# Center of Mass/Momentum

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1 Center of Mass/Momentum An L-shaped piece, represented by the shaded area on the figure, is cut from a metal plate of uniform thickness. The point that corresponds to the center of mass of the L-shaped piece is 3. The shaded area in the figure represents a uniformly thick sheet of metal. The center of mass of the sheet is closest to point 4. The figure shows a piece of sheet metal suspended in two positions by a string. From the way the metal hangs, you can see that the center of gravity is nearest point The center of mass of the system of particles shown in the diagram is at point 5. The condition necessary for the conservation of momentum in a given system is that A) energy is conserved. D) internal forces equal external forces. B) one body is at rest. E) None of these is correct. C) the net external force is zero. 6. Momentum is conserved in which of the following? A) elastic collisions D) collisions between automobiles B) inelastic collisions E) All of these are correct. C) explosions

2 7. A boy and girl on ice skates face each other. The girl has a mass of 20 kg and the boy has a mass of 30 kg. The boy pushes the girl backward at a speed of 3.0 m/s. As a result of the push, the speed of the boy is A) zero B) 2.0 m/s C) 3.0 m/s D) 4.5 m/s E) 9.0 m/s 8. Two identical bodies of mass M move with equal speeds v. The direction of their velocities is illustrated in the figure. The magnitude of the linear momentum of the system is A) 2Mv B) Mv C) 4Mv D) E) 9. A golfball and a Ping-Pong ball are dropped in a vacuum chamber. When they have fallen halfway to the floor, they have the same A) speed. B) potential energy. C) kinetic energy. D) momentum. E) speed, potential energy, kinetic energy, and momentum. 10. In any and all collisions of short duration and for which it is true that no external forces act on the collision participants, A) kinetic energy is conserved. B) both momentum and kinetic energy are conserved. C) neither momentum nor kinetic energy is conserved. D) the relative velocities before and after impact are equal and oppositely directed. E) momentum is conserved. 11. For a system consisting of two particles that undergo an elastic collision, A) momentum is conserved but the total energy is not conserved. B) neither the kinetic energy nor the momentum is conserved. C) neither the total energy nor the momentum is necessarily conserved. D) the mechanical energy is conserved but momentum is not conserved. E) both kinetic energy and momentum are conserved. 12. If a body moves in such a way that its linear momentum is constant, then A) its kinetic energy is zero. B) the sum of all the forces acting on it must be zero. C) its acceleration is greater than zero and is constant. D) its center of mass remains at rest. E) the sum of all the forces acting on the body is constant and nonzero. 13. If the momentum of a mass M is doubled, its kinetic energy will be multiplied by a factor of A) B) 2 C) D) 4 E) 14. An object of mass M 1 is moving with a speed v on a straight, level, frictionless track when it collides with another mass M 2 that is at rest on the track. After the collision, M 1 and M 2 stick together and move with a speed of A) v B) M 1 v C) (M 1 + M 2 )v/m 1 D) M 1 v/(m 1 + M 2 ) E) M 1 v/m 2

3 15. A 40-kg girl, standing at rest on the ice, gives a 60-kg boy, who is also standing at rest on the ice, a shove. After the shove, the boy is moving backward at 2.0 m/s. Ignore friction. The girl's speed is A) zero B) 1.3 m/s C) 2.0 m/s D) 3.0 m/s E) 6.0 m/s 16. A moving particle is stopped by a single head-on collision with a second, stationary particle, if the moving particle undergoes A) an elastic collision with a second particle of much smaller mass. B) an elastic collision with a second particle of much greater mass. C) an elastic collision with a second particle of equal mass. D) an inelastic collision with a second particle of any mass. 17. Two cars of equal mass travel in opposite directions at equal speeds. They collide in a perfectly inelastic collision. Just after the collision, their velocities are A) zero. B) equal to their original velocities. C) equal in magnitude but opposite in direction to their original velocities. D) less in magnitude and in the same direction as their original velocities. E) less in magnitude and opposite in direction to their original velocities. 18. Two equal masses travel in opposite directions with equal speed. If they collide in a perfectly elastic collision, then, just after the collision, their velocities will be A) zero. B) equal to their original velocities. C) equal in magnitude but opposite in direction to their original velocities. D) less in magnitude and in the same direction as their original velocities. E) less in magnitude and opposite in direction to their original velocities. 19. Two equal masses travel in opposite directions with equal speeds. They collide in a collision that is between elastic and inelastic. Just after the collision, their velocities are A) zero. B) equal to their original velocities. C) equal in magnitude but opposite in direction to their original velocities. D) less in magnitude and in the same direction as their original velocities. E) less in magnitude and opposite in direction to their original velocities. 20. In an elastic collision of two objects, A) momentum is not conserved. B) momentum is conserved, and the kinetic energy after the collision is less than its value before the collision. C) momentum is conserved, and the kinetic energy after the collision is the same as the kinetic energy before the collision. D) momentum is not conserved, and the kinetic energy of the system after the collision differs from the kinetic energy of the system before the collision. E) the kinetic energy of the system after the collision depends on the masses of the objects. 21. Two balls of equal mass are thrown against a massive wall with equal velocities. The first rebounds with a speed equal to its striking speed, and the second sticks to the wall. The impulse that the first ball transmits to the wall, relative to the second, is A) twice as great. D) four times as great. B) half as great. E) one-fourth as great. C) the same. 22. A ball of mass m strikes a wall that is perpendicular to its path at speed +v and rebounds in the opposite direction with a speed v. The impulse imparted to the ball by the wall is A) 2mv B) mv C) zero D) mv E) 2mv

4 23. The center of mass of a system of particles is so defined that A) it is always at rest. B) it is always at rest or moving with constant velocity. C) it always moves in a straight line even if the particles are rotating about it. D) the kinetic energy of the system is a maximum about any axis through the center of mass. E) its location depends only on the masses of the particles and their locations. Circular Motion/Gravity 24. When a particle moves in a circle with constant speed, its acceleration is A) constantly increasing. D) constant in magnitude. B) constant in direction. E) constant in magnitude and direction. C) zero. 25. An object traveling in a circle at constant speed A) is moving with constant velocity. B) may be slowing down or picking up speed. C) experiences no acceleration. D) experiences an acceleration toward the center of the circle. E) is described by none of the above statements. 26. A car going around a curve of radius R at a speed V experiences a centripetal acceleration a c. What is its acceleration if it goes around a curve of radius 3R at a speed of 2V? A) (2/3)a c B) (4/3)a c C) (2/9)a c D) (9/2)a c E) (3/2)a c 27. A car experiences both a centripetal and a tangential acceleration. For which of the following would this be true? A) It is going around a curve at a constant speed. B) It is going around a curve and slowing down. C) It is going along a straight road at a constant speed. D) It is going along a straight road and increasing its speed. E) It is going along a straight road and decreasing its speed. 28. The figure shows a top view of a ball on the end of a string traveling counterclockwise in a circular path. The speed of the ball is constant. If the string should break at the instant shown, the path that the ball would follow is A) 1 B) 2 C) 3 D) 4 E) impossible to tell from the given information. 29. The figure shows a top view of a ball on the end of a string traveling counterclockwise in a circular path. Assume that air resistance is negligible. The free-body diagram that best represents the net force acting on the ball is 30. The figure shows a top view of a ball on the end of a string traveling counterclockwise in a circular path. Assume that air resistance is negligible. The free-body diagram that best represents the acceleration of the ball is

5 31. If the mass of a satellite is doubled while the radius of its orbit remains constant, the speed of the satellite is A) increased by a factor of 8. D) reduced by a factor of 8. B) increased by a factor of 2. E) reduced by a factor of 2. C) not changed. 32. Of the satellites shown revolving around the earth, the one with the greatest speed is 33. If the mass of a planet is doubled while its radius and the radius of orbit of its moon remain constant, the speed of the moon is A) increased by a factor of. D) reduced by a factor of. B) increased by a factor of 2. E) reduced by a factor of 2. C) not changed. 34. A woman whose weight on earth is 500 N is lifted to a height of two earth radii above the surface of the earth. Her weight A) decreases to one-half of the original amount. B) decreases to one-quarter of the original amount. C) decreases to one-fifth of the original amount. D) decreases to one-third of the original amount. E) decreases to one-ninth of the original amount. 35. The acceleration due to gravity at the surface of the earth is g. The radius of the earth is R E. The distance from the center of the earth to a point where the acceleration due to gravity is g/9 is A) R E B) 9R E C) R E /3 D) 3R E E) None of these is correct. 36. At the surface of the moon, the acceleration due to the gravity of the moon is a. At a distance from the center of the moon equal to three times the radius of the moon, the acceleration due to the gravity of the moon is A) 9 a B) a /3 C) a /4 D) a /9 E) 27 a 37. Suppose a planet exists that has half the mass of earth and half its radius. On the surface of that planet, the acceleration due to gravity is A) twice that on earth. D) one-fourth that on earth. B) the same as that on earth. E) none of these. C) half that on earth. 38. You need an expression for the acceleration of the moon toward the earth. If the mass of the earth is M e, the mass of the moon M m, the separation of the earth and moon r, and the appropriate gravitational constant is G, the correct expression for the moon's acceleration is A) GM e M m /r 2 B) GM e M m 2 /r 2 C) GM m /r 2 D) GM e /r 2 E) GM e /r 2 M m 39. The acceleration due to gravity in the vicinity of the earth A) varies directly with the distance from the center of the earth. B) is a constant that is independent of altitude. C) varies inversely with the distance from the center of the earth. D) varies inversely with the square of the distance from the center of the earth. E) is described by none of these.

6 40. If a planet has a mass twice that of the earth and a radius four times that of the earth, the ratio of the acceleration due to gravity on the planet to that on the earth is A) 1/8 B) 1/2 C) 1/16 D) 2/1 E) 12/1 41. When two masses are a distance R apart, each exerts a force of magnitude F on the other. When the distance between them is changed to 4R, the force is changed to A) 16F B) 4F C) F/2 D) F/4 E) F/ According to Newton's law of universal gravitation, if the distance between two bodies is tripled, the gravitational force between them is A) unchanged. D) reduced to 1/3 its previous value. B) halved. E) None of these is correct. C) doubled. 43. The radius R of a stable, circular orbit for a satellite of mass m and velocity v about a planet of mass M is given by A) R = Gv/M B) R = Gv/mM C) R = GmM/v D) R = GM/mv E) R = GM/v 2 Simple Harmonic Motion 44. When an object is oscillating in simple harmonic motion in the vertical direction, its maximum speed occurs when the object A) is at its highest point. B) is at its lowest point. C) is at the equilibrium point. D) has the maximum net force exerted on it. E) has a position equal to its amplitude. 45. A mass m hanging on a spring with a spring constant k has simple harmonic motion with a period T. If the mass is doubled to 2m, the period of oscillation A) increases by a factor of 2. D) decreases by a factor of B) decreases by a factor of 2. E) is not affected. C) increases by a factor of 46. If F is the force, x the displacement, and k a particular constant, for simple harmonic motion we must have A) F = k/x 2 D) F = kx 2 B) F = k/x E) None of these is correct. C) F = (k/x 2 ) 1/2 47. A mass m hanging on a spring with a spring constant k executes simple harmonic motion with a period T. If the same mass is hung from a spring with a spring constant of 2k, the period of oscillation A) increases by a factor of 2. D) decreases by a factor of. B) decreases by a factor of 2. E) is not affected. C) increases by a factor of. 48. You want a mass that, when hung on the end of a spring, oscillates with a period of 1 s. If the spring has a spring constant of 10 N/m, the mass should be A) 10 kg D) 10/(4π 2 ) kg B) E) None of these is correct. C) 4π 2 (10) kg

7 49. Any body moving with simple harmonic motion is being acted on by a force that is A) constant. B) proportional to a sine or cosine function of the displacement. C) proportional to the inverse square of the displacement. D) directly proportional to the displacement. E) proportional to the square of the displacement. 50. The top graph represents the variation of displacement with time for a particle executing simple harmonic motion. Which curve in the bottom graph represents the variation of acceleration with time for the same particle? A) 1 B) 2 C) 3 D) 4 E) None of these is correct. 51. A body moving in simple harmonic motion has maximum acceleration when it has A) maximum velocity. D) minimum kinetic energy. B) maximum kinetic energy. E) zero displacement. C) minimum potential energy. 52. The displacement in simple harmonic motion is a maximum when the A) acceleration is zero. D) kinetic energy is a maximum. B) velocity is a maximum. E) potential energy is a minimum. C) velocity is zero. 53. In simple harmonic motion, the magnitude of the acceleration of a body is always directly proportional to its A) displacement. B) velocity. C) mass. D) potential energy. E) kinetic energy. 54. A system consists of a mass vibrating on the end of a spring. The total mechanical energy of this system A) varies as a sine or cosine function. B) is constant only when the mass is at maximum displacement. C) is a maximum when the mass is at its equilibrium position only. D) is constant, regardless of the displacement of the mass from the equilibrium position. E) is always equal to the square of the amplitude. 55. A body on a spring is vibrating in simple harmonic motion about an equilibrium position indicated by the dashed line. The figure that shows the body with maximum acceleration is 56. A body of mass M suspended from a spring oscillates with a period T. If the mass of the spring can be neglected, a body of mass 2M, suspended from the same spring, oscillates with a period of A) T/2 B) C) T D) E) 2T

8 57. A mass of 2.00 kg suspended from a spring 100 cm long is pulled down 4.00 cm from its equilibrium position and released. The amplitude of vibration of the resulting simple harmonic motion is A) 4.00 cm B) 2.00 cm C) 8.00 cm D) 1.04 cm E) 1.02 cm 58. If the length of a simple pendulum with a period T is reduced to half of its original value, the new period T is approximately A) 0.5T B) 0.7T C) T (unchanged) D) 1.4T E) 2T 59. To double the period of a pendulum, the length A) must be increased by a factor of 2. D) must be increased by a factor of 4. B) must be decreased by a factor of 2. E) need not be affected. C) must be increased by a factor of. 60. A clock keeps accurate time when the length of its simple pendulum is L. If the length of the pendulum is increased a small amount, which of the following is true? A) The clock will run slow. B) The clock will run fast. C) The clock will continue to keep accurate time. D) The answer cannot be determined without knowing the final length of the pendulum. E) The answer cannot be determined without knowing the percentage increase in the length of the pendulum. 61. You have landed your spaceship on the moon and want to determine the acceleration due to gravity using a simple pendulum of length 1.0 m. If the period of this pendulum is 5.0 s, what is the value of g on the moon? A) 1.3 m/s 2 B) 1.6 m/s 2 C) 0.80 m/s 2 D) 0.63 m/s 2 E) 2.4 m/s A simple pendulum on the earth has a period T. The period of this pendulum could be decreased by A) increasing the mass of the pendulum bob. B) taking the pendulum to the moon. C) taking the pendulum to the planet Jupiter (M Jupiter = 315M Earth ). D) decreasing the mass of the pendulum bob. E) increasing the length of the wire supporting the pendulum. Waves 63. A particle is subject to a wave motion. Its maximum distance from the equilibrium is called its A) amplitude B) displacement C) phase D) wavelength E) period 64. In which of the following is the speed of sound greatest? A) air B) water C) a vacuum D) wood E) steel 65. A string under tension carries transverse waves traveling at speed v. If the same string is under four times the tension, what is the wave speed? A) v B) 2v C) v/2 D) 4v E) v/4 66. A string under tension carries a transverse wave traveling at speed v. If the tension in the string is halved, what is the wave speed? A) The wave speed is unchanged. B) The wave speed is halved. C) The wave speed is quadrupled. D) The wave speed decreases to about 0.71 v. E) The wave speed increases by about 41%.

9 67. A traveling wave passes a point of observation. At this point, the time between successive crests is 0.2 s. Which of the following statements can be justified? A) The wavelength is 5 m. B) The frequency is 5 Hz. C) The velocity of propagation is 5 m/s. D) The wavelength is 0.2 m. E) There is not enough information to justify any of these statements The figure represents a string of length L, fixed at both ends, vibrating in several harmonics. The 4th harmonic is shown in The figure represents a tube of length L, vibrating in several harmonics. The 3 rd Answer: 2 harmonic is shown in 70. The fundamental frequency of a vibrating string is f 1. If the tension in the string is doubled, the fundamental frequency becomes A) f 1 /2 B) C) f 1 D) E) 2f The fundamental frequency of a vibrating string is f 1. If the tension in the string is quadrupled while the linear density is held constant, the fundamental frequency becomes A) f 1 B) 1.2f 1 C) 1.5f 1 D) 1.7f 1 E) 2f The figure shows several modes of vibration of a string fixed at both ends. The mode of vibration that represents the fifth harmonic is A) 1 B) 2 C) 3 D) 4 E) None of these is correct.

10 73. Of the sound sources shown, that which is vibrating with its first harmonic is A) the whistle. D) the vibrating rod. B) the organ pipe. E) None of these is correct. C) the vibrating string. 74 Of the sound sources shown, that which is vibrating with its first harmonic is the A) whistle. D) vibrating rod. B) organ pipe. E) vibrating spring. C) vibrating string. 74. A string fixed at both ends is vibrating in a standing wave. There are three nodes between the ends of the string, not including those on the ends. The string is vibrating at a frequency that is its A) fundamental. D) fourth harmonic. B) second harmonic. E) fifth harmonic. C) third harmonic. 75. On a standing-wave pattern, the distance between two consecutive nodes is d. The wavelength is A) d/2 B) d C) 3d/2 D) 2d E) 4d 76. In a pipe that is open at one end and closed at the other and that has a fundamental frequency of 256 Hz, which of the following frequencies cannot be produced? A) 768 Hz D) 19.7 khz B) 1.28 khz E) All of these can be produced. C) 5.12 khz 77. The fundamental frequency of a pipe that has one end closed is 256 Hz. When both ends of the same pipe are opened, the fundamental frequency is A) 64.0 Hz B) 128 Hz C) 256 Hz D) 512 Hz E) 1.02 khz 78. A 1.00 m string fixed at both ends vibrates in its fundamental mode at 440 Hz. What is the speed of the waves on this string? A) 220 m/s B) 440 m/s C) 660 m/s D) 880 m/s E) 1.10 km/s DC Circuits 79.. In a parallel circuit, A) the current is the same in every branch. B) the voltage is the sum of those in all branches. C) the voltage is the same for each element of the parallel circuit. D) the heat generated is the same in all branches. E) the resistance is the sum of the resistances of the branches. Ans: C

11 80. When two identical resistors are connected in parallel, the equivalent resistance, compared with that of the same two resistors connected in series, is A) exactly the same. D) four times as great. B) twice as great. E) one-fourth as much. C) one-half as much. Ans: E 81. If three resistors in the various configurations shown are placed in a simple circuit, the configuration in which all three resistors carry the same current is Ans: C 82. The circuit in the figure contains a cell of voltage E and four resistors connected as shown. Let the currents in these resistances be designated by I 1, I 2, I 3, I 4, respectively. Which of the following equations is correct? A) I 1 = I 2 B) I 2 = I 3 C) I 3 = I 4 D) I 1 = I 4 E) I 1 = I 2 + I 3 Ans: A

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