MECH + E&M - 1995

MECH + E&M - 1995 - 1995 The College Board Advanced...

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Unformatted text preview: 1995 The College Board Advanced Placement Examination PHYSICS C SECTION 11 TABLE OF INFORMATION 1 atomic mass unit, 1 u = 1.66 x 10'21 kilogram Rest mass of the proton. mp = 1.67 x 10‘27 kilogram Rest mass of the neutron. mn = 1.67 x 10'” kilogram Rest mass of the electron. me = 9.11 x 10'31 kilogram Magnitude of the electron charge, a = 1.60 X 10—19 coulomb Avogadro’s number. N0 = 6.02 x 1023 per mole Universal gas constant. R = 8.32 joules/(mole - K) Boltzmann’s constant. k3 = 1.33 x 10’23 joule/K Speed of light. c = 3.00 x 108 meters/second Planck‘s constant, in = 6.63 x 10‘3‘joule - second = 4.14 x 10'” eV - second he = 1.99 x 10’:5 joule - meter = 1240 eV - nanometers 1 electron volt. 1 W = 1.60 x 10'19joule Vacuum permittivity, Eu 2 8.85 x 10’12 coulombIKnewton v meter-2) Coulomb’s law constant, k = 1/41160 = 9.0 x 109 newtons - meter’lcoulomb2 Vacuum permeability. pa. = 4n x 10‘T weber/(ampere - meter) Magnetic constant. k' = Id:2 = [Jo/411: = 10'7 weber/(ampcre - meter) Acceleration due to gravity at the Earth’s surface. 3 = 9.8 meterst'second‘l Universal gravitational constant. G = 6.67 x 10"“ meter’Kkilogram - second!) 1 atmosphere pressure. 1 atm = 1.0 x 10"| newtonslmeter2 = 1.0 x 105 pascals (Pa) 1 angstrom, 1 131: 1 x 10"” meter 1 tesla. 1 T = 1 weberlmelerl The following conventions are used in this examination. I. Unless otherwise stated; the frame of reference of any problem is assumed to be inertial. Il. The direction of any electric current is the direction of flow of positive charge (conventional current). III. For any isolated electric charge, the electric potential is defined as zero at an infinite distance from the charge. Copyright © 1995 College Entrance Examination Board Educational Testing Service. All rights reserved. Princeton. NJ. 08541 1 995 PHYSICS C SECTION II, MECHANICS Time—45 minutes 3 Questions Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions, which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the pink booklet in the spaces provided after each part, NOT in this green insert. Mech. 1. Note: Figure not drawn to scale. A S-kilogram ball initially rests at the edge of a 2-meter-long, 1.2-meter—high frictionless table, as shown above. A hard plastic cube of mass 0.5 kilogram slides across the tabie at a speed of 26 meters per second and strikes the ball, causing the ball to leave the table in the direction in which the cube was moving. The figure below shows a graph of the force exerted on the ball by the cube as a function of time. III-IIIIII IIIIIIEIII Force (x 103 N) III-IIIEII Ill-IIIIEI III 2.0 4.0 6.0 8.0 10.0 Time (x 10‘3 s) GO ON TO THE NEXT PAGE (a) (b) (c) (d) (c) 1995 PHYSICS C — MECHANICS Determine the total impulse given to the ball. Determine the horizontal velocity of the ball immediately after the collisiOn. Determine the following for the cube immediately after the collision. i. Its speed ii. Its direction of travel (right or left), if moving Determine the kinetic energy dissipated in the collision. Determine the distance between the two points of impact of the objects with the floor. GO ON TO THE NEXT PAGE 1995 PHYSICS C —— MECHANICS Mech. 2. A particle of mass m moves in a conservative force field described by the potential energy function U (r) = a(r/b + b/r), where a and b are positive constants and r is the distance from the origin. The graph of U (r) has the following shape. rU 2r0 Bro 4r0 (a) In terms of the constants a and I), determine the following. i. The position r0 at which the potential energy is a minimum ii. The minimum potential energy U0 (b) Sketch the net force on the particle as a function of r on the graph below, considering a force directed away from the origin to be positive, and a force directed toward the origin to be negative. H- __L_J__J__L__ __'1"'T__!'__l‘- -_l—-‘1__T__l_"' ___L_+-_Lfl- I l I I ——4——+——F I I I I I The particle is released from rest at r = r012. (c) In terms of U0 and m, determine the speed of the particle when it is at r = r0 . (d) Write the equation or equations that could be used to determine where, if ever, the particle will again come to rest. It is not necessary to solve for this position. (e) Briefly and qualitatively describe the motion of the particle over along period of time. GU ON TO THE NEXT PAGE 1995 PHYSICS C — MECHANICS I \ x \ x \ x x I \\ I "'“-~ .I l”, ‘x \ r , ‘\ \ r r \ \ t I \ 'l l' l P l l ‘ 4—.—|—> l A * r, ’rb u \ a , I l \ , l‘ \ \ z -' \ ‘\ II t \\ \‘-—__..v” I, \ I \ / \ z \ / \ I x / Mech 3. Two stars, A and B, are in circular orbits of radii ra and rb, respectively, about their common center of mass at point P, as shown above. Each star has the same period of revolution T. Determine expressions for the following three quantities in terms of ra, rb, T, and fundamental constants. (a) The centripetal acceleration of star A (b) The mass Mb of star B (c) The mass Ma of star A Determine expressions for the following two quantities in terms of Ma, Mb, ra, rb, T, and fundamental constants. (C!) The moment of inertia of the two-star system about its center of mass (e) The angular momentum of the system about the center of mass STOP END OF SECTION II, MECHANICS 1995 PHYSICS C SECTION II, ELECTRICITY AND MAGNETISM Time—45 minutes 3 Questions Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions, which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the pink booklet in the spaces provided after each part, NOT in this green insert. E & M 1. A very long nonconducting rod of radius a has positive charge distributed throughout its volume. The charge distribution is cyclindrically symmetric, and the total charge per unit length of the rod is 7L . (it) Use Gauss’s law to derive an expression for the magnitude of the electric field E outside the rod. (b) The diagrams below represent cross sections of the rod. On these diagrams, sketch the following. i. Several equipotential lines in the region r > a ii. Several electric field lines in the region r > a GO ON TO THE NEXT PAGE 1995 PHYSICS C — E 8: M (c) In the diagram above, point C is a distance a from the center of the rod (i.e., on the rod’s surface), and point D is a distance 3a from the center on a radius that is 90° from point C. Determine the following. i. The potential difference VC ~ VD between points C and D ii. The work required by an external agent to move a charge + Q from rest at point D to rest at point C Inside the rod (r < a), the charge density p is a function of radial distance r from the axis of the rod 1’2 and is given by p = p9(rla) , where p0 is a constant. ((1) Determine the magnitude of the electric field E as a function of r for r < a. Express your answer in terms of pa, a. and fundamental constants. GO ON TO THE NEXT PAGE E&M2. 1995 PHYSICS c -— E a M A parallel-plate capacitor is made from two sheets of metal, each with an area of 1.0 square meter, separated by a sheet of plastic 1.0 millimeter (10'3 m) thick, as shown above. The capacitance is measured to be 0.05 microfarad (5 x 10—8 F) . (a) What is the dielectric constant of the plastic? (b) The uncharged capacitor is connected in series with a resistor R = 2 x 10'3 ohms, a 30-volt battery, and an open switch S . as shown ab0ve. The switch is then closed. i. What is the initial charging current when the switch S is closed? ii. What is the time constant for this circuit? iii. Determine the magnitude and sign of the final charge on the bottom plate of the fully charged capacitor. iv. How much electrical energy is stored in the fully charged capacitor? After the capacitor is fully charged, it is carefully disconnected, leaving the charged capacitor isolated in space. The plastic sheet is then removed from between the metal plates. The metal plates retain their original separation of 1.0 millimeter. (c) What is the new voltage across the plates? (d) If there is now more energy stored in the capacitor, where did it come from? If there is now less energy, what happened to it? GO ON TO THE NEXT PAGE E&M3. 1995 PHYSICS C — E 8! M I I I O O O O O O O O O B(out of page) - o - Glider The long, narrow rectangular loop of wire shown above has vertical height H, length D, and resistance R. The loop is mounted on an insulated stand attached to a glider, which moves on a fric- tionless horizontal air track with an initial speed of no to the right. The loop and glider have a com— bined mass m. The loop enters a long, narrow region of uniform magnetic field B directed out of the page toward the reader. Express your answers to the parts below in terms of B, D, H, R, m, and no . (a) What is the magnitude of the initial induced emf in the loop as the front end of the loop begins to enter the region containing the field? (b) What is the magnitude of the initial induced current in the loop? (c) State whether the initial induced current in the loop is clockwise or counterclockwise around the 100p. (d) Derive an expression for the ve10city of the glider as a function of time t for the interval after the front edge of the loop has entered the magnetic field but before the rear edge has entered the field. GO ON TO THE NEXT PAGE 1995. PHYSICS c —— E a M (e) Using the axes below. sketch qualitatively a graph of speed 1) versus time t for the glider. The front end of the loop enters the field at t = 0. At t; the back end has entered and the loop is completely inside the field. At t2 the loop begins to come out of the field. At :3 it is com- pletely out of the field. Continue the graph until :4, a short time after the loop is completely out of the field. These times may not be shown to scale on the t-axis below. STOP END OF SECTION II, ELECTRICITY AND MAGNETISM ...
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MECH + E&amp;amp;M - 1995 - 1995 The College Board Advanced...

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