2015 USA Physics Olympiad Exam AAPT UNITED STATES PHYSICS TEAM AIP 2015 USA Physics Olympiad Exam DO NOT DISTRIBUTE THIS PAGE Important Instructions for the Exam Supervisor This examination consists of two parts. Part A has four questions and is allowed 90 minutes. Part B has two questions and is allowed 90 minutes. The first page that follows is a cover sheet.Examinees may keep the cover sheet for both parts of the exam. The 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.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 go back to Part A. 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 9-11),and several answer sheets for one of the questions in Part A(pages 13-13).Examinees should be provided parts A and B individually,although they may keep the cover sheet.The answer sheets should be printed single sided! 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 April 15,2015. 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. Copyright C2015 American Association of Physics Teachers
2015 USA Physics Olympiad Exam 1 AAPT AIP 2015 UNITED STATES PHYSICS TEAM USA Physics Olympiad Exam DO NOT DISTRIBUTE THIS PAGE Important Instructions for the Exam Supervisor • This examination consists of two parts. • Part A has four questions and is allowed 90 minutes. • Part B has two questions and is allowed 90 minutes. • The first page that follows is a cover sheet. Examinees may keep the cover sheet for both parts of the exam. • The 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. 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 go back to Part A. • 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 9-11), and several answer sheets for one of the questions in Part A (pages 13-13). Examinees should be provided parts A and B individually, although they may keep the cover sheet. The answer sheets should be printed single sided! • 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 April 15, 2015. • 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. Copyright c 2015 American Association of Physics Teachers
2015 USA Physics Olympiad Exam Cover Sheet 2 AAPT UNITED STATES PHYSICS TEAM AIP 2015 USA Physics Olympiad 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 Part B 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 Part A 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 AAPT 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, AAPT 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. In order to maintain exam security,do not communicate any information about the questions (or their answers/solutions)on this contest until after April 15, 2015. Possibly Useful Information.You may use this sheet for both parts of the exam. g=9.8 N/kg G=6.67×10-11N·m2/kg2 k=1/4r60=8.99×109N.m2/C2 km=4o/4r=10-7T.m/A c=3.00×108m/s k3=1.38×10-23J/K NA=6.02×1023(mol)-1 R=Na=8.31J/(mol·K) o=5.67×10-8J/(s·m2.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+nx for<1 sin0≈0-ae3 for0l<1 cos0≈1-02forl0l<1 Copyright C2015 American Association of Physics Teachers
2015 USA Physics Olympiad Exam Cover Sheet 2 AAPT AIP 2015 UNITED STATES PHYSICS TEAM USA Physics Olympiad 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 Part B 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 Part A 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 AAPT 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, AAPT 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. • In order to maintain exam security, do not communicate any information about the questions (or their answers/solutions) on this contest until after April 15, 2015. Possibly Useful Information. You may use this sheet for both parts of the exam. g = 9.8 N/kg G = 6.67 × 10−11 N · m2/kg2 k = 1/4π0 = 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 2015 American Association of Physics Teachers
2015 USA Physics Olympiad Exam Part A 3 Part A Question Al Consider a particle of mass m that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity.The total mechanical energy of the particle is E and the acceleration of free fall is g.Treat the particle as a point mass and assume the motion is non-relativistic. a.An estimate for the regime where quantum effects become important can be found by simply considering when the deBroglie wavelength of the particle is on the same order as the height of a bounce.Assuming that the deBroglie wavelength is defined by the maximum momentum of the bouncing particle,determine the value of the energy Ea where quantum effects become important.Write your answer in terms of some or all of g,m,and Planck's constant h. b.A second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle.Assuming that the particle rises to a height H,we can write 2p=(+)h where p is the momentum as a function of height x above the ground,n is a non-negative integer,and h is Planck's constant. i.Determine the allowed energies En as a function of the integer n,and some or all of g, m,and Planck's constant h. ii.Numerically determine the minimum energy of a bouncing neutron.The mass of a neutron is mn =1.675x 10-27 kg=940 MeV/c2;you may express your answer in either Joules or ev. iii.Determine the bounce height of one of these minimum energy neutrons. c.Let Eo be the minimum energy of the bouncing neutron and f be the frequency of the bounce. Determine an order of magnitude estimate for the ratio E/f.It only needs to be accurate to within an order of magnitude or so,but you do need to show work! Copyright C2015 American Association of Physics Teachers
2015 USA Physics Olympiad Exam Part A 3 Part A Question A1 Consider a particle of mass m that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is E and the acceleration of free fall is g. Treat the particle as a point mass and assume the motion is non-relativistic. a. An estimate for the regime where quantum effects become important can be found by simply considering when the deBroglie wavelength of the particle is on the same order as the height of a bounce. Assuming that the deBroglie wavelength is defined by the maximum momentum of the bouncing particle, determine the value of the energy Eq where quantum effects become important. Write your answer in terms of some or all of g, m, and Planck’s constant h. b. A second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height H, we can write 2 Z H 0 p dx = n + 1 2 h where p is the momentum as a function of height x above the ground, n is a non-negative integer, and h is Planck’s constant. i. Determine the allowed energies En as a function of the integer n, and some or all of g, m, and Planck’s constant h. ii. Numerically determine the minimum energy of a bouncing neutron. The mass of a neutron is mn = 1.675×10−27 kg = 940 MeV/c 2 ; you may express your answer in either Joules or eV. iii. Determine the bounce height of one of these minimum energy neutrons. c. Let E0 be the minimum energy of the bouncing neutron and f be the frequency of the bounce. Determine an order of magnitude estimate for the ratio E/f. It only needs to be accurate to within an order of magnitude or so, but you do need to show work! Copyright c 2015 American Association of Physics Teachers
2015 USA Physics Olympiad Exam Part A 4 Question A2 Consider the circuit shown below.Is is a constant current source,meaning that no matter what device is connected between points A and B,the current provided by the constant current source is the same. 4R3 2R 2R 4R M● ● A B 2n a.Connect an ideal voltmeter between A and B.Determine the voltage reading in terms of any or all of R and Is. b.Connect instead an ideal ammeter between A and B.Determine the current in terms of any or all of R and Is. c.It turns out that it is possible to replace the above circuit with a new circuit as follows: Rt B From the point of view of any passive resistance that is connected between A and B the circuits are identical.You don't need to prove this statement,but you do need to find I:and Rt in terms of any or all of R and Is. Copyright C2015 American Association of Physics Teachers
2015 USA Physics Olympiad Exam Part A 4 Question A2 Consider the circuit shown below. Is is a constant current source, meaning that no matter what device is connected between points A and B, the current provided by the constant current source is the same. 2R 4R 4R 2R Is 4R 2R A B a. Connect an ideal voltmeter between A and B. Determine the voltage reading in terms of any or all of R and Is. b. Connect instead an ideal ammeter between A and B. Determine the current in terms of any or all of R and Is. c. It turns out that it is possible to replace the above circuit with a new circuit as follows: It Rt A B From the point of view of any passive resistance that is connected between A and B the circuits are identical. You don’t need to prove this statement, but you do need to find It and Rt in terms of any or all of R and Is. Copyright c 2015 American Association of Physics Teachers
2015 USA Physics Olympiad Exam Part A 5 Question A3 A large block of mass m is located on a horizontal frictionless surface.A second block of mass mt is located on top of the first block;the coefficient of friction (both static and kinetic)between the two blocks is given by u.All surfaces are horizontal;all motion is effectively one dimensional. A spring with spring constant k is connected to the top block only;the spring obeys Hooke's Law equally in both extension and compression.Assume that the top block never falls off of the bottom block;you may assume that the bottom block is very,very long.The top block is moved a distance A away from the equilibrium position and then released from rest. WWW mi mb a.Depending on the value of A,the motion can be divided into two types:motion that expe- riences no frictional energy losses and motion that does.Find the value Ac that divides the two motion types.Write your answer in terms of any or all of u,the acceleration of gravity g,the masses mt and mb,and the spring constant k. b.Consider now the scenario A>Ac.In this scenario the amplitude of the oscillation of the top block as measured against the original equilibrium position will change with time.Determine the magnitude of the change in amplitude,AA,after one complete oscillation,as a function of any or all of A,u,g,and the angular frequency of oscillation of the top block wt. c.Assume still that A>Ac.What is the maximum speed of the bottom block during the first complete oscillation cycle of the upper block? Copyright C2015 American Association of Physics Teachers
2015 USA Physics Olympiad Exam Part A 5 Question A3 A large block of mass mb is located on a horizontal frictionless surface. A second block of mass mt is located on top of the first block; the coefficient of friction (both static and kinetic) between the two blocks is given by µ. All surfaces are horizontal; all motion is effectively one dimensional. A spring with spring constant k is connected to the top block only; the spring obeys Hooke’s Law equally in both extension and compression. Assume that the top block never falls off of the bottom block; you may assume that the bottom block is very, very long. The top block is moved a distance A away from the equilibrium position and then released from rest. mb mt a. Depending on the value of A, the motion can be divided into two types: motion that experiences no frictional energy losses and motion that does. Find the value Ac that divides the two motion types. Write your answer in terms of any or all of µ, the acceleration of gravity g, the masses mt and mb, and the spring constant k. b. Consider now the scenario A Ac. In this scenario the amplitude of the oscillation of the top block as measured against the original equilibrium position will change with time. Determine the magnitude of the change in amplitude, ∆A, after one complete oscillation, as a function of any or all of A, µ, g, and the angular frequency of oscillation of the top block ωt . c. Assume still that A Ac. What is the maximum speed of the bottom block during the first complete oscillation cycle of the upper block? Copyright c 2015 American Association of Physics Teachers