2017 USA Physics Olympiad Exam AAPT UNITED STATES PHYSICS TEAM AIP 2017 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 3 parts:the cover sheet (page 2), Part A (pages 3-6),Part B (pages 8-10).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 15.2017. 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 C2017 American Association of Physics Teachers
2017 USA Physics Olympiad Exam 1 AAPT AIP 2017 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 3 parts: the cover sheet (page 2), Part A (pages 3-6), Part B (pages 8-10). 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, 2017. • 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 2017 American Association of Physics Teachers
2017 USA Physics Olympiad Exam Cover Sheet 2 AAPT UNITED STATES PHYSICS TEAM AIP 2017 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 proctor's AAPT ID,your AAPT ID, 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, Doe,Jamie student AAPT ID# proctor AAPT ID 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 8,2017. 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=1/4re0=8.99×109N·m2/C2 km Ho/47 10-7 T.m/A c=3.00×103m/s k3=1.38×10-23J/K NA=6.02×1023(mol)-1 R=NAkB =8.31 J/(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/e2(1+x)"≈1+n.r for<1 mp=1.673×10-27kg=938MeV/c2 ln(1+x)≈x for<1 sin0≈0-03forl9l<1 cos0≈1-号02forl9l<1 Copyright C2017 American Association of Physics Teachers
2017 USA Physics Olympiad Exam Cover Sheet 2 AAPT AIP 2017 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 proctor’s AAPT ID, your AAPT ID, 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, Doe, Jamie student AAPT ID # proctor AAPT ID # 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 8, 2017. 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 1 eV = 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 mp = 1.673 × 10−27 kg = 938 MeV/c 2 ln(1 + x) ≈ x 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 2017 American Association of Physics Teachers
2017 USA Physics Olympiad Exam Part A 3 Part A Question Al A pair of wedges are located on a horizontal surface.The coefficient of friction (both sliding and static)between the wedges is u,the coefficient of friction between the bottom wedge B and the horizontal surface is u,and the angle of the wedge is 6.The mass of the top wedge A is m,and the mass of the bottom wedge B is M=2m.A horizontal force F directed to the left is applied to the bottom wedge as shown in the figure. A B Determine the range of values for F so that the the top wedge does not slip on the bottom wedge.Express your answer(s)in terms of any or all of m,g,6,and u. Question A2 Consider two objects with equal heat capacities C and initial temperatures T1 and T2.A Carnot engine is run using these objects as its hot and cold reservoirs until they are at equal temperatures. Assume that the temperature changes of both the hot and cold reservoirs is very small compared to the temperature during any one cycle of the Carnot engine. a.Find the final temperature Tr of the two objects,and the total work W done by the engine. Now consider three objects with equal and constant heat capacity at initial temperatures Ti=100 K,T2=300 K,and T3 =300 K.Suppose we wish to raise the temperature of the third object. To do this,we could run a Carnot engine between the first and second objects,extracting work W.This work can then be dissipated as heat to raise the temperature of the third object.Even better,it can be stored and used to run a Carnot engine between the first and third object in reverse,which pumps heat into the third object. Assume that all work produced by running engines can be stored and used without dissipation. b.Find the minimum temperature TL to which the first object can be lowered. c.Find the maximum temperature Ty to which the third object can be raised. Copyright C2017 American Association of Physics Teachers
2017 USA Physics Olympiad Exam Part A 3 Part A Question A1 A pair of wedges are located on a horizontal surface. The coefficient of friction (both sliding and static) between the wedges is µ, the coefficient of friction between the bottom wedge B and the horizontal surface is µ, and the angle of the wedge is θ. The mass of the top wedge A is m, and the mass of the bottom wedge B is M = 2m. A horizontal force F directed to the left is applied to the bottom wedge as shown in the figure. B A F θ Determine the range of values for F so that the the top wedge does not slip on the bottom wedge. Express your answer(s) in terms of any or all of m, g, θ, and µ. Question A2 Consider two objects with equal heat capacities C and initial temperatures T1 and T2. A Carnot engine is run using these objects as its hot and cold reservoirs until they are at equal temperatures. Assume that the temperature changes of both the hot and cold reservoirs is very small compared to the temperature during any one cycle of the Carnot engine. a. Find the final temperature Tf of the two objects, and the total work W done by the engine. Now consider three objects with equal and constant heat capacity at initial temperatures T1 = 100 K, T2 = 300 K, and T3 = 300 K. Suppose we wish to raise the temperature of the third object. To do this, we could run a Carnot engine between the first and second objects, extracting work W. This work can then be dissipated as heat to raise the temperature of the third object. Even better, it can be stored and used to run a Carnot engine between the first and third object in reverse, which pumps heat into the third object. Assume that all work produced by running engines can be stored and used without dissipation. b. Find the minimum temperature TL to which the first object can be lowered. c. Find the maximum temperature TH to which the third object can be raised. Copyright c 2017 American Association of Physics Teachers
2017 USA Physics Olympiad Exam Part A 4 Question A3 A ship can be thought of as a symmetric arrangement of soft iron.In the presence of an external magnetic field,the soft iron will become magnetized,creating a second,weaker magnetic field.We want to examine the effect of the ship's field on the ship's compass,which will be located in the middle of the ship. Let the strength of the Earth's magnetic field near the ship be Be,and the orientation of the field be horizontal,pointing directly toward true north. The Earth's magnetic field Be will magnetize the ship,which will then create a second magnetic field Bs in the vicinity of the ship's compass given by B=Be（-K6cos06+Ksin0s） where K and Ks are positive constants,6 is the angle between the heading of the ship and magnetic north,measured clockwise,b and s are unit vectors pointing in the forward direction of the ship (bow)and directly right of the forward direction (starboard),respectively. Because of the ship's magnetic field,the ship's compass will no longer necessarily point North. a.Derive an expression for the deviation of the compass,60,from north as a function of Ko, Ks;and 0. b.Assuming that Ko and Ks are both much smaller than one,at what heading(s)0 will the deviation 60 be largest? A pair of iron balls placed in the same horizontal plane as the compass but a distance d away can be used to help correct for the error caused by the induced magnetism of the ship. A binnacle,protecting the ship's compass in the center,with two soft iron spheres to help correct for errors in the compass heading.The use of the spheres was suggested by Lord Kelvin. Just like the ship,the iron balls will become magnetic because of the Earth's field Be.As spheres,the balls will individually act like dipoles.A dipole can be thought of as the field produced by two magnetic monopoles of strength +m at two different points. Copyright C2017 American Association of Physics Teachers
2017 USA Physics Olympiad Exam Part A 4 Question A3 A ship can be thought of as a symmetric arrangement of soft iron. In the presence of an external magnetic field, the soft iron will become magnetized, creating a second, weaker magnetic field. We want to examine the effect of the ship’s field on the ship’s compass, which will be located in the middle of the ship. Let the strength of the Earth’s magnetic field near the ship be Be, and the orientation of the field be horizontal, pointing directly toward true north. The Earth’s magnetic field Be will magnetize the ship, which will then create a second magnetic field Bs in the vicinity of the ship’s compass given by B~ s = Be −Kb cos θ bˆ + Ks sin θ ˆs where Kb and Ks are positive constants, θ is the angle between the heading of the ship and magnetic north, measured clockwise, bˆ and ˆs are unit vectors pointing in the forward direction of the ship (bow) and directly right of the forward direction (starboard), respectively. Because of the ship’s magnetic field, the ship’s compass will no longer necessarily point North. a. Derive an expression for the deviation of the compass, δθ, from north as a function of Kb, Ks, and θ. b. Assuming that Kb and Ks are both much smaller than one, at what heading(s) θ will the deviation δθ be largest? A pair of iron balls placed in the same horizontal plane as the compass but a distance d away can be used to help correct for the error caused by the induced magnetism of the ship. A binnacle, protecting the ship’s compass in the center, with two soft iron spheres to help correct for errors in the compass heading. The use of the spheres was suggested by Lord Kelvin. Just like the ship, the iron balls will become magnetic because of the Earth’s field Be. As spheres, the balls will individually act like dipoles. A dipole can be thought of as the field produced by two magnetic monopoles of strength ±m at two different points. Copyright c 2017 American Association of Physics Teachers
2017 USA Physics Olympiad Exam Part A 5 The magnetic field of a single pole is B=士m2 where the positive sign is for a north pole and the negative for a south pole.The dipole magnetic field is the sum of the two fields:a north pole at y=+a/2 and a south pole at y=-a/2,where the y axis is horizontal and pointing north.a is a small distance much smaller than the radius of the iron balls;in general a KiBe where Ki is a constant that depends on the size of the iron sphere. North B 4 iron ball c.Derive an expression for the magnetic field Bi from the iron a distance d>a from the center of the ball.Note that there will be a component directed radially away from the ball and a component directed tangent to a circle of radius d around the ball,so using polar coordinates is recommended. d.If placed directly to the right and left of the ship compass,the iron balls can be located at a distance d to cancel out the error in the magnetic heading for any angle(s)where 60 is largest. Assuming that this is done,find the resulting expression for the combined deviation 60 due to the ship and the balls for the magnetic heading for all angles 6. Copyright C2017 American Association of Physics Teachers
2017 USA Physics Olympiad Exam Part A 5 The magnetic field of a single pole is B~ = ±m ˆr r 2 where the positive sign is for a north pole and the negative for a south pole. The dipole magnetic field is the sum of the two fields: a north pole at y = +a/2 and a south pole at y = −a/2, where the y axis is horizontal and pointing north. a is a small distance much smaller than the radius of the iron balls; in general a = KiBe where Ki is a constant that depends on the size of the iron sphere. φ North B~ i iron ball a c. Derive an expression for the magnetic field B~ i from the iron a distance d a from the center of the ball. Note that there will be a component directed radially away from the ball and a component directed tangent to a circle of radius d around the ball, so using polar coordinates is recommended. d. If placed directly to the right and left of the ship compass, the iron balls can be located at a distance d to cancel out the error in the magnetic heading for any angle(s) where δθ is largest. Assuming that this is done, find the resulting expression for the combined deviation δθ due to the ship and the balls for the magnetic heading for all angles θ. Copyright c 2017 American Association of Physics Teachers