2008 Semifinal Exam AAPT UNITEDSTATES PHYSICS TEAM AIP 2008 Semifinal Exam DO NOT DISTRIBUTE THIS PAGE Important Instructions for the Exam Supervisor This examination consists of three parts. Part A has four questions and is allowed 90 minutes. Part B has two questions and is allowed 90 minutes. Part C has one question and is allowed 20 minutes.The answer for Part C will not be used for team selection,but will be used for special recognition from the Optical Society of America. The first page that follows is a cover sheet.Examinees may keep the cover sheet for all three parts of the exam. The three parts are then identified by the center header on each page.Examinees are only allowed to do one part at a time,and may not work on other parts,even if they have time remaining. Allow 90 minutes to complete Part A.Do not let students look at Part B or Part C.Collect the answers to Part A before allowing the examinee to begin Part B.Examinees are allowed a 10 to 15 minutes break between parts A and B. Allow 90 minutes to complete Part B.Do not let students look at Part C or go back to Part A.Collect the answers to part B before allowing the examinee to begin Part C.Examinees are allowed a 10 to 15 minutes break between Parts B and C. Allow 20 minutes to complete Part C.This part is optional;scores on Part C will not be used to select the US Team.Examinees may not go back to Part A or B. Ideally the test supervisor will divide the question paper into 4 parts:the cover sheet (page 2),Part A (pages 3-7),Part B(pages 8-10),and Part C(page 11).Examinees should be provided the parts individually,although they may keep the cover sheet. The supervisor must collect all examination questions,including the cover sheet,at the end of the exam, as well as any scratch paper used by the examinees.Examinees may not take the exam questions.The examination questions may be returned to the students after March 31,2008. Examinees are allowed calculators,but they may not use symbolic math,programming,or graphic features of these calculators.Calculators may not be shared and their memory must be cleared of data and programs.Cell phones,PDA's or cameras may not be used during the exam or while the exam papers are present.Examinees may not use any tables,books,or collections of formulas. Please provide the examinees with graph paper for Part A. Copyright C2008 American Association of Physics Teachers
2008 Semifinal Exam 1 AAPT UNITED STATES PHYSICS TEAM AIP 2008 Semifinal Exam DO NOT DISTRIBUTE THIS PAGE Important Instructions for the Exam Supervisor • This examination consists of three parts. • Part A has four questions and is allowed 90 minutes. • Part B has two questions and is allowed 90 minutes. • Part C has one question and is allowed 20 minutes. The answer for Part C will not be used for team selection, but will be used for special recognition from the Optical Society of America. • The first page that follows is a cover sheet. Examinees may keep the cover sheet for all three parts of the exam. • The three parts are then identified by the center header on each page. Examinees are only allowed to do one part at a time, and may not work on other parts, even if they have time remaining. • Allow 90 minutes to complete Part A. Do not let students look at Part B or Part C. Collect the answers to Part A before allowing the examinee to begin Part B. Examinees are allowed a 10 to 15 minutes break between parts A and B. • Allow 90 minutes to complete Part B. Do not let students look at Part C or go back to Part A. Collect the answers to part B before allowing the examinee to begin Part C. Examinees are allowed a 10 to 15 minutes break between Parts B and C. • Allow 20 minutes to complete Part C. This part is optional; scores on Part C will not be used to select the US Team. Examinees may not go back to Part A or B. • Ideally the test supervisor will divide the question paper into 4 parts: the cover sheet (page 2), Part A (pages 3-7), Part B (pages 8-10), and Part C (page 11). Examinees should be provided the parts individually, although they may keep the cover sheet. • The supervisor must collect all examination questions, including the cover sheet, at the end of the exam, as well as any scratch paper used by the examinees. Examinees may not take the exam questions. The examination questions may be returned to the students after March 31, 2008. • Examinees are allowed calculators, but they may not use symbolic math, programming, or graphic features of these calculators. Calculators may not be shared and their memory must be cleared of data and programs. Cell phones, PDA’s or cameras may not be used during the exam or while the exam papers are present. Examinees may not use any tables, books, or collections of formulas. • Please provide the examinees with graph paper for Part A. Copyright c 2008 American Association of Physics Teachers
2008 Semifinal Exam Cover Sheet 2 AAPT UNITEDSTATES PHYSICS TEAM AIP 2008 Semifinal Exam INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN Work Part A first.You have 90 minutes to complete all four problems.Each question is worth 25 points.Do not look at Parts B or C during this time. After you have completed Part A you may take a break. Then work Part B.You have 90 minutes to complete both problems.Each question is worth 50 points. Do not look at Parts A or C during this time. Show all your work.Partial credit will be given.Do not write on the back of any page.Do not write anything that you wish graded on the question sheets. Start each question on a new sheet of paper.Put your school ID number,your name,the question number and the page number/total pages for this problem,in the upper right hand corner of each page.For example, School ID# Doe,Jamie A1-1/3 A hand-held calculator may be used.Its memory must be cleared of data and programs.You may use only the basic functions found on a simple scientific calculator.Calculators may not be shared.Cell phones,PDA's or cameras may not be used during the exam or while the exam papers are present. You may not use any tables,books,or collections of formulas. Questions with the same point value are not necessarily of the same difficulty. Part C is an optional part of the test.You will be given 20 additional minutes to complete Part C. Your score on Part C will not affect the selection for the US Team,but can be used for special prizes and recognition to be awarded by the Optical Society of America. In order to maintain exam security,do not communicate any information about the questions (or their answers/solutions)on this contest until after March 31,2008. Possibly Useful Information.You may use this sheet for all three parts of the exam. g=9.8 N/kg G=6.67×10-11N.m2/kg2 k=1/4xe0=8.99×109N.m2/C2 km Ho/4T 10-7 T.m/A c=3.00×108m/s kB=1.38×10-23J/K NA=6.02×1023(mol)-1 R=NAkB =8.31 J/(mol.K) σ=5.67×10-8J/(sm2.K4) e=1.602×10-19C 1eV=1.602×10-19J h=6.63×10-34J.s=4.14×10-15eV.s me=9.109×10-31kg=0.511MeV/c2(1+x)n≈1+nz for<1 sin0≈0-93forl0l<1 cos0≈1-号92for|9<1 Copyright C2008 American Association of Physics Teachers
2008 Semifinal Exam Cover Sheet 2 AAPT UNITED STATES PHYSICS TEAM AIP 2008 Semifinal Exam INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN • Work Part A first. You have 90 minutes to complete all four problems. Each question is worth 25 points. Do not look at Parts B or C during this time. • After you have completed Part A you may take a break. • Then work Part B. You have 90 minutes to complete both problems. Each question is worth 50 points. Do not look at Parts A or C during this time. • Show all your work. Partial credit will be given. Do not write on the back of any page. Do not write anything that you wish graded on the question sheets. • Start each question on a new sheet of paper. Put your school ID number, your name, the question number and the page number/total pages for this problem, in the upper right hand corner of each page. For example, School ID # Doe, Jamie A1 - 1/3 • A hand-held calculator may be used. Its memory must be cleared of data and programs. You may use only the basic functions found on a simple scientific calculator. Calculators may not be shared. Cell phones, PDA’s or cameras may not be used during the exam or while the exam papers are present. You may not use any tables, books, or collections of formulas. • Questions with the same point value are not necessarily of the same difficulty. • Part C is an optional part of the test. You will be given 20 additional minutes to complete Part C. Your score on Part C will not affect the selection for the US Team, but can be used for special prizes and recognition to be awarded by the Optical Society of America. • In order to maintain exam security, do not communicate any information about the questions (or their answers/solutions) on this contest until after March 31, 2008. Possibly Useful Information. You may use this sheet for all three parts of the exam. g = 9.8 N/kg G = 6.67 × 10−11 N · m2/kg2 k = 1/4πǫ0 = 8.99 × 109 N · m2/C 2 km = µ0/4π = 10−7 T · m/A c = 3.00 × 108 m/s kB = 1.38 × 10−23 J/K NA = 6.02 × 1023 (mol)−1 R = NAkB = 8.31 J/(mol · K) σ = 5.67 × 10−8 J/(s · m2 · K4 ) e = 1.602 × 10−19 C 1eV = 1.602 × 10−19 J h = 6.63 × 10−34 J · s = 4.14 × 10−15 eV · s me = 9.109 × 10−31 kg = 0.511 MeV/c 2 (1 + x) n ≈ 1 + nx for |x| ≪ 1 sin θ ≈ θ − 1 6 θ 3 for |θ| ≪ 1 cos θ ≈ 1 − 1 2 θ 2 for |θ| ≪ 1 Copyright c 2008 American Association of Physics Teachers
2008 Semifinal Exam Part A 3 Part A Question Al Four square metal plates of area A are arranged at an even spacing d as shown in the diagram.(Assume that Ad2.) Plate I Plate 3 ate4 Plates 1 and 4 are first connected to a voltage source of magnitude Vo,with plate 1 positive;plates 2 and 3 are then connected together with a wire.The wire is subsequently removed.Finally,the voltage source attached between plates 1 and 4 is replaced with a wire.The steps are summarized in the diagrams below Step I Step 2 Step 3 Find the resulting potential difference AVi2 between plates 1 and 2;like wise find AV23 and AV34,defined similarly. Assume,in each case,that a positive potential difference means that the top plate is at a higher potential than the bottom plate. Copyright C2008 American Association of Physics Teachers
2008 Semifinal Exam Part A 3 Part A Question A1 Four square metal plates of area A are arranged at an even spacing d as shown in the diagram. (Assume that A ≫ d 2 .) Plate 1 Plate 2 Plate 3 Plate 4 d d d Plates 1 and 4 are first connected to a voltage source of magnitude V0, with plate 1 positive; plates 2 and 3 are then connected together with a wire. The wire is subsequently removed. Finally, the voltage source attached between plates 1 and 4 is replaced with a wire. The steps are summarized in the diagrams below. Step 1 Step 2 Step 3 Find the resulting potential difference ∆V12 between plates 1 and 2; like wise find ∆V23 and ∆V34, defined similarly. Assume, in each case, that a positive potential difference means that the top plate is at a higher potential than the bottom plate. Copyright c 2008 American Association of Physics Teachers
2008 Semifinal Exam Part A Question A2 A simple heat engine consists of a moveable piston in a cylinder filled with an ideal monatomic gas.Initially the gas in the cylinder is at a pressure Po and volume Vo.The gas is slowly heated at constant volume.Once the pressure reaches 32Po the piston is released,allowing the gas to expand so that no heat either enters or escapes the gas as the piston moves.Once the pressure has returned to Po the outside of the cylinder is cooled back to the original temperature,keeping the pressure constant.For the monatomic ideal gas you should assume that the molar heat capacity at constant volume is given by Cy=R,where R is the ideal gas constant.You may express your answers in fractional form or as decimals.If you choose decimals,keep three significant figures in your calculations.The diagram below is not necessarily drawn to scale. 32P0 Po Vo Vmax Volume a.Let Vax be the maximum volume achieved by the gas during the cycle.What is Vax in terms of Vo?If you are unable to solve this part of the problem,you may express your answers to the remaining parts in terms of Vmax without further loss of points. b.In terms of Po and Vo determine the heat added to the gas during a complete cycle. c.In terms of Po and Vo determine the heat removed from the gas during a complete cycle. d.What is the efficiency of this cycle? Copyright C2008 American Association of Physics Teachers
2008 Semifinal Exam Part A 4 Question A2 A simple heat engine consists of a moveable piston in a cylinder filled with an ideal monatomic gas. Initially the gas in the cylinder is at a pressure P0 and volume V0. The gas is slowly heated at constant volume. Once the pressure reaches 32P0 the piston is released, allowing the gas to expand so that no heat either enters or escapes the gas as the piston moves. Once the pressure has returned to P0 the outside of the cylinder is cooled back to the original temperature, keeping the pressure constant. For the monatomic ideal gas you should assume that the molar heat capacity at constant volume is given by CV = 3 2R, where R is the ideal gas constant. You may express your answers in fractional form or as decimals. If you choose decimals, keep three significant figures in your calculations. The diagram below is not necessarily drawn to scale. Pressure P 0 0 V0 32P Vmax Volume a. Let Vmax be the maximum volume achieved by the gas during the cycle. What is Vmax in terms of V0? If you are unable to solve this part of the problem, you may express your answers to the remaining parts in terms of Vmax without further loss of points. b. In terms of P0 and V0 determine the heat added to the gas during a complete cycle. c. In terms of P0 and V0 determine the heat removed from the gas during a complete cycle. d. What is the efficiency of this cycle? Copyright c 2008 American Association of Physics Teachers
2008 Semifinal Exam Part A 5 Question A3 A certain planet of radius R is composed of a uniform material that,through radioactive decay,generates a net power P.This results in a temperature differential between the inside and outside of the planet as heat is transfered from the interior to the surface. The rate of heat transfer is governed by the thermal conductivity.The thermal conductivity of a ma- terial is a measure of how quickly heat flows through that material in response to a temperature gradient. Specifically,consider a thin slab of material of area A and thickness Ax where one surface is hotter than the other by an amount AT.Suppose that an amount of heat AQ flows through the slab in a time At.The thermal conductivity k of the material is then k=414x △tA△T It is found that k is approximately constant for many materials;assume that it is constant for the planet. For the following assume that the planet is in a steady state;temperature might depend on position,but does not depend on time. a.Find an expression for the temperature of the surface of the planet assuming blackbody radiation,an emissivity of 1,and no radiation incident on the planet surface.You may express your answer in terms of any of the above variables and the Stephan-Boltzmann constant o. b.Find an expression for the temperature difference between the surface of the planet and the center of the planet.You may express your answer in terms of any of the above variables;you do not need to answer part (a)to be able to answer this part. Copyright C2008 American Association of Physics Teachers
2008 Semifinal Exam Part A 5 Question A3 A certain planet of radius R is composed of a uniform material that, through radioactive decay, generates a net power P. This results in a temperature differential between the inside and outside of the planet as heat is transfered from the interior to the surface. The rate of heat transfer is governed by the thermal conductivity. The thermal conductivity of a material is a measure of how quickly heat flows through that material in response to a temperature gradient. Specifically, consider a thin slab of material of area A and thickness ∆x where one surface is hotter than the other by an amount ∆T . Suppose that an amount of heat ∆Q flows through the slab in a time ∆t. The thermal conductivity k of the material is then k = ∆Q ∆t 1 A ∆x ∆T . It is found that k is approximately constant for many materials; assume that it is constant for the planet. For the following assume that the planet is in a steady state; temperature might depend on position, but does not depend on time. a. Find an expression for the temperature of the surface of the planet assuming blackbody radiation, an emissivity of 1, and no radiation incident on the planet surface. You may express your answer in terms of any of the above variables and the Stephan-Boltzmann constant σ. b. Find an expression for the temperature difference between the surface of the planet and the center of the planet. You may express your answer in terms of any of the above variables; you do not need to answer part (a) to be able to answer this part. Copyright c 2008 American Association of Physics Teachers