T A V A

T C V C ) T C

2T A

580 K Q C!A

n p (T A T C )

7 2 nR T A

7 2 8:31 J K 1 290 K

8:43 kJ v alido qualunque sia lo stato B. P ag. 3 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 9 Ap rile 2010

Topic: Thermodynamics Metodi: Ideal Gas Law, First Law of Thermodynamics Competenze: Mathematical Modeling, Physical Reasoning Objects: Gas Fonte: Testo (PDF) — p.3 Soluzione: Soluzioni (PDF)

Question No n. 1. P er la the transformation Other, of a kind used for the manufacture of goods T A V A

T C V C ) T C

2T A

580 K Q C!A

n p (T A T C )

7 2 nR T A

7 2 8:31 J K 1 290 K

8:43 kJ v Other any It’s either lo State of the Union B. P ag. 3 The following information is provided: { The Olympics di The following is a list of Ga ra National team: The Commission of the Try it . T eo ri a { The following is the list of countries: { 9 Ap Reels 2010

Quesito n. 2. La temp eratura T B nello stato B dev e essere maggiore o uguale a T C , o vv ero, nel piano p V , B dev e stare sopra l’isoterma p er C. Infatti nell’espansione adiabati a il la v oro

e p ositiv o a sp ese dell’energia in terna e dunque la temp eratura pu o solo dimin uire o restare uguale (p er la v oro n ullo, ome aso estremo). Ne segue

he (T B ) min

T C

580 K Al temp o stesso, p oi h

e le adiabati he rev ersibili sono urv e iso en tropi he, il pun to B dev e stare sotto l’adiabati a rev ersibile p er C dato

he l’en tropia pu o solo aumen tare o al massimo restare ostan te (se la trasformazione del gas fosse un’adiabati a rev ersibile). Il massimo, o meglio, l’estremo sup eriore p er la temp eratura T B pu o essere al olato su una trasformazione adiabati a rev ersibile o v e T V

1

ost, do v e

p

V

7=5 nel aso di gas biatomi o. Quindi (T B ) sup

V C V B

1 T C

2 2=5 T C

765 K Il ragionamen to sv olto sopra pu o essere sviluppato a partire dalle stesse premesse dis utendo i div ersi on tributi di alore e la v oro p er determinare il v alore minimo di T B e al olando la v ariazione di en tropia p er il v alore massimo. Nel seguito si indi heranno on

p e

V i alori molari, risp ettiv amen te a pressione e v olume ostan te. P er un i lo termo dinami o U

0 ) Q

L Q A!B

n V (T B T A )

5 2 nR (T B T A ) assorbito duran te l’iso ora Q B!C

0 essendo un’adiabati a Q C!A

n p (T A T C )

7 2 nR (T A T C ) eduto duran te l’isobara L A!B

0 p er h

e non ’

e v ariazione di v olume L B!C

0 la v oro fatto v erso l’esterno nell’abiabati a irrev ersibile L C!A

p V

nR (T A T C ) < 0 p er h

e il gas viene ompresso P er il primo prin ipio si ha dunque Q A!B + Q C!A

L B!C + L C!A ) L B!C

n V (T B T A ) + n p (T A T C ) nR (T A T C )

= n V T B + n( V +

p R )T A n( p R )T C

n V (T B T C ) T rattandosi di un’espansione il la v oro dev e essere non negativ o p er ui (T B T C )

0 ) T B

T C P er l’espansione adiabati a irrev ersibile dev e essere S

0, o vv ero S B!C

nR ln V C V B + n V ln T C T B

ln 2 + 5 2 ln T C T B

0 ) ) ln T C T B

2 5 ln 2 ) T B < 2 2=5 T C In de nitiv a, p er la temp eratura dello stato B v algono le ondizioni 580 K

T B < 765 K P ag. 4 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 9 Ap rile 2010

Topic: Thermodynamics Metodi: First Law of Thermodynamics, Thermodynamic Cycle Analysis, Ideal Gas Law Competenze: Mathematical Modeling, Physical Reasoning Objects: Gas Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

Question No n. 2. La Temp Other T B In the State of the Union B It must be e to be major o equal to a T C , o vv I was, In the

It ‘s not easy . p V , B It must be e to be above the isothermal p er C. In fact, In the expansion the following: il la v gold

e p Other o a sp Exports energy in Other e So, what? la Temp Other pu o only Minimum number of days Other o Staying equal to (p er la v gold n the name of the person concerned, human beings Other Extreme . Ne follows

he (T B ) Minimum number of days

T C

580 K Al Temp o itself, p oi h

e le Other, not further worked than hot-rolled he The Commission shall adopt implementing acts. Other, of a kind used for the manufacture of goods I am the following information is provided: e Other en Tropical rainforests he, il the following points are added: to B It must be e to be below The following are the The Commission shall adopt implementing acts. Other, not further worked than cut p er C given

he l’en Other pu o only Other Other o al Not more than Staying Other te (se la the transformation of the Gas If it were a diaphragm of The Commission shall adopt implementing acts. The following is the list of the products: Il maximum, o I’m better off. The extreme Supplementary Other p er la Temp Other T B pu o to be to the wave su One of them. the transformation the following: The Commission shall adopt implementing acts. Other, not further worked than cut o v e T V

1

the following: do v e

p

V

7=5 In the Other di Gas The biomaterials are not. So, what? (T B ) Supplementary

V C V B

1 T C

2 2=5 T C

765 K Il Reasonamen to sv I’m going to above pu o to be developed a to leave from The same the following conditions are fulfilled: Say using i Div Other on Taxes di Other e la v gold p er determining the il v Other Minimum di T B e The Dutch la v Air conditioning di en Other p er il v Other

That’s the best. In the followed si Indians herring on

p e

V i Other molars, Responsibility Other I will not let you go. te a pressure e v Other, of a kind used for the manufacture of foodstuffs Other te. P er un i lo term Other, of a kind used for the manufacture of goods U

0 ) Q

L Q A!B

n V (T B T A )

5 2 nR (T B T A ) absorbed Durand te the isotope Now Q B!C

0 being a diaphragm of Q C!A

n p (T A T C )

7 2 nR (T A T C ) Educated Durand te the isobarbaric L A!B

0 p er h

e Not ’

e v Air conditioning di v Other, of a kind used for the manufacture of foodstuffs L B!C

0 la v gold made v Other the outside In the abyss of Other Other, not further worked than cut L C!A

p V

nR (T A T C ) < 0 p er h

e il Gas He ‘s coming . Other, of a thickness of not more than 10 mm P er il first The first si ha So, what? Q A!B + Q C!A

L B!C + L C!A ) L B!C

n V (T B T A ) + n p (T A T C ) nR (T A T C )

= n V T B + n( V +

p R )T A n( p R )T C

n V (T B T C ) T ♪ ♪ and it’s all over ♪ di an expansion il la v gold It must be e to be Not Negative o p er ui (T B T C )

0 ) T B

T C P er The expansion the following: Other Other, not further worked than cut It must be e to be S

0, o vv I was S B!C

nR ln V C V B + n V ln T C T B

ln 2 + 5 2 ln T C T B

0 ) ) ln T C T B

2 5 ln 2 ) T B < 2 2=5 T C In de Other a, p er la Temp Other of the State of the Union B v Other le Other articles of heading No. 580 K

T B < 765 K P ag. 4 The following information is provided: { The Olympics di The following is a list of Ga ra National team: The Commission of the Try it . T eo ri a { The following is the list of countries: { 9 Ap Reels 2010

Quesito n. 3. Il i lo sv olto dal gas

e stato de nito termi o se il gas ompie la v oro sull’esterno: L

0 e quindi an he Q

0; in altre parole il gas assorb e pi

u alore di quan to ne ede; al on trario se il la v oro

e fatto sul gas dall’esterno (L < 0 e Q < 0) allora esegue un i lo frigorifero. Il v alore parti olare di T B p er ui il i lo am bia da frigorifero a termi o si ottiene quindi imp onendo

he Q

Q A!B + Q C!A

0 o vv ero Q A!B

Q C!A 5 2 nR (T B T A )

7 2 nR (T C T A ) ) 2T A + 5T B 7T C

0 ) T B

12 5 T A

696 K : Alternativ amen te: n V (T B T A )

n p (T C T A ) ) T B

V T A +

p T A

= ( 1 +

) T A

12 5 T A detta T 0 tale temp eratura, si ha

he p er T B < T 0 il i lo

e frigorifero, men tre p er T B

T 0 il i lo

e termi o. Come aso sup eriore estremo si ha quello di un’adiabati a rev ersibile

he rende rev ersibile l’in tero i lo: notare

he solo in questo aso il la v oro

e al olabile attra v erso l’area del piano p V delimitata dalla urv a

he rappresen ta il i lo e solo in questo aso il v erso orario di p er orrenza del i lo onferma

he si tratta di un i lo termi o.

Topic: Thermodynamics Metodi: Thermodynamic Cycle Analysis, First Law of Thermodynamics, Ideal Gas Law Competenze: Mathematical Modeling Objects: Gas Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)

Question No n. 3. Il i lo sv I’m going to from Gas

e State of the Union de Other Other, of a kind used for the manufacture of goods se il Gas Other la v gold on the outside: L

0 e So, what do you mean? an he Q

0; in Other Words il Gas Other, of a kind used for the manufacture of goods e pi

u Other di When to ne the following: al on Other se il la v gold

e made on the Gas from the outside (L < 0 e Q < 0) Then It shall be executed un i lo The fridge. Il v Other Other, of a kind used for the manufacture of goods di T B p er ui il i lo am Other da Other, of a kind used for the manufacture of refrigerators a Other, of a kind used for the manufacture of goods si The result is So, what do you mean? Employment by the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight of the weight

he Q

Q A!B + Q C!A

0 o vv I was Q A!B

Q C!A 5 2 nR (T B T A )

7 2 nR (T C T A ) ) 2T A + 5T B 7T C

0 ) T B

12 5 T A

696 K : Other I will not let you go. te: n V (T B T A )

n p (T C T A ) ) T B

V T A +

p T A

= ( 1 +

) T A

12 5 T A the following: T 0 such Temp the following: si ha

he p er T B < T 0 il i lo

e refrigerator, Men and women three p er T B

T 0 il i lo

e The term ‘sopher’ How Other Supplementary Other Extreme si ha That one. di a diaphragm of The Commission shall adopt implementing acts. Other, not further worked than cut

he It makes The Commission shall adopt implementing acts. Other, not further worked than cut l’in Other i lo: Notes

he only in This one. Other il la v gold

e The oval the following points are added: v Other the area of the

It ‘s not easy . p V Delimited from the following information is provided: a

he representing ta il i lo e only in This one. Other il v Other time di p The following is the list of the following: of the i lo sickness

he si The Commission shall adopt implementing acts in accordance with Article 21 of this Regulation. di un i lo The term ‘sopher’

Quesito n. 4. Il rendimen to di un i lo termi o

= L Q 1

Q Q 1

Q 1 + Q 2 Q 1

Q 1 jQ 2 j Q 1

1 jQ 2 j Q 1 do v e Q

e il alore omplessiv amen te s am biato, men tre Q 1 e Q 2 sono quelli e ettiv amen te assorbito e eduto (Q 1

0; Q 2 < 0). Il rendimen to

e quindi una funzione della temp eratura T B :

(T B )

1 Q C!A Q A!B

1 n

p (T C T A ) n

V (T B T A )

p T A

V (T B T A )

1 T B =T A 1 L’espressione del rendimen to, v alida p er 696 K < T B < 765 K,

e una funzione monotona res en te di T B . Numeri amen te risulta:

(696 K)

0 ed

(765 K)

0:15 L’eÆ ienza frigorifera “

Q 1 jLj

Q 1 j Qj

Q 1 jQ 2 j Q 1

1 jQ 2 j =Q 1 1

e pure una funzione monotona res en te di T B : “(T B )

1 jQ C!A j =Q A!B 1

p T A n

V (T B T A ) 1

1

T B =T A 1 1

1 Si ri a v a: “(580 K)

2:5 ed “(T B ! 696 K) ! 1 : I gra i dei due parametri sono rip ortati qui sotto. P ag. 5 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 9 Ap rile 2010 PROBLEMA n. 3 { Esplo rando il fondo 50 Pun ti

Topic: Thermodynamics Metodi: Thermodynamic Cycle Analysis, First Law of Thermodynamics, Ideal Gas Law Competenze: Mathematical Modeling, Physical Reasoning Objects: Gas Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)

Question No n. 4. Il The Commission shall adopt implementing acts. to di un i lo Other, of a kind used for the manufacture of goods

= L Q 1

Q Q 1

Q 1 + Q 2 Q 1

Q 1 jQ 2 j Q 1

1 jQ 2 j Q 1 do v e Q

e il Other Other I will not let you go. te s am the beaten, Men and women three Q 1 e Q 2 I am The other e Other I will not let you go. te absorbed e Educated (Q 1

0; Q 2 < 0). Il The Commission shall adopt implementing acts. to

e So, what do you mean? One of them. function of the Temp Other T B :

(T B )

1 Q C!A Q A!B

1 n

p (T C T A ) n

V (T B T A )

p T A

V (T B T A )

1 T B =T A 1 The expression of the The Commission shall adopt implementing acts. to, v other than p er 696 K < T B < 765 K,

e One of them. function The following is the list of the categories of products: The following is a list of the te di T B . Number of the te It is:

(696 K)

0 ed

(765 K)

0:15 The Commission has already taken a decision on the Other refrigerating equipment “

Q 1 The following is the list of the Member States:

Q 1 j Qj

Q 1 jQ 2 j Q 1

1 jQ 2 j =Q 1 1

e I’m not sure. One of them. function The following is the list of the categories of products: The following is a list of the te di T B : “(T B )

1 jQ C!A j =Q A!B 1

p T A n

V (T B T A ) 1

1

T B =T A 1 1

1 Si ri a v a: “(580 K)

2:5 ed “(T B ! 696 K) ! 1 : I The following is the list of the i of the two Parameters I am RIP Other articles of heading No. Here .

Down here. P ag. 5 The following information is provided: { The Olympics di The following is a list of Ga ra National team: The Commission of the Try it . T eo ri a { The following is the list of countries: { 9 Ap Reels 2010 The problem n. 3 { The Commission shall adopt implementing acts. I ‘m going to il Fund 50 The Commission ti

Quesito n. 1. Si onsideri un arbitrario pun to A del fondo della v as a, a distanza x dal piede della p erp endi olare ondotta dal pun to di osserv azione O, e il raggio di lu e

he partendo da questo pun to arriv a nello stesso pun to O. A ausa della rifrazione subita dalla lu e nel passaggio dall’a qua all’aria, esso sem bra essere emesso da un pun to A 0 , ome mostrato in

gura. In termini dell’angolo ’ mostrato in

gura, si ha x

L tg ’. Il raggio

he parte da A, on un angolo ’ risp etto alla v erti ale, passa p er O; in v e e un raggio adia en te

he parte on angolo ’ + d’ giunge alla sup er ie di separazione in un pun to distan te da O di una quan tit a dx data da x + dx

L tg (’ + d’) ) dx

L [tg (’ + d’) tg ’℄ ) dx

tg (’ + d’) tg ’ d’

d’ e, onsiderando

he p er d’ ! 0 l’espressione tra paren tesi quadra

e la deriv ata della funzione tg ’, si ha dx

L os 2 ’ d’ (1) Nota: Si p otev a giungere allo stesso risultato di erenziando l’espressione iniziale: x

L tg ’ ) dx

L d(tg ’) ) dx

L 1 os 2 ’ d’ Analogamen te, indi ando on ` la profondit a del pun to A 0 , si ha

he x

` os 2

d# (2) Di erenziando la Legge di Snell, sen

= n sen ’, si ottiene d (sen #)

nd (sen ’) ) os

d#

n os ’ d’ ) d’

1 n os

os ’ d# : Sostituendo nella (1) l’espressione p er d’ app ena otten uta, si ri a v a dx

L n os

os 3 ’ d# (3) ed uguagliando le espressioni (2) e (3) si ottiene in ne `

L n

os ‘

3 (4) Sempre da x

L tg ’ si ri a v a

he x L

p 1 os 2 ’ os ’ ) os 2 ‘

1 1 + (x=L) 2 P ag. 6 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 9 Ap rile 2010 D’altra parte, appli ando la Legge di Snell, si pu o an he s riv ere x L

sen #=n p 1 (sen #=n) 2 )

x L

2

1 os 2

n 2 1 + os 2

) os 2

= 1 + (1 n 2 )(x=L) 2 1 + (x=L) 2 In ultima analisi

os ‘

2

1 (n 2 1)

x L

he sostituita nella (4) fornis e `

L n ” r 1 (n 2 1)

x L

3 (5) ||||||||||| Soluzione alternativ a Come prima, oltre al raggio us en te da A e passan te p er O, si onsideri un se ondo raggio

he forma on il primo un angolo pi olo

he attra v ersa la sup er ie dell’a qua in un pun to O 0 . La profondit a ` di A 0 si pu o esprimere in funzione della profondit a e ettiv a L, della distanza x e dell’indi e di rifrazione n onsiderando

he il segmen to OO 0

e un lato om une dei due triangoli OHO 0 e OH 0 O 0 . Da sempli i onsiderazioni geometri he si ri a v a

he d OO 0 H

’ e d OO 0 H 0

: Dalla geometria del sistema si ri a v ano su essiv amen te A 0 O 0

` os (#

) ; O 0 H 0

A 0 O 0 sen

= ` sen

os (#

) ; OO 0

O 0 H 0 os

= ` sen

os (#

) os

: (1 0 ) Analogamen te A O 0

L os (’

) ; O 0 H

A O 0 sen

= L sen

os (’

) ; OO 0

O 0 H os ’

L sen

os (’

) os ’ : (2 0 ) Uguagliando la (1 0 ) e la (2 0 ) si ottiene `

L sen

os (#

) sen

os ’ os (’

) : Ri ordando

he la div ergenza del fas io

e molto pi ola si pu o approssimare sen

, sen

e os

1; p ertan to l’espressione pre eden te div en ta `

os ’

: (3 0 ) P ag. 7 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 9 Ap rile 2010 La legge di Snell appli ata nei pun ti O e O 0 fornis e risp ettiv amen te sen

sen ’

n ; e sen (#

) sen (’

)

n : Con l’approssimazione in tro dotta e tenendo presen te la prima espressione, dalla se onda si ottiene sen

= n sen ’ n

os ’ )

= n

os ’ )

= n os ’ os

: Sostituendo quest’ultima nella (3’) si ottiene `

L n

os ’

3 (4 0 )

he oin ide on la (4). La legge di Snell, espressa in termini dei oseni degli angoli si pu o s riv ere 1 os 2

1 os 2 ’

n 2 ) os 2

= 1 n 2 (1 os 2 ’)

n 2 os 2 ’ (n 2 1) Dunque

os# os ‘

2 n 2 n 2 1 os 2 ‘

n 2 (n 2 1)(1 + tg 2 ’)

1 (n 2 1) tg 2 ’ In ne, ri ordando

he tg ’

x=L, dall’espressione pre eden te si ha os

os ’

r 1 (n 2 1) +

x L

2 e la (4’) { in funzione di L, n ed x { si s riv e ome `

L n ” r 1 ( n 2 1)

x L

3 : (5 0 )

Topic: Geometric Optics Metodi: Snell’s Law, Ray Tracing, Calculus-Integration Competenze: Mathematical Modeling, Diagrammatic Reasoning Objects: Container Fonte: Testo (PDF) — p.6 Soluzione: Soluzioni (PDF)

Question No n. 1. Si Other un the amount of the contribution the following points are added: to A of the Fund of the v as a, a distance x from foot of the p Other Other, of a kind used for the manufacture of goods Wavy from the following points are added: to di See also Action O, e il Radius di lu e

he I ‘m leaving da This one. the following points are added: to I ‘m coming a In the of the same the following points are added: to O. A Other of the the refraction Subtitles from lu e In the Passage From here. in the air, It We are Good . to be issued da un the following points are added: to A 0 , human beings shown in

I’m going to go. In the terms of the corner ’ shown in

the mouth, si ha x

L tg ’. Il Radius

he Part da A, on un angle ’ Responsibility Other to the v high-winged, Go ahead . p er O; in v e e un Radius The Commission shall adopt the following measures: te

he Part on angle ’ + d’ It comes to the Supplementary er ie di Separation in un the following points are added: to Distance te da O di One of them. When Title a dx Date of the date da x + dx

L tg (’ + d’) ) dx

L [tg (’ + d’) tg ’℄ ) dx

tg (’ + d’) tg ’ d’

d’ e, and weighing

he p er d’ ! 0 the expression between the following points are added: The Commission Court

e la derivatives Other of the function tg ’, si ha dx

L os 2 ’ d’ (1) Note: Si p Other a to arrive The following is the list of the of the same result di by herring the expression the initial: x

L tg ’ ) dx

L d(tg ’) ) dx

L 1 os 2 ’ d’ Other te, Indians on ` la depth a of the the following points are added: to A 0 , si ha

he x

` os 2

d# (2) Di by herring la Law di I’m going to get you. Other

= n Other ’, si The result is d (without #)

nd (without ’) ) os

d#

n os ’ d’ ) d’

1 n os

os ’ d# : Substituting In the (1) the expression p er d’ App Other obtained the following: si ri a v a dx

L n os

os 3 ’ d# (3) ed Equalizing le Expressions (2) e (3) si The result is in ne `

L n

os ‘

3 (4) Always da x

L tg ’ si ri a v a

he x L

p 1 os 2 ’ os ’ ) os 2 ‘

1 1 + (x=L) 2 P ag. 6 The following information is provided: { The Olympics di The following Ga ra National team: The Commission of the Try it . T eo ri a { The following is the list of countries: { 9 Ap Reels 2010 Other part, The following is a list of the la Law di I’m going to get you. si pu o an he The following is the list of the The following are the x L

Other #=n p 1 (without #=n) 2 )

x L

2

1 os 2

n 2 1 + os 2

) os 2

= 1 + (1 n 2 )(x=L) 2 1 + (x=L) 2 In Last of the The following table shows the results of the analysis:

os ‘

2

1 (n 2 1)

x L

he Replaced In the (4) supplies and `

L n ” r 1 (n 2 1)

x L

3 (5) ||||||||||| The solution Other a How Before that, Other al Radius us en te da A e Other te p er O, si Other un If the wave Radius

he Form on il first un angle Other

he the following points are added: v Other la Supplementary er ie I’m not sure. in un the following points are added: to O 0 . La depth a ` di A 0 si pu o to express in function of the depth a e Other a L, of the distance x e of the Indian and di the refraction n by weeping

he il segmentation to OO 0

e un side om One of them of the two Triangles OHO 0 e OH 0 O 0 . Da The following is the list of the The Commission shall adopt the following measures: Other geometric he si ri a v a

he d OO 0 H

’ e d OO 0 H 0

: From her The following is the list of the following: of the system si ri a v

What? on the exiv I will not let you go. te A 0 O 0 = ` os (#

) ; O 0 H 0

A 0 O 0 Other

= ` Other

os (#

) ; OO 0

O 0 H 0 os

= ` Other

os (#

) os

: (1 0 ) Other te A O 0

L os (’

) ; O 0 H

A O 0 Other

= L Other

os (’

) ; OO 0

O 0 H os ’

L Other

os (’

) os ’ : (2 0 ) Equalizing la (1 0 ) e la (2 0 ) si The result is `

L Other

os (#

) Other

os ’ os (’

) : I ‘m ordering again .

he la Div emergency of the I do .

e Very much Other si pu o Approaching Other

, Other

e os

1; p Other to the expression for Eden te Div en ta `

os ’

: (3 0 ) P ag. 7 The following information is provided: { The Olympics di The following is a list of Ga ra National team: The Commission of the Try it . T eo ri a { The following is the list of countries: { 9 Ap Reels 2010 La Law di The following is the list of the countries: Applications for the In the the following points are added: ti O e O 0 supplies and Responsibility Other I will not let you go. te Other

Other ’

n ; e Other (#

) Other (’

)

n : With the approximation in The Commission Other e holding the present te la Before the expression, from If it flows si The result is Other

= n Other ’ n

os ’ )

= n

os ’ )

= n os ’ os

: Substituting This last one In the (3’) si The result is `

L n

os ’

3 (4 0 )

he The Commission shall adopt a decision on the on la (4). La Law di I’m going to get you. expressed in the terms of the You dare to of the Corners si pu o The following is the list of the The Commission 1 os 2

1 os 2 ’

n 2 ) os 2

= 1 n 2 (1 os 2 ’)

n 2 os 2 ’ (n 2 1) So, what?

os# os ‘

2 n 2 n 2 1 os 2 ‘

n 2 (n 2 1)(1 + tg 2 ’)

1 (n 2 1) tg 2 ’ In ne, Re-ordering

he tg ’

x=L, from the expression for Eden te si ha os

os ’

r 1 (n 2 1) +

x L

2 e la (4’) { in function di L, n ed x { si The following is the list of the e human beings `

L n ” r 1 ( n 2 1)

x L

3 : (5 0 )

Quesito n. 2. Come si pu o an he omprendere dalla

gura, la p osizione geometri a dell'immagine A 0 dimin uis e al res ere di #, quindi si ha min quando

= 90 Æ e ` max quando

= 0 Æ . P er

= 90 Æ , os

= 0, sen ‘

1=n e os ‘

p n 2 1=n, p er ui dalla (4) si ha ` min

P er

= 0 Æ an he ‘

0 Æ o vv ero x

0, p er ui dalla (5) si ha ` max

L=n.

Topic: Geometric Optics Metodi: Snell’s Law, Ray Tracing Competenze: Mathematical Modeling, Physical Reasoning Objects: Container Fonte: Testo (PDF) — p.8 Soluzione: Soluzioni (PDF)

Question No n. 2. How si pu o an he the following paragraphs are added: from

the mouth, la p The Commission shall adopt implementing acts. Geometers a of the image A 0 Minimum number of days Uis and al Other di #, So, what do you mean? si ha Minimum number of days When

= 90 Æ e ` Max When

= 0 Æ . P er

= 90 Æ , os

= 0, Other ‘

1=n e os ‘

p n 2 1=n, p er ui from (4) si ha ` Minimum number of days

P er

= 0 Æ an he ‘

0 Æ o vv I was x

0, p er ui from (5) si ha ` Max

L=n.

Quesito n. 3. Il massimo v alore di x si ha quando il v alore sotto la radi e quadrata nella (5) si ann ulla (non pu o essere negativ o p er ` reale) quindi x max

L p n 2 1 e l’area della sup er ie omplessiv a del fondale

he pu o essere osserv ata

e S

x 2 max

L 2 n 2 1 P ag. 8 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 9 Ap rile 2010

Topic: Geometric Optics Metodi: Snell’s Law, Ray Tracing Competenze: Mathematical Modeling Objects: Container Fonte: Testo (PDF) — p.8 Soluzione: Soluzioni (PDF)

Question No n. 3. Il Not more than v Other di x si ha When il v Other below la radi and square In the (5) si The Commission The following is the list of the products: (not pu o to be Negative o p er ` (Real) So, what do you mean? x Max

L p n 2 1 e the area of the Supplementary er ie Other a of the The following table shows the following:

he pu o to be See also Other

e S

x 2 Max

L 2 n 2 1 P ag. 8 The following information is provided: { The Olympics di The following is a list of Ga ra National team: The Commission of the Try it . T eo ri a { The following is the list of countries: { 9 Ap Reels 2010

Topic: Geometric Optics Metodi: Snell’s Law, Ray Tracing Competenze: Mathematical Modeling Objects: Container Fonte: Testo (PDF) — p.8 Soluzione: Soluzioni (PDF)

PROBLEMA n. 4 { Co rrenti in un anello 70 Pun ti

Topic: Circuits, Magnetism Metodi: Physical Modeling Competenze: Physical Reasoning Objects: Wire Fonte: Testo (PDF) — p.9 Soluzione: Soluzioni (PDF)

The problem n. 4 { Co Other in un ring 70 The Commission ti

Topic: Circuits, Magnetism Metodi: Physical Modeling Competenze: Physical Reasoning Objects: Wire Fonte: Testo (PDF) — p.9 Soluzione: Soluzioni (PDF)

Quesito n. 1. La resistenza equiv alen te tra i pun ti A e B

e data da R ?

R 1 R 2 R 1 + R 2 on R i

` i

essendo

l’area della sezione del

lo; essendo R 1 + R 2

L= si ha R ?

`1` 2 L

x(1 x) L

A orren te ostan te (I 0 ) la massima d.d.p. si ha quando

e massima la resistenza equiv alen te io

e p er x

1=2, o vv ero quando B

e diametralmen te opp osto ad A. Si ha R ? max

1 4

= V 0 I 0 )

L I 0 4V 0 da ui in ne d

2r

2 r

= r

L I 0

V 0

0:624 mm

Topic: Circuits Metodi: Kirchhoff’s Laws, Equivalent Circuit Reduction Competenze: Mathematical Modeling, Diagrammatic Reasoning Objects: Wire Fonte: Testo (PDF) — p.9 Soluzione: Soluzioni (PDF)

Question No n. 1. La resistance Equivalent to Other te between i the following points are added: ti A e B

e Date of the date da R ?

R 1 R 2 R 1 + R 2 on R i

` i

being

the area of the Section of the

lo; being R 1 + R 2

L= si ha R ?

`1` 2 L

x(1 x) L

A Other articles te Other te (I 0 ) la maximum d.d.p. si ha When

e maximum la resistance Equivalent to Other te io

e p er x

1=2, o vv I was When B

e Other, of a width of not more than 600 mm te Op. Other ad A. Si ha R ? Max

1 4

= V 0 I 0 )

L I 0 4V 0 da ui in ne d

2r

2 r

= r

L I 0

V 0

0:624 mm

Quesito n. 2. La d.d.p.

e prop orzionale alla resistenza equiv alen te: V (x)

I 0 R ? (x)

x(1 x)I 0 R on R

R 1 + R 2

L=

1:6

Notare

he la d.d.p. pu o essere v alutata in due mo di equiv alen ti V

R 1 I 1

R 2 I 2 da ui ` 1 I 1

` 2 I 2 (servir a dop o) Il gra o ri hiesto di V (x)

e rip ortato a

an o.

Question No n. 2. La d.d.p.

e Prop the following table is inserted: to the resistance Equivalent to Other te: V (x)

I 0 R ? (x)

x(1 x)I 0 R on R

R 1 + R 2

L=

1:6

Notes

he la d.d.p. pu o to be v Other in two mo di Equivalent to Other ti V

R 1 I 1

R 2 I 2 da ui ` 1 I 1

` 2 I 2 (to serve a after o) Il The following is the list of the o ri Heist di V (x)

e RIP Other, not elsewhere specified or included a

an o.

Quesito n. 3. P osto

he R 1

xR ed R 2

(1 x)R , la p otenza dissipata nel primo ar o

e W 1

R 1 I 2 1 on I 1

I 0 R 2 R 1 + R 2

I 0 R 2 R (partitore di orren te). Dunque W 1 (x)

xR I 2 0 (1 x) 2 R 2 R 2

x(1 x) 2 I 2 0 R Notare

he W 1 (x) non

e simmetri a risp etto a x

1=2. Il massimo di W 1 si tro v a deriv ando dW 1 (x) dx

(1 x) 2 2x(1 x)

I 2 0 R

(1 x)(1 3x)I 2 0 R

0 p er x

1 3 W 1;max

4 27 I 2 0 R

14:8 mW quando i

li formano un angolo di 120 Æ . An he qui, a

an o

e rip ortato il gra o della funzione W 1 (x).

Topic: Circuits Metodi: Kirchhoff’s Laws, Calculus-Integration Competenze: Mathematical Modeling Objects: Wire Fonte: Testo (PDF) — p.9 Soluzione: Soluzioni (PDF)

Question No n. 3. P Other

he R 1

xR ed R 2

(1 x)R , la p the authorisation Dispersed In the first ar o

e W 1

R 1 I 2 1 on I 1

I 0 R 2 R 1 + R 2

I 0 R 2 R (supply) di Other articles te). So, what? W 1 (x)

xR I 2 0 (1 x) 2 R 2 R 2

x(1 x) 2 I 2 0 R Notes

he W 1 (x) Not

e symmetry to Responsibility Other a x

1=2. Il Not more than di W 1 si The Commission v a derivatives I ‘m going dW 1 (x) dx

(1 x) 2 2x(1 x)

I 2 0 R

(1 x)(1 3x)I 2 0 R

0 p er x

1 3 W 1,max

4 27 I 2 0 R

14:8 mW When i

li form un angle di 120 Æ . An he Here, this is the a

an o

e RIP Other, not elsewhere specified or included il The following is the list of the o of the function W 1 (x).

Topic: Circuits Metodi: Kirchhoff’s Laws, Calculus-Integration Competenze: Mathematical Modeling Objects: Wire Fonte: Testo (PDF) — p.9 Soluzione: Soluzioni (PDF)

Quesito n. 4. Il amp o magneti o al en tro dell’anello

e n ullo in ogni aso. Infatti, ri ordando l’espressione di Lapla e p er il al olo del amp o magneti o, ~ B

0 I 4 Z

~ d`

^ r r 2 il on tributo dei due

li rettilinei

e n ullo essendo questi allineati ol en tro dell’anello (e quindi ~ d`

^ r

e p er i due ar hi p er orsi da orren ti in v erso opp osto, detto a il raggio dell’anello [ a

L=(2 ) ℄, si ha B

0 I 1 2a ` 1 L

0 I 2 2a ` 2 L

L 2 ( `1 I 1` 2 I 2 ) ma `1 I 1` 2 I 2

0 ome si

e visto al pun to 2. P ag. 9 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 9 Ap rile 2010

Topic: Magnetism, Circuits Metodi: Biot-Savart Law, Symmetry Argument Competenze: Mathematical Modeling, Physical Reasoning Objects: Wire Fonte: Testo (PDF) — p.9 Soluzione: Soluzioni (PDF)

Question No n. 4. Il amp o Magnets or al en The Commission of a kind used for the manufacture of goods

e n The following is the list of the countries of the European Union: in Each I’m not going to. In fact, Re-ordering the expression di Lapla and p er il the oil of the amp o magnetic or, ~ B

0 I 4 Z

~ d`

^ r r 2 il on Tribute of the two

li Other, of a width of not more than 600 mm

e n The following is the list of the countries of the European Union: being These aligned ol en The Commission of a kind used for the manufacture of goods (e So, what do you mean? ~ d`

^ r

e p er i two ar hi p Bears da Other articles ti in v Other Op. This, said a il Radius of a kind used for the manufacture of goods [ a

L=(2 ) ℄, si ha B

0 I 1 2a ` 1 L

0 I 2 2a ` 2 L

L 2 ( `1 I 1` 2 I 2 ) ma `1 I 1` 2 I 2

0 human beings si

e See also al the following points are added: to 2. P ag. 9 The following information is provided: { The Olympics di The following is a list of Ga ra National team: The Commission of the Try it . T eo ri a { The following is the list of countries: { 9 Ap Reels 2010

Quesito n. 5. Le risp oste 1, 2 e 3 non am biano; am bia in v e e il amp o magneti o al en tro dell’anello. Adesso esso

e do vuto al

lo p erp endi olare ed

e quindi nel piano dell’anello; il suo mo dulo

e om unque indip enden te dalla p osizione di B (quindi da x) e pu o essere ri a v ato dall’espressione di Lapla e data sopra, o pi

u sempli emen te on onsiderazioni di simmetria, osserv ando

he due tratti in nitesimi di

lo disp osti simmetri amen te risp etto ad un piano ortogonale al

lo stesso, danno uguali on tributi al amp o al olato in un pun to qualunque del piano. Dunque il mo dulo del amp o

e met a di quello di un

lo rettilineo inde nito e v ale B

1 2

0 I 0 2 a

0 I 0 2L

0:157 T Materiale pro dotto dal grupp o OLIMPIADI PROGETTO PROGETTO OLIMPIADI Segreteria Olimpiadi Italiane della Fisi a p resso Li eo S ienti o \U. Mo rin” VENEZIA MESTRE fax: 041.584.1272 e-mail: olifis libero.i t P ag. 10

Topic: Magnetism Metodi: Biot-Savart Law, Symmetry Argument Competenze: Mathematical Modeling, Physical Reasoning Objects: Wire Fonte: Testo (PDF) — p.10 Soluzione: Soluzioni (PDF)

Question No n. 5. Le Responsibility Other 1, 2 e 3 Not am white; am Other in v e e il amp o Magnets or al en The Commission The ring. Now, now. It

e do I ‘m not sure al

lo p Other Other, of a kind used for the manufacture of goods ed

e So, what do you mean? In the

It ‘s not easy . the ring; il his mo Other

e om Other Indip The Commission shall adopt implementing acts. te from p The Commission shall adopt implementing acts. di B (this is the case for da x) e pu o to be ri a v the act from the expression di Lapla and Date of the date above, o pi

u Other te on The Commission shall adopt the following measures: di the symmetry, See also I ‘m going

he two Tracts in Other di

lo Other Other ♪ I ‘m going to be a little bit more ♪ te Responsibility Other ad un

It ‘s not easy . orthodox al

lo itself, Damages equal to on Taxes al amp o to the wave in un the following points are added: to any of the

I’m not going to. So, what? il mo Other of the amp o

e The Commission a di That one. di un

lo Other, of a kind used for the manufacture of goods Other Other e v Other B

1 2

0 I 0 2 a

0 I 0 2L

0:157 T Other materials for The Commission shall adopt the following: from Group o Olympic Games Project Project Olympic Games Secretariat The Olympics Italian of the The following is a list of p I’m not sure. Li eo The Commission o \U. Mo ‘R’ The Commission The Master The fax: 041.584.1272 E-mail: Free olives t P ag. 10

Raccolta Gare di Fisica

Esplora

OII 2010 Nazionale Teorica

he il amp o in terno alla distribuzione uniforme di ari a negativ a v ale ~ E (r )

e il mo dulo del amp o p er r

R : In analogia a vremo ~ g (r )

he il amp o in Other to the distribution uniform di the following points are inserted: Negative a v Other ~ E (r )

e il mo Other of the amp o p er r

R : In The following is the list of the products concerned: a We will. ~ g (r )

x

x + ! 2 x

0 on ! 2

g 0 R ) T o

e met a del p erio do: T

p R =g 0

x

x + ! 2 x

0 on ! 2

g 0 R ) T o

e The Commission a of the p the air do: T

p R =g 0

Quesito n. 3. La v elo it a massima si tro v a on la onserv azione dell’energia o on la legge oraria: 1 2 mv 2 max

e funzione solo di r : U (r )

~ dr

Z r 0 m g (r ) dr

m(g 0 =2R )r 2 a v endo p osto ~ g (r )

g (r ) ^ r

(g 0 =R ) ~ r e U (0)

0 : Essendo p oi r max

R ed r min

R sen ’ si tro v a U

U max U min

= 1 2 mg 0 R os 2 ’ e quindi v max

p 2 U =m

p g 0 R os ‘

3:95 km=s : Oppure x(t)

A os ! t on A

R os ’ ) v (t)

R ! os ’ sen ! t ) ) v max

R ! os ’

Question No n. 3. La v I’m going to it a maximum si The Commission v a on la onserv Action energy o on la Law time: 1 2 mv 2 Max

e function only di r : U (r )

~ dr

Z r 0 m g (r ) dr

m(g 0 =2R )r 2 a v Other p Other ~ g (r )

g (r ) ^ r

(g 0 =R ) ~ r e U (0)

0 : Being p oi r Max

R ed r Minimum number of days

R Other ’ si The Commission v a U

U Max U Minimum number of days

= 1 2 mg 0 R os 2 ’ e So, what do you mean? v Max

p 2 U =m

p g 0 R os ‘

3:95 km=s : Or x(t)

A os ! t on A

R os ’ ) v (t)

R ! os ’ Other ! t ) ) v Max

R ! os ’

x

2 )℄x ) ! 0 2

2 ) T 0 o

2 Numeri amen te g 0 =R

2

4 2 T 2 g

10 5 ) 2

2

1

2

1

2 T 0

1 2 T 0 o

1

r g 0 R

e quindi Æ T

he nel riferimen to non inerziale si do vrebb e onsiderare an he la forza di Coriolis (F Cor

x

2 )℄x ) ! 0 2

2 ) T 0 o

2 Number of the te g 0 =R