OII 2008 Nazionale Teorica

PROBLEMA n. 1 { T re Cilindri molto s ivolosi 100 Pun ti Quesito n. 1. La

gura qui sotto rappresen ta il sistema di forze agen ti sui tre ilindri iden ti ati ome n. 1, 2 e 3: ad essa si fa riferimen to p er i sim b oli. NOT A: p er motivi di hiar ezza le intensit a del le forze in questa

gur a non sono ripr o dotte in s ala. Nelle

gure seguen ti sono mostrati i diagrammi di orp o lib ero dei ilindri n. 3 e n. 1 (quello p er il ilindro n. 2

e simmetri o del se ondo). Le forze ~ N 1 e ~ N 2 e le loro reazioni, ~ N 0 1 e ~ N 0 2 ( he, p er il terzo prin ipio della dinami a, sono risp ettiv amen te uguali in mo dulo alle prime, on v erso opp osto) sono in linate di 60 Æ sul piano orizzon tale dato

he il triangolo

he ha p er v erti i i en tri delle sezioni ir olari dei ilindri in on tatto tra loro

e

hiaramen te equilatero; ne segue

he le loro proiezioni, orizzon tale e v erti ale, si ottengono moltipli ando i mo duli risp ettiv amen te p er os 60 Æ

1=2 e sen 60 Æ

p 3 =2. Detto ~ P

M ~ g il p eso di ogni ilindro, le ondizioni di equilibrio del ilindro n. 3, tenendo on to

he p er simmetria N 1

N 2 , p ortano all’equazione seguen te, relativ a alla omp onen te v erti ale delle forze 2 p 3 2 N 1 P

0 : Quindi N 1

P p 3 : P ag. 1 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 18 Ap rile 2008 P er l’equilibrio del sistema basta studiare il ilindro n. 1 essendo il sistema di forze equiv alen te a quello del ilindro n. 2. La ondizione di equilibrio p er traslazioni orizzon tali

e suÆ ien te p er determinare la tensione T in ias una delle due orde. 2T N 3 1 2 N 1

0 da ui 2T N 3

P 2 p 3 : Il v alore minimo di T si ha p er N 3

0 p er ui T min

P 4 p 3 :

Topic: Rigid Body Statics, Newtonian Mechanics Metodi: Free-Body Diagram, Vector Decomposition, Symmetry Argument Competenze: Diagrammatic Reasoning, Mathematical Modeling, Physical Reasoning Objects: Cylinder, String Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

The problem n. 1 { T re Other, of a kind used for the manufacture of goods Very much The following is the list of the countries of the European Union: 100 The Commission ti Question No n. 1. La

Gurus Here . below representing ta il system di Forces Other ti on the three Other, of a kind used for the manufacture of goods Other ti Other human beings n. 1, 2 e 3: ad It si fa References to p er i Yes b oil. Not A: p er reasons di the following is the list of the following: Other le intensity a of the le Forces in This one

The Commission shall adopt implementing acts in accordance with Article 21 of this Regulation. a Not I am The Commission shall adopt implementing acts. o The Commission shall adopt implementing acts. in s The wing. In the

Other following ti I am shown i The following table shows the following: di orp o The Commission shall adopt implementing acts. I was of the Other, of a kind used for the manufacture of goods n. 3 e n. 1 (that p er il Other, of a kind used for the manufacture of goods n. 2

e symmetries or of the If the wave is Le Forces ~ N 1 e ~ N 2 e le They the reaction, ~ N 0 1 e ~ N 0 2 ( he, p er il third The first of the the dynamics a, I am Responsibility Other I will not let you go. te equal to in mo Other The Commission first, on v Other Op. (including I am in strips di 60 Æ on the

• It ‘s not easy . The following is the list of the countries: such given

he il Triangle

he ha p er v The Commission shall adopt the following measures: i en three of the Sections Other of the Other, of a kind used for the manufacture of goods in on The touch between They

e

the following is the list of the following: te the equilateral; ne follows

he le They projections, The following is the list of the countries: such e v high-winged, si They get Multiple IDE i mo Other Responsibility Other I will not let you go. te p er os 60 Æ

1=2 e Other 60 Æ

p 3 =2. That said, ~ P

M ~ g il p That’s it. di Each the cylinder, le Other articles of heading No. di Balance of the Other, of a kind used for the manufacture of goods n. 3, holding on to

he p er The following table shows the following: N 1

N 2 , p Other, of a kind used for the manufacture of goods to the equation following te, Relative a to the The Commission shall adopt implementing acts. Other te v High-winged of the Forces 2 p 3 2 N 1 P

0 : So, what? N 1

P p 3 : 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: { 18 Ap Reels 2008 P er The balance of the system That ‘s enough . study il Other, of a kind used for the manufacture of goods n. 1 being il system di Forces Equivalent to Other te a That one. of the Other, of a kind used for the manufacture of goods n. 2. La the condition di Balance p er translations The following is the list of the countries: of which:

e The Commission te p er determining the la The following conditions shall apply: T in Ias one of the two The Horde. 2T N 3 1 2 N 1

0 da ui 2T N 3

P 2 p 3 : Il v Other Minimum di T si ha p er N 3

0 p er ui T Minimum number of days

P 4 p 3 :

Topic: Rigid Body Statics, Newtonian Mechanics Metodi: Free-Body Diagram, Vector Decomposition, Symmetry Argument Competenze: Diagrammatic Reasoning, Mathematical Modeling, Physical Reasoning Objects: Cylinder, String Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

Quesito n. 2. In assenza di attrito, p oi h

e le rette d’azione di tutte le forze presen ti tra i ilindri e tra il piano e i ilindri in terse ano gli assi di ias un ilindro, tutti i momen ti assiali sono n ulli; se dunque inizialmen te i ilindri sono in quiete, non si a vr a nessun moto rotatorio: i tre ilindri traslano senza rotolare. P oi h

e si assume

he il sistema a elerato non ollassi, esso si m uo v e rigidamen te e dunque, p er il se ondo prin ipio, l’a elerazione a pu o essere subito determinata essendo F

3M a. Si onsiderano p oi le equazioni di moto, separatamen te p er ias un ilindro (p er i sim b oli fare riferimen to ai relativi diagrammi di orp o lib ero), ome segue. Risp etto al aso stati o studiato prima, il sistema di forze non

e pi

u simmetri o: adesso le reazioni vin olari non hanno lo stesso mo dulo, men tre, p er il prin ipio di azione e reazione, resta v ero

he N 1

N 0 1 , N 2

N 0 2 , N 3

N 0 3 . P er il ilindro n. 1 8

<

: F N 3 1 2 N 1

M a N 4 P p 3 2 N 1

0 P er il ilindro n. 2 8

<

: N 0 3 + 1 2 N 2

M a N 5 P p 3 2 N 2

0 P er il ilindro n. 3 8

<

: 1 2 N 0 1 1 2 N 0 2

M a p 3 2 N 0 2 + p 3 2 N 0 1 P

0 P ag. 2 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 18 Ap rile 2008 L’insieme delle 6 equazioni ostituis ono un sistema nelle in ognite N 1;::: ;5 ed a. Quest’ultima, an-

he senza la onsiderazione fatta sopra, si sarebb e p otuta tro v are sommando mem bro a mem bro le tre equazioni relativ e alle omp onen ti orizzon tali del moto (le prime di ogni oppia), p er ui si pu o sostituire M a

F =3 Il sistema di forze sul ilindro n. 3 onsen te di ri a v are N 1 e N 2 . Risolv endo il sistema, infatti, si tro v a N 1

P p 3 + F 3 N 2

P p 3 F 3 Una prima ondizione su F si ottiene allora osserv ando

he il sistema non ollassa se risulta N 2

Da qui si ri a v a F < p 3P . La se onda ondizione su F si ottiene dalle equazioni del moto di uno degli altri due ilindri. In parti olare, dall’equazione della omp onen te orizzon tale del moto del ilindro n. 2, si ri a v a N 3

F 3 N 2 2

F 2 P 2 p 3 An he in questo aso dev e essere N 3

Da qui si ri a v a F

P p 3 . In on lusione, il sistema non ollassa se viene a elerato nel mo do des ritto da una forza F tale

he M g p 3 < F < p 3 M g : PROBLEMA n. 2 { T ri- i lo: : : termo dinami o 35 Pun ti Il rendimen to

di una ma hina termi a

e dato dal rapp orto tra il la v oro netto s am biato nel i lo e il alore assorbito. Il la v oro si al ola ome segue. Nell’isoterma quasi stati a 1{2, essendo V 2

2V 1 , il la v oro

e dato da: L 12

Z V 2 V 1 p(V ) dV

Z V 2 V 1 n R T V dV

n R T ln

V 2 V 1

= p 1 V 1 ln 2 Nel ra reddamen to isobaro quasi stati o 2{3, tenendo on to del fatto

he p 2

p 1 =2 e V 3

V 1 , il la v oro

e: L 23

Z V 3 V 2 p 2 dV

p 2 (V 3 V 2 )

1 2 p 1 V 1 In ne, nell’iso ora 2{3, il la v oro

e n ullo. Il la v oro netto del i lo risulta quindi: L i lo

p 1 V 1

ln 2 1 2

Si al ola ora il alore assorbito nel i lo: il sistema assorb e alore nell’espansione isoterma 1{2 e nel ris aldamen to iso oro 3{1. Nell’espansione isoterma, p oi h

e l’energia in terna di un gas p erfetto dip ende solo dalla temp eratura, U

0, e dunque Q 12

L 12 : Nel ris aldamen to iso oro, L

0, e dunque Q

U

n V T : P oi h

e il gas

e monoatomi o,

V

(3=2)R e risulta Q 31

3 2 n R T P ag. 3 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 18 Ap rile 2008 Dall’equazione di stato dei gas p erfetti, tenendo on to

he il v olume non v aria, si ha V p

n R T e dunque Q 31

3 2 V 1 p

3 4 V 1 p 1 In de nitiv a: Q ass

Q 12 + Q 31

p 1 V 1

ln 2 + 3 4

In ne il rendimen to risulta:

= L i lo Q ass

ln 2 1=2 ln 2 + 3=4

0:134

13:4 %

Topic: Thermodynamics Metodi: Thermodynamic Cycle Analysis, First Law of Thermodynamics, Ideal Gas Law, Calculus-Integration Competenze: Mathematical Modeling, Physical Reasoning Objects: Gas, Heat Engine Fonte: Testo (PDF) — p.2 Soluzione: Soluzioni (PDF)

Question No n. 2. In Absence di the friction, p oi h

e le The Commission shall adopt implementing acts. The Commission di All of them le Forces the present ti between i Other, of a kind used for the manufacture of goods e between il

• It ‘s not easy . e i Other, of a kind used for the manufacture of goods in Tese year The Commission Other, not further worked than hot-rolled di The Commission shall adopt a decision on the the cylinder, All of them. i Moments ti Other I am n the oils; se So, what? Initially te i Other, of a kind used for the manufacture of goods I am in You’re going to be quiet. Not si a vr a No one Motorcycles Rotary: i three Other, of a kind used for the manufacture of goods They’re moving. without rolling. P oi h

e si He takes

he il system Other, not further worked than cold-rolled Not The following is the list of the following: It si m uo v e The following is a list of the measures taken: te e So, what’s the point? p er il If the wave the first horse, The following is a list of the types of products: a pu o to be Right now. determined being F

3M a. Si They are all p oi le Equations di motorcycle, Separation of the te p er The Commission shall adopt a decision on the Other, of a kind used for the manufacture of goods (p er i Yes b oils to do References to ai Relative The following table shows the following: di orp o The Commission shall adopt implementing acts. I was . human beings Go on. Responsibility Other al Other States or studied Before that, il system di Forces Not

e pi

u symmetries or: Now le reactions Olary wine Not They have lo of the same mo the following: Men and women three, p er il The first di Action e reaction, Stay here v I was

he N 1

N 0 1 , N 2

N 0 2 , N 3

N 0 3 . P er il Other, of a kind used for the manufacture of goods n. 1 8

<

: F N 3 1 2 N 1

M a N 4 P p 3 2 N 1

0 P er il Other, of a kind used for the manufacture of goods n. 2 8

<

: N 0 3 + 1 2 N 2

M a N 5 P p 3 2 N 2

0 P er il Other, of a kind used for the manufacture of goods n. 3 8

<

: 1 2 N 0 1 1 2 N 0 2

M a p 3 2 N 0 2 + p 3 2 N 0 1 P

0 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: { 18 Ap Reels 2008 The whole of the 6 Equations The Commission shall adopt a decision on the un system In the in days N 1;::: ;5 ed a. The latter, an-

he without la The Commission shall adopt implementing acts. made above, si I would e p Other The Commission v are by adding The Commission Bro a The Commission Bro le three Equations Relative e The Commission The Commission shall adopt implementing acts. Other ti The following is the list of the countries: of which: of the Motorcycles (le Other di Each the opium), p er ui si pu o replace M a

F =3 Il system di Forces on the Other, of a kind used for the manufacture of goods n. 3 onsen te di ri a v are N 1 e N 2 . Resolved Other il system, In fact, si The Commission v a N 1

P p 3 + F 3 N 2

P p 3 F 3 One of them. Before the condition su F si The result is Then See also I ‘m going

he il system Not Other, of a thickness of not more than 10 mm se The following is the list of the N 2

Da Here . si ri a v a F < p 3P . La If it flows the condition su F si The result is from Equations of the Motorcycles di One of them. of the Other two The other two. In Wave parts, from the equation of the The Commission shall adopt implementing acts. Other te The following is the list of the countries: such of the Motorcycles of the Other, of a kind used for the manufacture of goods n. 2, si ri a v a N 3

F 3 N 2 2

F 2 P 2 p 3 An he in This one. Other It must be e to be N 3

Da Here . si ri a v a F

P p 3 . In On lust, il system Not Other, of a thickness of not more than 10 mm se He ‘s coming . with a diameter of not more than 30 mm In the mo do of the following: da One of them.

• What ? F such

he M g p 3 < F < p 3 M g : The problem n. 2 { T ri- i lo: : : term Other, of a kind used for the manufacture of goods 35 The Commission ti Il The Commission shall adopt implementing acts. to

di One of them. ma Other Other

e given from REP vegetable between il la v gold Net s am Other In the i lo e il Other I’m absorbed by it. Il la v gold si to the wave human beings Go on. In the isotherm Almost The following are the statistics: 1{2, being V 2

2V 1 , il la v gold

e given da: L 12

Z V 2 V 1 p(V ) dV

Z V 2 V 1 n R T V dV

n R T ln

V 2 V 1

= p 1 V 1 ln 2 In the ra The following is the list of the categories of persons who are not members of the European Parliament: to Other, of a kind used for the manufacture of goods Almost States or 2{3, holding on to of the made

he p 2

p 1 =2 e V 3

V 1 , il la v gold

e: L 23

Z V 3 V 2 p 2 dV

p 2 (V 3 V 2 )

1 2 p 1 V 1 In ne, in the ice Now 2{3, il la v gold

e n I’m going to go. Il la v gold Net of the i lo The following is the list of the So, what? L i lo

p 1 V 1

ln 2 1 2

Si to the wave Now il Other absorbed In the i lo: il system Other, of a kind used for the manufacture of goods e Other In the expansion Other, of a kind used for the manufacture of goods 1{2 e In the Other articles of heading No 5 to Other gold 3{1. In the expansion isothermal, p oi h

e energy in Other di un Gas p of which: Dip Other only from Temp the following: U

0, e So, what? Q 12

L 12 : In the Other articles of heading No 5 to Other gold, L

0, e So, what? Q

U

n V T : P oi h

e il Gas

e monoatoms or,

V

(3=2)R e The following is the list of the Q 31

3 2 n R T 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: { 18 Ap Reels 2008 From the equation di State of the Union of the Gas p the heirs, holding on to

he il v Other, of a kind used for the manufacture of foodstuffs Not v air, si ha V p

n R T e So, what? Q 31

3 2 V 1 p

3 4 V 1 p 1 In de Other a: Q Other

Q 12 + Q 31

p 1 V 1

ln 2 + 3 4

In ne il The Commission shall adopt implementing acts. to It is:

= L i lo Q Other

ln 2 1=2 ln 2 + 3=4

0:134

13:4 %

Topic: Thermodynamics Metodi: Thermodynamic Cycle Analysis, First Law of Thermodynamics, Ideal Gas Law, Calculus-Integration Competenze: Mathematical Modeling, Physical Reasoning Objects: Gas, Heat Engine Fonte: Testo (PDF) — p.2 Soluzione: Soluzioni (PDF)

PROBLEMA n. 3 { Prove di elasti it

a 100 Pun ti

Topic: Elasticity & Materials Metodi: Physical Modeling, Stress-Strain Analysis Competenze: Physical Reasoning Objects: Atom Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

The problem n. 3 { The evidence di You have it

a 100 The Commission ti

Topic: Elasticity & Materials Metodi: Physical Modeling, Stress-Strain Analysis Competenze: Physical Reasoning Objects: Atom Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

Quesito n. 1. Si rip ortano qui a

an o i gra i ri hiesti. Quello dell’energia p otenziale U (r ) pu o essere tra iato { in mo do qualitativ o { al olando direttamen te la funzione p er una de ina di v alori del rapp orto r =r 0 , oppure disegnando separatamen te i due rami ip erb oli i (1=r 2 e 1=r 4 ) e v alutando \gra amen te” la di erenza, o an ora on uno studio sempli ato di funzione ( asi limite, asin toti, massimi e minimi). Dalla de nizione di di erenza di energia p otenziale, ome l’opp osto del la v oro della forza al v ariare della distanza r ,

he p er sp ostamen ti in-

nitesimi si s riv e Æ U

~ F

~ Æ r , si pu o dedurre la relazione in v ersa

he serv e in questo aso p er determinare la forza in teratomi a: f (r )

dU (r ) dr

4 U 0 r 4 0 r 5 4 U 0 r 2 0 r 3 : Il gra o della forza si pu o quindi ottenere sempli emen te onsiderando la deriv ata, io

e osserv ando nel primo gra o la p endenza e i pun ti di minimo e di

esso, o altrimen ti si pu o utilizzare l’espressione espli ita della forza, in analogia on quan to fatto on quello dell’energia.

Topic: Elasticity & Materials, Newtonian Mechanics Metodi: Calculus-Integration, Energy Conservation Method, Physical Modeling Competenze: Mathematical Modeling, Diagrammatic Reasoning, Physical Reasoning Objects: Atom Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

Question No n. 1. Si RIP Other, of a kind used for the manufacture of goods Here . a

an o i The following is the list of the i ri I’m not going to. That one. energy p The following information shall be provided: U (r ) pu o to be between yachts { in mo do The quality o { The Dutch Directors te la function p er One of them. of in di v Other of the REP vegetable r =r 0 , or Drawing Separation of the te i two Other ip Herbs and spices Oil and gas (1=r 2 e 1=r 4 ) e v by highlighting The Commission I will not let you go. te” la di the herring, o

• I’m not going to. on One of them. study Other the act di function (a) the the limit, Other the total of the Maximum e The Commission shall adopt implementing acts in accordance with Article 21 of this Regulation. From her . de Other di di the herence di energy p the following conditions shall apply: human beings the op Other of the la v gold of the
• What ? al v Airing of the distance r ,

he p er sp Other articles of heading No. ti in-

Other si The following is the list of the e Æ U

~ F

~ Æ r , si pu o the following points shall be added: la Report by the Commission in v Other

he Serving e in This one. Other p er determining the la

• What ? in a. f (r ) = dU (r ) dr = 4 U 0 r 4 0 r 5 4 U 0 r 2 0 r 3 : Il The following is the list of the o of the
• What ? si pu o So, what do you mean? to obtain Other te by weeping la derivatives the following: io

e See also I ‘m going In the first The following is the list of the o la p Other, of a kind used for the manufacture of goods e i the following points are added: ti di Minimum e di

It is, o Other ti si pu o use the expression Explain the following of the Come on, you guys. in The following is the list of the products concerned: on When to made on That one. The energy.

Topic: Elasticity & Materials, Newtonian Mechanics Metodi: Calculus-Integration, Energy Conservation Method, Physical Modeling Competenze: Mathematical Modeling, Diagrammatic Reasoning, Physical Reasoning Objects: Atom Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

Quesito n. 2. Da quan to sopra si evin e

he p er r

r 0 l’energia p otenziale ha un minimo, e la forza

e n ulla. Questa

e p ertan to la distanza di equilibrio (stabile). Nell’espressione pre eden te della forza il primo termine

e repulsiv o e il se ondo

e attrattiv o; si ome il primo termine

e maggiore del se ondo in v alore assoluto quando r < r 0 , l’in terazione

e repulsiv a p er r < r 0 e attrattiv a p er r

r 0 . In ondizioni di equilibrio l’energia p otenziale in teratomi a

e U 0 . In un mo dello pi

u realisti o gli esp onen ti sarebb ero pi

u alti, ma il mo dello

e stato qui s elto in mo do da essere fa ilmen te trattabile. P ag. 4 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 18 Ap rile 2008

Topic: Elasticity & Materials, Newtonian Mechanics Metodi: Energy Conservation Method, Physical Modeling Competenze: Physical Reasoning, Mathematical Modeling Objects: Atom Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

Question No n. 2. Da When to above si (i) the following:

he p er r

r 0 energy p The following information shall be provided: ha un at least, e la

• What ?

e n I’m going to see you. This one

e p Other to la distance di Balance (stable) In the expression for Eden te of the

• What ? il first term

e Rejecting o e il If the wave

e attractive o;

• Yes, I did. il first term

e major of the If the wave in v Other The total When r < r 0 , l’in Tanning

e Rejecting a p er r < r 0 e attractive a p er r

r 0 . In Other articles of heading No. di Balance energy p The following information shall be provided: in The following table shows the results of the tests:

e U 0 . In un mo of the pi

u realistic or The Commission External trade Other ti I would I was pi

u high, ma il mo of the

e State of the Union Here . The Commission shall adopt a decision on the in mo do da to be It’s a pain. te It’s not treatable. 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: { 18 Ap Reels 2008

Topic: Elasticity & Materials, Newtonian Mechanics Metodi: Energy Conservation Method, Physical Modeling Competenze: Physical Reasoning, Mathematical Modeling Objects: Atom Fonte: Testo (PDF) — p.4 Soluzione: Soluzioni (PDF)

Quesito n. 3. A seguito dell’appli azione di una forza esterna, il ristallo si deforma sp ostandosi dalla ondizione di equilibrio, ed ogni distanza in teratomi a nella direzione della forza passa da r 0 a r

r 0 + Æ (nelle direzioni ortogonali alla forza la distanza in teratomi a rimane r 0 p er h

e non i sono in terazioni oi se ondi vi ini). In tal mo do si eser ita una forza

he bilan ia omplessiv amen te quella esterna. P er studiare le pi ole deformazioni si p onga r

r 0 + Æ on Æ

r 0 ; si tras urano p oi i termini in (Æ =r 0 ) 2 risp etto a quelli in Æ =r 0 . La forza diviene f

4 U 0 r 4 0 (r 0 + Æ ) 5 4 U 0 r 2 0 (r 0 + Æ ) 3 e sviluppando i binomi di Newton a denominatore e fermandosi al prim’ordine in Æ =r 0 , si ha f

4 U 0 1 r 0 (1 + 5Æ =r 0 ) 4 U 0 1 r 0 (1 + 3Æ =r 0 ) : Nello stesso ordine di approssimazione si pu o s riv ere

he 1 1 + 5Æ =r 0

1 5 Æ r 0 e analogamen te p er l’altro termine. Si p otev a giungere a questa espressione an he in un solo passo essendo, (1 + x)

1 +

x on

= 5. Si ottiene quindi f

4 U 0 r 0

1 5 Æ r 0

1 3 Æ r 0

= 8 U 0 Æ r 2 0 : Ogni legame viene allungato di una quan tit a Æ nella direzione di appli azione della forza esterna, quindi L=L

Æ =r 0 . Le forze f in terne si bilan iano, e rimangono le forze esterne al b ordo del ristallo,

he si sommano in prop orzione al n umero n di legami p er unit a di sup er ie; essendo F

nAf ed n

1=r 2 0 si ha F A

jnf j A

jf j r 2 0 da ui segue

he E

f =r 2 0

Æ =r 0

8 U 0 r 3 0 : Sostituendo i v alori n umeri i e ri ordando

he 1 eV

1:6

10 19 J, si ha E

2:37

10 11 N=m 2 . L’ordine di grandezza otten uto

e ragionev ole (p er l’a iaio

e E

2

10 11 N=m 2 ).

Topic: Elasticity & Materials Metodi: Approximation & Series Expansion, Stress-Strain Analysis, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Atom Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)

Question No n. 3. A followed of the application di One of them.

• What ? the external, il food si Deformation sp Other, of a kind used for the manufacture of goods from the condition di the balance, ed Each distance in The following table shows the results of the tests: In the Direction of the
• What ? Go ahead . da r 0 a r = r 0

Æ (in the Directions orthogenous to the

• What ? la distance in The following table shows the results of the tests: It stays r 0 p er h

e Not i I am in Other, of a kind used for the manufacture of goods oi If you go The Commission shall adopt implementing acts in accordance with Article 21 of this Regulation. In such mo do si Exercise One of them.

• What ?

he Balance sheets Other I will not let you go. te That one. The outside. P er study le Other Deformations si p Other r

r 0 + Æ on Æ

r 0 ; si Trans uranium p oi i the terms in (Æ =r 0 ) 2 Responsibility Other a The other in Æ =r 0 . La

• What ? It becomes f = 4 U 0 r 4 0 (r 0

Æ ) 5 4 U 0 r 2 0 (r 0 + Æ ) 3 e developing i Other di The Commission shall adopt implementing acts in accordance with Article 21 of this Regulation. a Name of the manufacturer e By stopping al First order in Æ =r 0 , si ha f

4 U 0 1 r 0 (1 + 5Æ =r 0 ) 4 U 0 1 r 0 (1 + 3Æ =r 0 ) : In the middle of the same Order di Approximation si pu o The following is the list of the The following are the

he 1 1 + 5Æ =r 0

1 5 Æ r 0 e Other te p er The other The end. Si p Other a to arrive a This one The following is the list of the following: an he in un only I ‘m going through being, (1 + x)

1 +

x on

= 5. Si The result is So, what do you mean? f

4 U 0 r 0

1 5 Æ r 0

1 3 Æ r 0

= 8 U 0 Æ r 2 0 : Each the link He ‘s coming . of a width of not more than 30 mm di One of them. When Title a Æ In the Direction di The Commission of the

• What ? the external, So, what do you mean? L=L = Æ =r 0 . Le Forces f in Other si the balance sheet, e They stay. le Forces Other al b I ‘m not a doctor . of the the food,

he si They add up in Prop Other al n Other n di binding p er Unit a di Supplementary er ie; being F

nF ed n

1=r 2 0 si ha F A

Other j A

jf j r 2 0 da ui follows

he E

f =r 2 0

Æ =r 0

8 U 0 r 3 0 : Substituting i v Other n The following is the list of e Re-ordering

he 1 eV

1:6

10 19 J, si ha E

2:37

10 11 N=m 2 . The Order di size obtained Other

e That ‘s right . Oil and gas (p er The Commission shall adopt the following measures:

e E

2

10 11 N=m 2 ).

Topic: Elasticity & Materials Metodi: Approximation & Series Expansion, Stress-Strain Analysis, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Atom Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)

Quesito n. 4. P er tro v are le ondizioni di rottura, si pu o osserv are

he p er v alori molto grandi di r la forza f tende a zero, p er ui o orre tro v are quando l’in tensit a della forza in teratomi a attrattiv a f ha un massimo in funzione di r : al di l a non pu o pi

u omp ensare la forza esterna e il ristallo si sp ezza. Questa ondizione si ha quando d f dr

0 ) 20 U 0 r 4 0 r 6 12 U 0 r 2 0 r 4

0 io

e r

p 5=3 r 0

1:291 r 0 : La deformazione, o allungamen to relativ o, in queste ondizioni

e L L

r r 0 r 0

0:291 : Naturalmen te questo v alore non

e realisti o p er h

e un solido reale on tiene sempre impurezze

he ne determinano la rottura prima. P ag. 5 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 18 Ap rile 2008

Topic: Elasticity & Materials Metodi: Calculus-Integration, Physical Modeling, Approximation & Series Expansion Competenze: Mathematical Modeling, Physical Reasoning Objects: Atom Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)

Question No n. 4. P er The Commission v are le Other articles of heading No. di the breakdown, si pu o See also are

he p er v Other Very much large di r la

• What ? f Tents a the value of the underlying equation is zero, p er ui o Horrors The Commission v are When l’in Tensile a of the
• What ? in The following table shows the results of the tests: attractive a f ha un Not more than in function di r : al di l a Not pu o pi

u The Commission shall adopt implementing acts. the test la

• What ? Other e il food si sp I’m not going to. This one the condition si ha When d f dr = 0 ) 20 U 0 r 4 0 r 6 12 U 0 r 2 0 r 4 = 0 io

e r

p 5=3 r 0

1:291 r 0 : La deformation, o Other to Relative o, in These Other articles of heading No.

e L L

r r 0 r 0

0:291 : Naturalmen te This one. v Other Not

e realistic or p er h

e un Other, not further worked than hot-rolled real on It is Always Impurities

he ne They determine la Other, not further worked than hot-rolled I’m not going to. 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: { 18 Ap Reels 2008

Topic: Elasticity & Materials Metodi: Calculus-Integration, Physical Modeling, Approximation & Series Expansion Competenze: Mathematical Modeling, Physical Reasoning Objects: Atom Fonte: Testo (PDF) — p.5 Soluzione: Soluzioni (PDF)

Quesito n. 5. A questo pun to p er ottenere lo sforzo di rottura basta sostituire r

p 5=3 r 0 nella form ula di f e p oi ri ordare

he lo sforzo

e F

A

jf j =r 2 0 . Si ottiene quindi f

4 U 0 r 0 ”

3 5

5=2

3 5

3=2

= 0:7436 U 0 r 0 e in ultima analisi F

A

0:7436 (U 0 =r 3 0 )

2:2

10 10 N=m 2 : An he questo v alore non

e realisti o p er la stessa ragione pre eden te (p er l’a iaio

e 50 v olte meno, ma l’a iaio non

e ristallino). PROBLEMA n. 4 { Condensato ri semp re pi

u pi oli 65 Pun ti

Topic: Elasticity & Materials Metodi: Stress-Strain Analysis, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Atom, Capacitor Fonte: Testo (PDF) — p.6 Soluzione: Soluzioni (PDF)

Question No n. 5. A This one. the following points are added: to p er to obtain lo The amount of effort di Other, not further worked than hot-rolled That ‘s enough . replace r

p 5=3 r 0 In the form The Commission shall adopt a decision on the di f e p oi Re-order

he lo The amount of effort

e F

A

jf j =r 2 0 . Si The result is So, what do you mean? f

4 U 0 r 0 ”

3 5

5=2

3 5

3=2

= 0:7436 U 0 r 0 e in Last of the The following is the list of the samples: F

A

0:7436 (U 0 =r 3 0 )

2:2

10 10 N=m 2 : An he This one. v Other Not

e realistic or p er la itself Right for Eden te (p er The Commission shall adopt a decision on the

e 50 v Other less, ma The Commission shall adopt a decision on the Not

e (Some of the above mentioned products are not available for sale) The problem n. 4 { Other, of a kind used for the manufacture of goods ri The Commission shall adopt the following: re pi

u Other oils 65 The Commission ti

Topic: Elasticity & Materials Metodi: Stress-Strain Analysis, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Atom, Capacitor Fonte: Testo (PDF) — p.6 Soluzione: Soluzioni (PDF)

Quesito n. 1. In un ondensatore sferi o, detta Q la ari a distribuita uniformemen te sul onduttore in terno di raggio R 1 , il amp o tra le armature si determina appli ando il teorema di Gauss (?) I S (r ) ~ E

^ n ds

I S (r ) E (r ) ds

4 r 2 E (r )

Q ” 0 ) E (r )

Q 4 ” 0 r 2 e la d.d.p. tra le armature, direttamen te dalla de nizione, essendo R 2 il raggio del onduttore sferi o esterno. V

V (R 1 ) V (R 2 )

Z R 2 R 1 E (r ) dr

Q 4 ” 0

1 R 1 1 R 2

Si ri a v a quindi la apa it a C

Q V

4 ” 0 1=R 1 1=R 2

4 ” 0 R 1 R 2 R 2 R 1 : P onendo adesso R 1

R e R 2

R + Æ R on Æ R

R si ottiene C

4 ” 0 R (R + Æ R ) R + Æ R R

” 0 4 R 2 Æ R

” 0 S d

he oin ide appun to on l’espressione della apa it a di un ondensatore piano nel vuoto.

Topic: Electrostatics Metodi: Gauss’s Law, Electric Potential Method, Approximation & Series Expansion Competenze: Mathematical Modeling, Physical Reasoning Objects: Capacitor Fonte: Testo (PDF) — p.6 Soluzione: Soluzioni (PDF)

Question No n. 1. In un Other, of a kind used for the manufacture of goods spheres or, the following: Q la the following points are inserted: distributed The uniforms te on the Other, of a kind used for the manufacture of goods in Other di Radius R 1 , il amp o between le Other articles of iron or steel si determines the The following is a list of the il Theorem di Gauss (?) I S (r ) ~ E

^ n ds

I S (r ) E (r ) ds

4 r 2 E (r )

Q ” 0 ) E (r )

Q 4 ” 0 r 2 e la d.d.p. between le Armour, Directors te from de the Commission, being R 2 il Radius of the Other, of a kind used for the manufacture of goods spheres or The outside. V

V (R 1 ) V (R 2 )

Z R 2 R 1 E (r ) dr

Q 4 ” 0

1 R 1 1 R 2

Si ri a v a So, what do you mean? la

• What? a C = Q V = 4 ” 0 1=R 1 1=R 2 = 4 ” 0 R 1 R 2 R 2 R 1 : 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 Now R 1 = R e R 2 = R

Æ R on Æ R

R si The result is C

4 ” 0 R (R + Æ R ) R + Æ R R

” 0 4 R 2 Æ R

” 0 S d

he The Commission shall adopt a decision on the The Commission has already adopted a proposal. to on the expression of the

• What? a di un Other, of a kind used for the manufacture of goods
• It ‘s not easy . In the It’s empty.

Topic: Electrostatics Metodi: Gauss’s Law, Electric Potential Method, Approximation & Series Expansion Competenze: Mathematical Modeling, Physical Reasoning Objects: Capacitor Fonte: Testo (PDF) — p.6 Soluzione: Soluzioni (PDF)

Quesito n. 2. La apa it a del ondensatore e la tensione di la v oro sono date da C

” 0 ” r S d V

V max

E m d P osto

= 2S d da ui S

=(2 d) C

” 0 ” r

2 d 2 on d 2

V 2 max E 2 m p er ui, eliminando d, si ottiene C

” 0 ” r E 2 m

2 V 2 max )

min

2 C V 2 max ” 0 ” r E 2 m (?) Il sim b olo H indi a

he l’in tegrale di sup er ie viene al olato su una sup er ie

hiusa, in questo aso una sfera di raggio r . P ag. 6 AIF { Olimpiadi di Fisi a Ga ra Nazionale: SOLUZIONE della Prova T eo ri a { Senigallia { 18 Ap rile 2008

Topic: Electrostatics Metodi: Electric Potential Method, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Capacitor Fonte: Testo (PDF) — p.6 Soluzione: Soluzioni (PDF)

Question No n. 2. La

• What? a of the Other, of a kind used for the manufacture of goods e la The following conditions shall apply: di la v gold I am Date of the date da C = ” 0 ” r S d V

V Max

E m d P Other

= 2S d da ui S

=(2 d) C

” 0 ” r

2 d 2 on d 2

V 2 Max E 2 m p er ui, by eliminating d, si The result is C

” 0 ” r E 2 m

2 V 2 Max )

Minimum number of days

2 C V 2 Max ” 0 ” r E 2 m (?) Il Yes b The following is the list of the countries of the European Union: H Indies and

he l’in Other di Supplementary er ie He ‘s coming . to the wave su One of them. Supplementary er ie

the hive, in This one. Other One of them. The sphere di Radius r . P ag. 6 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: { 18 Ap Reels 2008

Topic: Electrostatics Metodi: Electric Potential Method, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning Objects: Capacitor Fonte: Testo (PDF) — p.6 Soluzione: Soluzioni (PDF)

Quesito n. 3. Dall’espressione pre eden te si dedu e

he p er ottenere il v olume minimo, a parit a di apa it a e tensione massima di la v oro, o orre massimizzare il pro dotto ” r E 2 m . Dall’analisi della tab ella

he segue si ri a v a

he tra i materiali itati quello

he onsen te di realizzare ondensatori pi

u pi oli

e la mi a. Materiale ” r E m [ kV/mm ℄ ” r E 2 m [ 10 15 V 2 =m 2 ℄ 1. Carta paraÆnata 2:5 50 6:25 2. Cerami a 60 15 13:5 3. Mi a 8 90 64:8 4. P olistirolo 2:6 50 6:50 5. P or ellana 6 25 3:75 6. Resina ep ossidi a 4 35 4:90 7. T e on 2:2 20 0:88

Topic: Electrostatics Metodi: Physical Modeling, Experimental Data Analysis Competenze: Physical Reasoning, Mathematical Modeling Objects: Capacitor Fonte: Testo (PDF) — p.7 Soluzione: Soluzioni (PDF)

Question No n. 3. From the expression for Eden te si and deductions

he p er to obtain il v Other, of a kind used for the manufacture of foodstuffs at least, a The same a di

• What? a e The following conditions shall apply: maximum di la v gold, o Horrors Maximizing il for The Commission ” r E 2 m . From the analysis of the tab She ‘s a

he follows si ri a v a

he between i Other materials Other, not further worked than cut That one.

he onsen te di The Commission shall adopt the following measures: Other, of a kind used for the manufacture of goods pi

u Other oils

e la mi a. Other materials ” r E m [ kV/mm ℄ ” r E 2 m [ 10 15 V 2 =m 2 ℄ 1. Paper

• The first 2:5 50 6:25

Look at me 60 15 13:5 3. Mi a 8 90 64:8 4. P Other, of a thickness of not more than 10 mm 2:6 50 6:50 5. P or ellana 6 25 3:75 6. Other, of a kind used for the manufacture of goods ep Other, of a kind used for the manufacture of textile materials 4 35 4:90 7. T e on 2:2 20 0:88

Topic: Electrostatics Metodi: Physical Modeling, Experimental Data Analysis Competenze: Physical Reasoning, Mathematical Modeling Objects: Capacitor Fonte: Testo (PDF) — p.7 Soluzione: Soluzioni (PDF)

Quesito n. 4. In v ertendo la relazione pre eden te si tro v a ” r E 2 m

2 C V 2 max ” 0

= 13:3

10 15 V 2 =m 2 Appare ragionev ole

he il ondensatore sia erami o, dato

he utilizzando la mi a a vrebb e p otuto essere an ora pi

u pi olo. 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. 7

Topic: Electrostatics Metodi: Physical Modeling, Experimental Data Analysis Competenze: Physical Reasoning, Mathematical Modeling Objects: Capacitor Fonte: Testo (PDF) — p.7 Soluzione: Soluzioni (PDF)

Question No n. 4. In v with a diameter of not more than 30 mm la Report by the Commission for Eden te si The Commission v a ” r E 2 m

2 C V 2 Max ” 0

= 13:3

10 15 V 2 =m 2 Appears That ‘s right . Oil and gas

he il Other, of a kind used for the manufacture of goods It’s either were or, given

he using la mi a a The Commission shall adopt implementing acts. e p 8 to be

• I’m not going to. pi

u I’m not going to. Other materials for The Commission from Group o Olympic Games Project Project Olympic Games The Secretariat The Olympics Italian of the The following is a list of p Results Li eo The Commission o \U. Mo ‘R’ The Commission The Master The fax: 041.584.1272 E-mail: Free olive oil t P ag. 7

Topic: Electrostatics Metodi: Physical Modeling, Experimental Data Analysis Competenze: Physical Reasoning, Mathematical Modeling Objects: Capacitor Fonte: Testo (PDF) — p.7 Soluzione: Soluzioni (PDF)