AAPT UNITEDSTATES PHYSICS TEAM AIP 2007 2007 Semi-Final 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. After you have completed Part A,you may take a break. Then work Part B.You have 90 minutes to complete both problems. Show all your work.Partial credit will be given. Start each question on a new sheet of paper.Be sure to put your name in the upper right- hand corner of each page,along with the question number and the page number/total pages for this problem.For example, 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. .Do not discuss the contents of this exam with anyone until after March 27th Good luck! Possibly Useful Information-(Use for both part A and for part B) Gravitational field at the Earth's surface g =9.8 N/kg Newton's gravitational constant =6.67xi0-11Nm2kg2 Coulomb's constant k =1/4π8。=8.99x10Nm21c2 Biot-Savart constant Km =μo/4π=10~7Tm/A Speed of light in a vacuum =3.00x108m/s Boltzmann's constant 阳=1.38x102J/K Avogadro's number N=6.02x1023(mo Ideal gas constant R =NAkB =8.31 J/(mol-K) Stefan-Boltzmann constant o=5.67x108J/(sm2K4 Elementary charge e=1.602x1019C 1 electron volt 1eV=1.602x1019J Planck's constant h =6.63x1034Js=4.14x10-15eVs Electron mass m =9.109 x 103T kg=0.511 MeV/e2 Binomial expansion (1+x)P≈1+x forx<1 Small angle approximations sin0≈0 cos0≈1-'202 Copyright 2007,American Association of Physics Teachers
2007 Semi-Final 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. • After you have completed Part A, you may take a break. • Then work Part B. You have 90 minutes to complete both problems. • Show all your work. Partial credit will be given. • Start each question on a new sheet of paper. Be sure to put your name in the upper righthand corner of each page, along with the question number and the page number/total pages for this problem. For example, 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. • Do not discuss the contents of this exam with anyone until after March 27th. • Good luck! Possibly Useful Information - (Use for both part A and for part B) Gravitational field at the Earth’s surface g = 9.8 N/kg Newton’s gravitational constant G = 6.67 x 10-11 N·m 2 /kg2 Coulomb’s constant k = 1/4πεο = 8.99 x 109 N·m2 /C2 Biot-Savart constant km = µο/4π = 10-7 T·m/A Speed of light in a vacuum c = 3.00 x 108 m/s Boltzmann’s constant kB = 1.38 x 10-23 J/K Avogadro’s number NA = 6.02 x 1023 (mol)-1 Ideal gas constant R = NAkB = 8.31 J/(mol·K) Stefan-Boltzmann constant σ = 5.67 x 10-8 J/(s·m2 ·K4 ) Elementary charge e = 1.602 x 10-19 C 1 electron volt 1 eV = 1.602 x 10-19 J Planck’s constant h = 6.63 x 10-34 J·s = 4.14 x 10-15 eV·s Electron mass m = 9.109 x 10-31 kg = 0.511 MeV/c2 Binomial expansion (1 + x) n ≈ 1 + nx for |x| << 1 Small angle approximations sin θ ≈ θ cos θ ≈ 1 – 1 /2 θ 2 Copyright © 2007, American Association of Physics Teachers
AAPT UNITEDSTATES PHYSICS TEAM AIP 2007 Semi-Final Exam Part A A1.A group of 12 resistors is arranged along the edges of a cube as shown in the diagram below. The vertices of the cube are labeled a-h. a.(13 pts)The resistance between each pair of vertices is as follows: Rab =Rac Rae =3.0 Rcg Rer=Rbd=8.0 Rcd Rbf=Reg 12.0 Rdh =Rm=Rgh=1.0 What is the equivalent resistance between points a and h? b.(12 pts)The three 12.0 resistors are replaced by identical capacitors.Ced=Cor=Ceg= 15.0 uF.A 12.0 V battery is attached across points a and h and the circuit is allowed to operate for a long period of time.What is the charge(Ocd.Os Oeg)on each capacitor after this long period of time? Copyright 2007,American Association of Physics Teachers
Copyright © 2007, American Association of Physics Teachers Semi-Final Exam Part A A1. A group of 12 resistors is arranged along the edges of a cube as shown in the diagram below. The vertices of the cube are labeled a-h. a b c d f g h e a. (13 pts) The resistance between each pair of vertices is as follows: Rab = Rac = Rae = 3.0 Ω Rcg = Ref = Rbd = 8.0 Ω Rcd = Rbf = Reg = 12.0 Ω Rdh = Rfh = Rgh = 1.0 Ω What is the equivalent resistance between points a and h? b. (12 pts) The three 12.0 Ω resistors are replaced by identical capacitors. Ccd = Cbf = Ceg = 15.0 μF. A 12.0 V battery is attached across points a and h and the circuit is allowed to operate for a long period of time. What is the charge (Qcd, Qbf, Qeg) on each capacitor after this long period of time?
AAPT UNITEDSTATES PHYSICS TEAM AIP 2007 A2.A simple gun can be made from a uniform cylinder of length Lo and inside radius re.One end of the cylinder is sealed with a moveable plunger and the other end is plugged with a cylindrical cork bullet.The bullet is held in place by friction with the walls of the cylinder The pressure outside the cylinder is atmospheric pressure,P The bullet will just start to slide out of the cylinder if the pressure inside the cylinder exceeds P. a.There are two ways to launch the bullet:either by heating the gas inside the cylinder and keeping the plunger fixed,or by suddenly pushing the plunger into the cylinder.In either case, assume that an ideal monatomic gas is inside the cylinder,and that originally the gas is at temperature To,the pressure inside the cylinder is P,and the length of the cylinder is Lo. (8 pts)i.Assume that we launch the bullet by heating the gas without moving the plunger. Find the minimum temperature of the gas necessary to launch the bullet.Express your answer in terms of any or all of the variables:ToLo P,and P. (8 pts)ii.Assume,instead that we launch the bullet by pushing in the plunger,and that we do so quickly enough so that no heat is transferred into or out of the gas.Find the length of the gas column inside the cylinder when the bullet just starts to move. Express your answer in terms of any or all of the variables:r,To,Lo,Po,and P b.(9 pts)It is necessary to squeeze the bullet to get it into the cylinder in the first place.The bullet normally has a radius r that is slightly larger than the inside radius of the cylinder, =Ar,is small compared to r.The bullet has a length hLo.The walls of the cylinder apply a pressure to the cork bullet.When a pressure P is applied to the bullet along a given direction,the bullet's dimensions in that direction change by △r-P x E for a constant E known as Young's modulus.You may assume that compression along one direction does not cause expansion in any other direction.(This is true if the so-called Poisson ratio is close to zero,which is the case for cork. If the coefficient of static friction between the cork and the cylinder is ut,find an expression for Pr.Express your answer in terms of any or all of the variables:Po,u,h,E,Ar,and r. Copyright 2007,American Association of Physics Teachers
Copyright © 2007, American Association of Physics Teachers , A2. A simple gun can be made from a uniform cylinder of length L0 and inside radius rc. One end of the cylinder is sealed with a moveable plunger and the other end is plugged with a cylindrical cork bullet. The bullet is held in place by friction with the walls of the cylinder. The pressure outside the cylinder is atmospheric pressure, . The bullet will just start to slide out of the cylinder if the pressure inside the cylinder exceeds . P0 Pcr a. There are two ways to launch the bullet: either by heating the gas inside the cylinder and keeping the plunger fixed, or by suddenly pushing the plunger into the cylinder. In either case, assume that an ideal monatomic gas is inside the cylinder, and that originally the gas is at temperature , the pressure inside the cylinder is , and the length of the cylinder is T0 P0 0 L . (8 pts) i. Assume that we launch the bullet by heating the gas without moving the plunger. Find the minimum temperature of the gas necessary to launch the bullet. Express your answer in terms of any or all of the variables: and . 000 ,,,, crT L P Pcr (8 pts) ii. Assume, instead that we launch the bullet by pushing in the plunger, and that we do so quickly enough so that no heat is transferred into or out of the gas. Find the length of the gas column inside the cylinder when the bullet just starts to move. Express your answer in terms of any or all of the variables: and . 000 ,,, crT L P Pcr b. (9 pts) It is necessary to squeeze the bullet to get it into the cylinder in the first place. The bullet normally has a radius that is slightly larger than the inside radius of the cylinder; , is small compared to . The bullet has a length The walls of the cylinder apply a pressure to the cork bullet. When a pressure is applied to the bullet along a given direction, the bullet’s dimensions in that direction change by br b c rr r − =Δ cr 0 h L . P x P x E Δ − = for a constant E known as Young’s modulus. You may assume that compression along one direction does not cause expansion in any other direction. (This is true if the so-called Poisson ratio is close to zero, which is the case for cork.) If the coefficient of static friction between the cork and the cylinder is μ , find an expression for . Express your answer in terms of any or all of the variables: Pcr 0 P hE r , ,, , , μ Δ and . cr
AAPT UNITEDSTATES PHYSICS TEAM AIP 2007 A3.A volume V of fluid with uniform charge density p is sprayed into a room,forming spherical drops.As they float around the room,the drops may break apart into smaller drops or coalesce into larger ones.Suppose that all of the drops have radius R.Ignore inter-drop forces and assume that VR. (10 pts)a.Calculate the electrostatic potential energy of a single drop.(Hint:suppose the sphere has radius r.How much work is required to increase the radius by dr?). (4 pts)b.What is the total electrostatic energy of the drops? Your answer to(b)should indicate that the total energy increases with R.In the absence of surface tension,then,the fluid would break apart into infinitesimally small drops.Suppose, however,that the fluid has a surface tension y.(This value is the potential energy per unit surface area,and is positive.) (4 pts)c.What is the total energy of the drops due to surface tension? (7 pts)d.What is the equilibrium radius of the drops? Copyright 2007,American Association of Physics Teachers
Copyright © 2007, American Association of Physics Teachers A3. A volume Vf of fluid with uniform charge density ρ is sprayed into a room, forming spherical drops. As they float around the room, the drops may break apart into smaller drops or coalesce into larger ones. Suppose that all of the drops have radius R. Ignore inter-drop forces and assume that 3 . V R f (10 pts) a. Calculate the electrostatic potential energy of a single drop. (Hint: suppose the sphere has radius r. How much work is required to increase the radius by dr?). (4 pts) b. What is the total electrostatic energy of the drops? Your answer to (b) should indicate that the total energy increases with R. In the absence of surface tension, then, the fluid would break apart into infinitesimally small drops. Suppose, however, that the fluid has a surface tension γ . (This value is the potential energy per unit surface area, and is positive.) (4 pts) c. What is the total energy of the drops due to surface tension? (7 pts) d. What is the equilibrium radius of the drops?
AAPT UNITEDSTATES PHYSICS TEAM AIP 2007 A4.A nonlinear circuit element can be made out of a parallel plate capacitor and small balls, each of mass m,that can move between the plates.The balls collide inelastically with the plates,dissipate all kinetic energy as thermal energy,and immediately release the charge they are carrying to the plate.Almost instantaneously,the balls then pick up a small charge of magnitude g from the plate;the balls are then repelled directly toward the other plate under electrostatic forces only.Another collision happens,kinetic energy is dissipated,the balls give up the charge,collect a new charge,and the cycle repeats.There are no balls per unit surface area of the plate.The capacitor has a capacitance C.The separation d between the plates is much larger than the radius r of the balls.A battery is connected to the plates in order to maintain a constant potential difference V.Neglect edge effects and assume that magnetic forces and gravitational forces may be ignored. (5)a.Determine the time it takes for one ball to travel between the plates in terms of any or all of the following variables:m,g,d,and V. (5)b.Calculate the kinetic energy dissipated as thermal energy when one ball collides inelastically with a plate surface in terms of any or all of the following variables:m, g,d,and V. (5)c.Derive an expression for the current between the plates in terms of the permittivity of free space,5o,and any or all of the following variables:m,g,no,C,and V. (5)d.Derive an expression for the effective resistance of the device in terms of 6o,and any or all of the following variables:m,g,no.C,and V. (5)e.Calculate the rate at which the kinetic energy of the balls is converted into thermal energy in terms of so,and any or all of the following variables:m,g,no,C,and V. Copyright C 2007,American Association of Physics Teachers
Copyright © 2007, American Association of Physics Teachers A4. A nonlinear circuit element can be made out of a parallel plate capacitor and small balls, each of mass m, that can move between the plates. The balls collide inelastically with the plates, dissipate all kinetic energy as thermal energy, and immediately release the charge they are carrying to the plate. Almost instantaneously, the balls then pick up a small charge of magnitude q from the plate; the balls are then repelled directly toward the other plate under electrostatic forces only. Another collision happens, kinetic energy is dissipated, the balls give up the charge, collect a new charge, and the cycle repeats. There are n0 balls per unit surface area of the plate. The capacitor has a capacitance C. The separation d between the plates is much larger than the radius r of the balls. A battery is connected to the plates in order to maintain a constant potential difference V. Neglect edge effects and assume that magnetic forces and gravitational forces may be ignored. (5) a. Determine the time it takes for one ball to travel between the plates in terms of any or all of the following variables: m, q, d, and V. (5) b. Calculate the kinetic energy dissipated as thermal energy when one ball collides inelastically with a plate surface in terms of any or all of the following variables: m, q, d, and V. (5) c. Derive an expression for the current between the plates in terms of the permittivity of free space, 0 ε , and any or all of the following variables: m, q, n0, C, and V. (5) d. Derive an expression for the effective resistance of the device in terms of 0 ε , and any or all of the following variables: m, q, n0, C, and V. (5) e. Calculate the rate at which the kinetic energy of the balls is converted into thermal energy in terms of 0 ε , and any or all of the following variables: m, q, n0, C, and V