【AP物理C】【真题】1995-2008力学解答题

AP® Physics C Mechanics FILLIN "Enter Subject" \* MERGEFORMAT SUBJECT \* MERGEFORMAT 1995-2008 Free response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought from the Advanced Placement Program®. Teachers may reproduce them, in whole or in part, in limited quantities, for face-to-face teaching purposes but may not mass distribute the materials, electronically or otherwise. These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here. This permission does not apply to any third-party copyrights contained herein. These materials were produced by Educational Testing Service® (ETS®), which develops and administers the examinations of the Advanced Placement Program for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and their programs, services, and employment policies are guided by that principle. The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity. Founded in 1900, the association is composed of more than 4,200 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges, through major programs and services in college admission, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of equity and excellence, and that commitment is embodied in all of its programs, services, activities, and concerns. APIEL is a trademark owned by the College Entrance Examination Board. PSAT/NMSQT is a registered trademark jointly owned by the College Entrance Examination Board and the National Merit Scholarship Corporation. Educational Testing Service and ETS are registered trademarks of Educational Testing Service. 1995M1. A 5‑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 table 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. a. Determine the total impulse given to the ball. b. Determine the horizontal velocity of the ball immediately after the collision. c. Determine the following for the cube immediately after the collision. i. Its speed ii. Its direction of travel (right or left), if moving d. Determine the kinetic energy dissipated in the collision. e. Determine the distance between the two points of impact of the objects with the floor. 1995M2. 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. a. In terms of the constants a and b, determine the following. i. The position ro at which the potential energy is a minimum ii. The minimum potential energy Uo 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. The particle is released from rest at r = ro/2 c. In terms of Uo and m, determine the speed of the particle when it is at r = ro . 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 a long period of time. 1995M3. 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. d. 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. 1996M1. A thin, flexible metal plate attached at one end to a platform, as shown above, can be used to measure mass. When the free end of the plate is pulled down and released, it vibrates in simple harmonic motion with a period that depends on the mass attached to the plate. To calibrate the force constant, objects of known mass are attached to the plate and the plate is vibrated, obtaining the data shown below. a. Fill in the blanks in the data table. b. On the graph below, plot T2 versus mass. Draw on the graph the line that is your estimate of the best straight‑line fit to the data points. c. An object whose mass is not known is vibrated on the plate, and the average time for ten vibrations is measured to be 16.1 s. From your graph, determine the mass of the object. Write your answer with a reasonable number of significant digits. d. Explain how one could determine the force constant of the metal plate. e. Can this device be used to measure mass aboard the space shuttle Columbia as it orbits the Earth? Explain briefly. f. If Columbia is orbiting at 0.3 x 106 m above the Earth's surface, what is the acceleration of Columbia due to the Earth's gravity? (Radius of Earth = 6.4 x 106 m, mass of Earth = 6.0 x 1024 kg) g. Since the answer to part (f) is not zero, briefly explain why objects aboard the orbiting Columbia seem weightless. 1996M2. A 300‑kg box rests on a platform attached to a forklift, shown above. Starting from rest at at time = 0, the box is lowered with a downward acceleration of 1.5m/s2 a. Determine the upward force exerted by the horizontal platform on the box as it is lowered. At time t = 0, the forklift also begins to move forward with an acceleration of 2 m/s2 while lowering the box as described above. The box does not slip or tip over. b. Determine the frictional force on the box. c. Given that the box does not slip, determine the minimum possible coefficient of friction between the box and the platform. d. Determine an equation for the path of the box that expresses y as a function of x (and not of t), assuming that, at time t = 0, the box has a horizontal position x = 0 and a vertical position y = 2 m above the ground, with zero velocity. e. On the axes below sketch the path taken by the box 1996M3. Consider a thin uniform rod of mass M and length l, as shown above. a. Show that the rotational inertia of the rod about an axis through its center and perpendicular to its length is Ml2/12 . The rod is now glued to a thin hoop of mass M and radius R/2 to form a rigid assembly, as shown above. The centers of the rod and the hoop coincide at point P. The assembly is mounted on a horizontal axle through point P and perpendicular to the page. b. What is the rotational inertia of the rod‑hoop assembly about the axle? Several turns of string are wrapped tightly around the circumference of the hoop. The system is at rest when a cat, also of mass M, grabs the free end of the string and hangs vertically from it without swinging as it unwinds, causing the rod‑hoop assembly to rotate. Neglect friction and the mass of the string. c. Determine the tension T in the string. d. Determine the angular acceleration a of the rod‑hoop assembly. e. Determine the linear acceleration of the cat. f. After descending a distance H = 5l/3, the cat lets go of the string. At that instant, what is the angular momentum of the cat about point P ? 1997M1. A nonlinear spring is compressed horizontally. The spring exerts a force that obeys the equation F(x) = Ax½, where x is the distance from equilibrium that the spring is compressed and A is a constant. A physics student records data on the force exerted by the spring as it is compressed and plots the two graphs below, which include the data and the student's best‑fit curves. a. From one or both of the given graphs, determine A. Be sure to show your work and specify the units. b. i. Determine an expression for the work done in compressing the spring a distance x. ii. Explain in a few sentences how you could use one or both of the graphs to estimate a numerical answer to part (b)i for a given value of x. c. The spring is mounted horizontally on a countertop that is 1.3 m high so that its equilibrium position is just at the edge of the countertop. The spring is compressed so that it stores 0.2 J of energy and is then used to launch a ball of mass 0.10 kg horizontally from the countertop. Neglecting friction, determine the horizontal distance d from the edge of the countertop to the point where the hall strikes the floor 1997M2. An open‑top railroad car (initially empty and of mass Mo) rolls with negligible friction along a straight horizontal track and passes under the spout of a sand conveyor. When the car is under the conveyor, sand is dispensed from the conveyor in a narrow stream at a steady rate (M/(t = C and falls vertically from an average height h above the floor of the railroad car. The car has initial speed vo and sand is filling it from time t = 0 to t = T. Express your answers to the following in terms of the given quantities and g. a. Determine the mass M of the car plus the sand that it catches as a function of time t for 0 < t < T. b. Determine the speed v of the car as a function of time t for 0 < t < T. c. i. Determine the initial kinetic energy Ki of the empty car. ii. Determine the final kinetic energy Kf of the car and its load. iii. Is kinetic energy conserved? Explain why or why not. d. Determine expressions for the normal force exerted on the car by the tracks at the following times. i. Before t = 0 ii. For 0 < t < T iii. After t = T 1997M3. A solid cylinder with mass M, radius R, and rotational inertia ½MR2 rolls without slipping down the inclined plane shown above. The cylinder starts from rest at a height H. The inclined plane makes an angle ( with the horizontal. Express all solutions in terms of M, R, H, (, and g. a. Determine the translational speed of the cylinder when it reaches the bottom of the inclined plane. b. On the figure below, draw and label the forces acting on the cylinder as it rolls down the inclined plane Your arrow should begin at the point of application of each force. c. Show that the acceleration of the center of mass of the cylinder while it is rolling down the inclined plane is (2/3)g sin(. d. Determine the minimum coefficient of friction between the cylinder and the inclined plane that is required for the cylinder to roll without slipping. e. The coefficient of friction ( is now made less than the value determined in part (d), so that the cylinder both rotates and slips. i. Indicate whether the translational speed of the cylinder at the bottom of the inclined plane is greater than, less than, or equal to the translational speed calculated in part (a). Justify your answer. ii. Indicate whether the total kinetic energy of the cylinder at the bottom of the inclined plane is greater than, less than, or equal to the total kinetic energy for the previous case of rolling without slipping. Justify your answer. 1998M1. Two gliders move freely on an air track with negligible friction, as shown above. Glider A has a mass of 0.90 kg and glider B has a mass of 0.60 kg. Initially, glider A moves toward glider B, which is at rest. A spring of negligible mass is attached to the right side of glider A. Strobe photography is used to record successive positions of glider A at 0.10 s intervals over a total time of 2.00 s, during which time it collides with glider B. The following diagram represents the data for the motion of glider A. Positions of glider A at the end of each 0.10s interval are indicated by the symbol A against a metric ruler. The total elapsed time t after each 0.50 s is also indicated. a. Determine the average speed of glider A for the following time intervals. i. 0.L0 s to 0.30 s ii. 0.90 s to 1.10 s iii. 1.70 s to 1.90 s b. On the axes below, sketch a graph, consistent with the data above, of the speed of glider A as a function of time t for the 2.00 s interval. [Continued on next page] c. i. Use the data to calculate the speed of glider B immediately after it separates from the spring. ii. On the axes below, sketch a graph of the speed of glider B as a function of time t. A graph of the total kinetic energy K for the two‑glider system over the 2.00 s interval has the following shape. Ko is the total kinetic energy of the system at time t = 0. d. i. Is the collision elastic? Justify your answer. ii. Briefly explain why there is a minimum in the kinetic energy curve at t = 1.00 s. 1998M2. A space shuttle astronaut in a circular orbit around the Earth has an assembly consisting of two small dense spheres, each of mass m, whose centers are connected by a rigid rod of length l and negligible mass. The astronaut also has a device that will launch a small lump of clay of mass m at speed v0 . Express your answers in terms of m, v0 l. and fundamental constants. a. Initially, the assembly is "floating" freely at rest relative to the cabin, and the astronaut launches the clay lump so that it perpendicularly strikes and sticks to the midpoint of the rod, as shown above. i. Determine the total kinetic energy of the system (assembly and clay lump) after the collision. ii. Determine the change in kinetic energy as a result of the collision. b. The assembly is brought to rest, the clay lump removed, and the experiment is repeated as shown above, with the clay lump striking perpendicular to the rod but this time sticking to one of the spheres of the assembly. i. Determine the distance from the left end of the rod to the center of mass of the system (assembly and clay lump) immediately after the collision. (Assume that the radii of the spheres and clay lump are much smaller than the separation of the spheres.) ii. On the figure above, indicate the direction of the motion of the center of mass immediately after the collision. iii. Determine the speed of the center of mass immediately after the collision. iv. Determine the angular speed of the system (assembly and clay lump) immediately after the collision. v. Determine the change in kinetic energy as a result of the collision. 1998M3. Block 1 of mass m1 is placed on block 2 of mass m2 which is then placed on a table. A string connecting block 2 to a hanging mass M passes over a pulley attached to one end of the table, as shown above. The mass and friction of the pulley are negligible. The coefficients of friction between blocks 1 and 2 and between block 2 and the tabletop are nonzero and are given in the following table. Express your answers in terms of the masses, coefficients of friction, and g, the acceleration due to gravity. a. Suppose that the value of M is small enough that the blocks remain at rest when released. For each of the following forces, determine the magnitude of the force and draw a vector on the block provided to indicate the direction of the force if it is nonzero. i. The normal force N1 exerted on block 1 by block 2 ii. The friction force f1 exerted on block 1 by block 2 iii. The force T exerted on block 2 by the string iv. The normal force N2 exerted on block 2 by the tabletop v. The friction force f2 exerted on block 2 by the tabletop [Continued next page] b. Determine the largest value of M for which the blocks can remain at rest. c. Now suppose that M is large enough that the hanging block descends when the blocks are released. Assume that blocks 1 and 2 are moving as a unit (no slippage). Determine the magnitude a of their acceleration. d. Now suppose that M is large enough that as the hanging block descends, block 1 is slipping on block 2. Determine each of the following. i. The magnitude a1 of the acceleration of block 1 ii. The magnitude a2 of the acceleration of block 2 1999M1 In a laboratory experiment, you wish to determine the initial speed of a dart just after it leaves a dart gun. The dart, of mass m, is fired with the gun very close to a wooden block of mass M0 which hangs from a cord of length l and negligible mass, as shown above. Assume the size of the block is negligible compared to l, and the dart is moving horizontally when it hits the left side of the block at its center and becomes embedded in it. The block swings up to a maximum angle from the vertical. Express your answers to the following in terms of m, M0, l, (max, and g. a. Determine the speed v0 of the dart immediately before it strikes the block. b. The dart and block subsequently swing as a pendulum. Determine the tension in the cord when it returns to the lowest point of the swing. c. At your lab table you have only the following additional equipment. Meter stick Stopwatch Set of known masses Protractor 5 m of string Five more blocks of mass M0 Spring Without destroying or disassembling any of this equipment, design another practical method for determining the speed of the dart just after it leaves the gun. Indicate the measurements you would take, and how the speed could be determined from these measurements. d. The dart is now shot into a block of wood that is fixed in place. The block exerts a force F on the dart that is proportional to the dart's velocity v and in the opposite direction, that is F = ‑bv, where b is a constant. Derive an expression for the distance L that the dart penetrates into the block, in terms of m, v0, and b. 1999 M2 A spherical, nonrotating planet has a radius R and a uniform density ( throughout its volume. Suppose a narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above, in which a small ball of mass m could move freely under the influence of gravity. Let r be the distance of the ball from the center of the planet. a. Show that the magnitude of the force on the ball at a distance r < R from the center of the planet is given by F = ‑Cr. where C = 4/3((G(m). b. On the axes below, sketch the force F on the ball as a function of distance r from the center of the planet. The ball is dropped into the tunnel from rest at point P at the planet's surface. c. Determine the work done by gravity as the ball moves from the surface to the center of the planet. d. Determine the speed of the ball when it reaches the center of the planet. e. Fully describe the subsequent motion of the ball from the time it reaches the center of the planet. f. Write an equation that could be used to calculate the time it takes the ball to move from point P to the center of the planet. It is not necessary to solve this equation. 1999M3 As shown above, a uniform disk is mounted to an axle and is free to rotate without friction. A thin uniform rod is rigidly attached to the disk so that it will rotate with the disk. A block is attached to the end of the rod. Properties of the disk, rod, and block are as follows. Disk: mass = 3m, radius = R, moment of inertia about center ID = 1.5mR2 R