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
AAPT UNITEDSTATES PHYSICS TEAM AIP 2007 Semi-Final Exam Part B B1.A certain mechanical oscillator can be modeled as an ideal massless spring connected to a moveable plate on an incline.The spring has spring constant k,the plate has mass m,and the incline makes an angle 0 with the horizontal.When the system is operating correctly,the plate oscillates between points A4 and B in the figure,located a distance L apart.When the plate reaches point 4 it has zero kinetic energy,but then trips a small lever that instantaneously loads a block of mass Monto the plate.The block and plate then move down the incline to point B, where the force from the spring stops the plate.At this point,the block falls through a hole in the incline,allowing the plate to move back up under the force of the spring.Upon returning to point 4 it collects another block,and the cycle repeats.Both the plate and the block have a coefficient of friction u with the incline for both kinetic and static friction.It is reasonable that the motion in either direction is simple harmonic in nature. blocks load here moving plate B blocks fall here (10 pts)a.Let u be the critical value of the coefficient of friction where the block will just start to slide under the force of gravity on an incline(without the spring acting on it).Then let Findin terms ofg,the acceleration of free fall,and any 2 or all of the following variables:and M. Copyright 2007,American Association of Physics Teachers
Copyright © 2007, American Association of Physics Teachers Semi-Final Exam Part B B1. A certain mechanical oscillator can be modeled as an ideal massless spring connected to a moveable plate on an incline. The spring has spring constant k, the plate has mass m, and the incline makes an angle θ with the horizontal. When the system is operating correctly, the plate oscillates between points A and B in the figure, located a distance L apart. When the plate reaches point A it has zero kinetic energy, but then trips a small lever that instantaneously loads a block of mass M onto the plate. The block and plate then move down the incline to point B, where the force from the spring stops the plate. At this point, the block falls through a hole in the incline, allowing the plate to move back up under the force of the spring. Upon returning to point A it collects another block, and the cycle repeats. Both the plate and the block have a coefficient of friction μ with the incline for both kinetic and static friction. It is reasonable that the motion in either direction is simple harmonic in nature. (10 pts) a. Let μc be the critical value of the coefficient of friction where the block will just start to slide under the force of gravity on an incline (without the spring acting on it). Then let 2 μc μ = . Find μ in terms of g, the acceleration of free fall, and any or all of the following variables: θ and M
AAPT UNITEDSTATES PHYSICS TEAM AIP 2007 (14 pts)b.In order for this system to work correctly,it is necessary to have the correct ratio between the mass of the block and the mass of the plate.These masses are chosen so that the downward moving block and plate just stop at point B while the upward moving plate just stops at point.Find the ratioRM m (13 pts)c.The system delivers blocks to point B with period To,until the blocks run out. the plate alone oscillates with a period Find the (13 pts)d.The plate only oscillates a few times after delivering the last block.At what distance up the incline,measured from point B,does the plate come to a permanent stop? Copyright 2007,American Association of Physics Teachers
Copyright © 2007, American Association of Physics Teachers (14 pts) b. In order for this system to work correctly, it is necessary to have the correct ratio between the mass of the block and the mass of the plate. These masses are chosen so that the downward moving block and plate just stop at point B while the upward moving plate just stops at point A. Find the ratio . M R m = (13 pts) c. The system delivers blocks to point B with period , until the blocks run out. After that, the plate alone oscillates with a period T T0 ′ . Find the ratio T0 T′ . (13 pts) d. The plate only oscillates a few times after delivering the last block. At what distance up the incline, measured from point B, does the plate come to a permanent stop?
AAPT UNITEDSTATES PHYSICS TEAM AIP 2007 B2.A model of the magnetic properties of materials is based upon small magnetic moments generated by each atom in the material.One source of this magnetic moment is the magnetic field generated by the electron in its orbit around the nucleus.For simplicity,we will assume that each atom consists of a single electron of charge-e and mass m,a single proton of charge +e and mass m>m,and that the electron orbits in a circular orbit of radius R about the proton. a.Magnetic Moments. Assume that the electron orbits in the x-y plane. (3 pts)i.Calculate the net electrostatic force on the electron from the proton.Express your answer in terms of any or all of the following parameters:e,m,m,R,and the permittivity of free space,where 1 604πk (k is the Coulomb's Law constant). (5 pts)ii.Determine the angular velocity of the electron around the proton in terms of any or all of the following parameters:e,me,R,and 8. (8 pts)iii.Derive an expression for the magnitude of the magnetic field B.due to the orbital motion of the electron at a distanceRfrom the x-y plane along the axis of orbital rotation of the electron.Express your answer in terms of any or all of the following parameters:e,mR,,and the permeability of free space (4 pts)iv.A small bar magnet has a magnetic field far from the magnet given by B=to m 2π23 where z is the distance from the magnet on the axis connecting the north and south poles,m is the magnetic dipole moment,and to is the permeability of free space.Assuming that an electron orbiting a proton acts like a small bar magnet, find the dipole moment m for an electron orbiting an atom in terms of any or all of the following parameters:e,m.,R,and Copyright 2007,American Association of Physics Teachers
Copyright © 2007, American Association of Physics Teachers e B2. A model of the magnetic properties of materials is based upon small magnetic moments generated by each atom in the material. One source of this magnetic moment is the magnetic field generated by the electron in its orbit around the nucleus. For simplicity, we will assume that each atom consists of a single electron of charge –e and mass , a single proton of charge +e and mass , and that the electron orbits in a circular orbit of radius R about the proton. me m m p a. Magnetic Moments. Assume that the electron orbits in the x-y plane. (3 pts) i. Calculate the net electrostatic force on the electron from the proton. Express your answer in terms of any or all of the following parameters: e, , R, and the permittivity of free space, me , mp 0 ε , where 0 1 4 k ε π = . (k is the Coulomb’s Law constant). (5 pts) ii. Determine the angular velocity ω0 of the electron around the proton in terms of any or all of the following parameters: e, , R, and me 0 ε . (8 pts) iii. Derive an expression for the magnitude of the magnetic field Be due to the orbital motion of the electron at a distance from the x-y plane along the axis of orbital rotation of the electron. Express your answer in terms of any or all of the following parameters: e, , R, z R me ω0 , z, and the permeability of free space μ0 . (4 pts) iv. A small bar magnet has a magnetic field far from the magnet given by 0 3 , 2 m B z μ π = where z is the distance from the magnet on the axis connecting the north and south poles, m is the magnetic dipole moment, and μ0 is the permeability of free space. Assuming that an electron orbiting a proton acts like a small bar magnet, find the dipole moment m for an electron orbiting an atom in terms of any or all of the following parameters: e, , me R, andω0
AAPT UNITEDSTATES PHYSICS TEAM AIP 2007 b.Diamagnetism. We model a diamagnetic substance to have all atoms oriented so that the electron orbits are in the x-y plane,exactly half of which are clockwise and half counterclockwise when viewed from the positive z axis looking toward the origin.Some substances are predominantly diamagnetic. (3 pts)i.Calculate the total magnetic moment of a diamagnetic substance with N atoms. Write your answer in terms of any or all of the following parameters: e,m,R,N,and uo. (6 pts)ii.An external magnetic field Bo=Bo2 is applied to the substance.Assume that the introduction of the external field doesn't change the fact that the electron moves in a circular orbit of radius R.Determine Ao,the change in angular velocity of the electron,for both the clockwise and counterclockwise orbits.Throughout this entire problem you can assume that A<.Write your answer in terms of e,m.,andB。only. (6 pts)iii.Assume that the external field is turned on at a constant rate in a time interval At. That is to say,when t=0the external field is zero and when t At the external field is B.Determine the induced emf experienced by the electron.Write your answer in terms of any or all of the following parameters:e,m,R,N,Bo,@o,and Lo- (6 pts)iv.Verify that the change in the kinetic energy of the electron satisfies AK=e8. This justifies our assumption in(ii)that R does not change. (6 pts)v.Determine the change in the total magnetic moment Am for the N atoms when the external field is applied,writing your answer in terms of e,m,R,N,uo and Bo. (3 pts)vi.Suppose that the uniform magnetic field used in the previous parts of this problem is replaced with a bar magnet.Would the diamagnetic substance be attracted or repelled by the bar magnet?How does your answer show this? Copyright C 2007,American Association of Physics Teachers
Copyright © 2007, American Association of Physics Teachers , b. Diamagnetism. We model a diamagnetic substance to have all atoms oriented so that the electron orbits are in the x-y plane, exactly half of which are clockwise and half counterclockwise when viewed from the positive z axis looking toward the origin. Some substances are predominantly diamagnetic. (3 pts) i. Calculate the total magnetic moment of a diamagnetic substance with N atoms. Write your answer in terms of any or all of the following parameters: , ,, and e em RN 0 μ . (6 pts) ii. An external magnetic field 0 0 B = B zˆ G is applied to the substance. Assume that the introduction of the external field doesn’t change the fact that the electron moves in a circular orbit of radius R. Determine Δω , the change in angular velocity of the electron, for both the clockwise and counterclockwise orbits. Throughout this entire problem you can assume that Δω ω0 . Write your answer in terms of e m, e , and B0 only. (6 pts) iii. Assume that the external field is turned on at a constant rate in a time interval Δt . That is to say, when t = 0 the external field is zero and when the external field is t = Δt 0 B . G Determine the induced emf E experienced by the electron. Write your answer in terms of any or all of the following parameters: , , , , e em RN B0 , ω0 , and 0 μ . (6 pts) iv. Verify that the change in the kinetic energy of the electron satisfies E. This justifies our assumption in (ii) that R does not change. Δ = K e (6 pts) v. Determine the change in the total magnetic moment Δm for the N atoms when the external field is applied, writing your answer in terms of 0 , ,, , e em RN μ and 0 B . (3 pts) vi. Suppose that the uniform magnetic field used in the previous parts of this problem is replaced with a bar magnet. Would the diamagnetic substance be attracted or repelled by the bar magnet? How does your answer show this?
