Chapter 1. Oscillations. Oscillations

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1 Oscillations 1. 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 2. The frequency of a simple harmonic motion is s 1. The oscillation starts (t = 0) when the displacement has its maximum positive value of cm. The earliest possible time at which the particle can be found at x = cm is A) s D) s B) s E) s C) s 3. 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. 4. 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 5. The instantaneous speed of a mass undergoing simple harmonic motion on the end of a spring depends on A) the amplitude of oscillation. B) The frequency of oscillation. C) the period of oscillation. D) the time at which the speed is measured. E) all of these. Page 1

2 6. A particle is oscillating with simple harmonic motion. The frequency of the motion is 10 Hz and the amplitude of the motion is 5.0 cm. As the particle passes its central equilibrium position, the acceleration of the particle is A) 100 cm/s 2 D) zero B) cm/s 2 E) cm/s 2 C) cm/s 2 7. 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. 8. A body moves with simple harmonic motion according to the equation x = (2/π) sin(4πt + π/3) where the units are SI. At t = 2 s, the speed of the body is A) 1/3 m/s B) 1/π m/s C) D) 4 m/s E) 9. A spring vibrates in simple harmonic motion according to the equation x = 15 cos πt where x is in centimeters and t is in seconds. The total number of vibrations this body makes in 10 s is A) 0.5 B) 10 C) π D) 15 E) A spring vibrates in simple harmonic motion according to the equation x = 0.15 cos πt where the units are SI. The period of the motion is A) 0.67 s B) 1.0 s C) 2.0 s D) π s E) 3.2 s 11. A body of mass 5.0 kg moves in simple harmonic motion according to the equation x = sin(30t + π/6) where the units are SI. The period of this motion is A) 1/30 s B) π/15 s C) π/6 s D) 15/π s E) 30 s Page 2

3 12. A body of mass 5.0 kg moves in simple harmonic motion according to the equation x = sin(30t + π/6) where the units are SI. The maximum speed of this body is approximately A) m/s B) 0.40 m/s C) 0.60 m/s D) 1.2 m/s E) 30 m/s 13. A body of mass 0.50 kg moves in simple harmonic motion with a period of 1.5 s and an amplitude of 20 mm. Which of the following equations correctly represents this motion? A) x = 40 cos(t/1.5) mm D) x = 20 sin(1.5πt) mm B) x = 40 cos(2πt/1.5) mm E) x = 20 sin(2πt/1.5) mm C) x = 20 sin(t/1.5) mm 14. A particle moves in one dimension with simple harmonic motion according to the equation d 2 x/dt 2 = 4π 2 x where the units are SI. Its period is A) 4π 2 s B) 2π s C) 1 s D) 1/(2π) s E) 1/(4π 2 ) s 15. 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. Page 3

4 16. In the following equations, a is acceleration, r is a fixed distance, s is displacement, and m is mass. Which equation describes simple harmonic motion? A) a = kr 2 B) a = πr 2 C) a = ks 1 D) a = 4πmr 2 /3 E) a = 4πms/3 17. In simple harmonic motion, the displacement x = A cos ωt and the acceleration a = ω 2 x. If A = 0.25 m and the period is 0.32 s, the acceleration when t = 0.12 s is A) zero B) +3.9 m/s 2 C) 3.9 m/s 2 D) +6.8 m/s 2 E) 6.8 m/s 2 Use the following to answer questions The object in the diagram is in circular motion. Its position at t = 0 was (A, 0). Its frequency is f. The y component of its position is given by A) y = y 0 + v 0y t +½ at 2 D) y = A sin 2πft B) y = A cos 2πft E) y = A cos ft C) y = A sin ft 19. The object in the diagram is in circular motion with frequency f. At t = 0 it was at (A, 0). The y component of its velocity is given by A) v y 2 = v 0y 2 + 2a(y y 0 ) D) vy = 2πfA sin 2πft B) v y = 2πfA cos 2πft E) v y = A cos ft C) v y = A sin ft 20. The object in the diagram is in circular motion with frequency f. At t = 0 it was at (A, 0). The y component of its acceleration is given by A) a y = (v y v 0y )/t D) a y = (2πf) 2 A sin 2πft B) a y = (2πf) 2 A cos 2πft E) a y = (2π) 2 A cos 2πt C) a y = (2π) 2 A sin 2πt Page 4

5 21. A body of mass M is executing simple harmonic motion with an amplitude of 8.0 cm and a maximum acceleration of 100 cm/s 2. When the displacement of this body from the equilibrium position is 6.0 cm, the magnitude of the acceleration is approximately A) 8.7 cm/s 2 B) 21 cm/s 2 C) 35 cm/s 2 D) 17 cm/s 2 E) 1.3 m/s A light spring stretches 0.13 m when a 0.35 kg mass is hung from it. The mass is pulled down from this equilibrium position an additional 0.15 m and then released. Determine the maximum speed of the mass. A) 1.10 m/s B) 2.75 m/s C) 11.4 m/s D) 1.25 m/s E) 0.02 m/s 23. A 2.50-kg object is attached to a spring of force constant k = 4.50 kn/m. The spring is stretched 10.0 cm from equilibrium and released. What is the maximum kinetic energy of this system? A) 45.0 J B) 22.5 J C) 56.0 J D) J E) 4.50 J 24. A mass attached to a spring has simple harmonic motion with an amplitude of 4.0 cm. When the mass is 2.0 cm from the equilibrium position, what fraction of its total energy is potential energy? A) one-quarter B) one-third C) one-half D) two-thirds E) three-quarters 25. When the compression of a spring is doubled, the potential energy stored in the spring is A) the same as before. D) increased by a factor of 8. B) doubled. E) None of these is correct. C) tripled. 26. The energy of a simple harmonic oscillator could be doubled by increasing the amplitude by a factor of A) 0.7 B) 1.0 C) 1.4 D) 2.0 E) The force constant for a simple harmonic motion is k and the amplitude of the motion is A. The maximum value of the potential energy of a mass m oscillating with simple harmonic motion is A) B) ½ ka 2 C) ka 2 D) ka E) None of these is correct. 28. When the displacement of an object in simple harmonic motio n is one-quarter of the amplitude A, the potential energy is what fraction of the total energy? A) ¼ B) ½ C) 1/16 D) Too little information is given to answer correctly. E) None of these is correct. Ans: C Page 5

6 29. If the amplitude of a simple harmonic oscillator is doubled, the total energy is A) unchanged. D) doubled. B) one-fourth as large. E) quadrupled. C) half as large. 30. Which of the following statements is true of a particle that is moving in simple harmonic motion? A) The momentum of the particle is constant. B) The kinetic energy of the particle is constant. C) The potential energy of the earth particle system is constant. D) The acceleration of the particle is constant. E) The force the particle experiences is a negative restoring force. 31. 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. 32. 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. 33. In simple harmonic motion, the magnitude of the acceleration of a body is always directly proportional to its A) displacement. D) potential energy. B) velocity. E) kinetic energy. C) mass. 34. The displacement of a body moving with simple harmonic motion is given by the equation y = A sin(2πt + ½π) After one-quarter of a period has elapsed since t = 0, which of the following statements is correct? A) Half the total energy of the body is kinetic energy and half is potential energy. B) The kinetic energy is a maximum. C) The potential energy is a maximum. D) The total energy is a negative maximum. E) Both kinetic and potential energies are maxima. Page 6

7 35. 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. 36. A 2-kg mass oscillates in one dimension with simple harmonic motion on the end of a massless spring on a horizontal frictionless table according to x = (6/π) cos(½πt+ 3π) where the units are SI. The total mechanical energy of this system is A) 1 J B) 3 J C) 5 J D) 7 J E) 9 J 37. A kg block starts from rest and slides 3.05 m down a frictionless plane inclined at 53º to the horizontal. At the bottom it slides 9.14 m over a horizontal frictionless plane before compressing a spring (k = 14.7 N/m) a distance x and coming momentarily to rest. The value of x is approximately A) m B) 1.52 m C) 2.44 m D) 1.83 m E) 1.21 m 38. 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 A) 1 B) 2 C) 3 D) 4 E) 5 Page 7

8 39. A 0.5-kg mass is suspended from a massless spring that has a force constant of 79 N/m. The mass is displaced 0.1 m down from its equilibrium position and released. If the downward direction is negative, the displacement of the mass as a function of time is given by A) y = 0.1 cos(158t π) D) y = 0.2 cos(12.6t + π) B) y = 0.2 cos(158t π) E) y = 0.1 cos(2t + π) C) y = 0.1 cos(12.6t π) 40. A spring is cut in half. The ratio of the force constant of one of the halves to the force constant of the original spring is A) ½ B) 1 C) 2 D) 4 E) ¼ 41. The mass on the end of the spring (which stretches linearly) is in equilibrium as shown. It is pulled down so that the pointer is opposite the 11-cm mark and then released. The mass experiences its maximum upward velocity at which of the following positions? A) 3-cm mark D) 11-cm mark B) 7-cm mark E) None of these is correct. C) 1-cm mark Page 8

9 42. 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 43. Both a mass spring system and a simple pendulum have a period of 1 s. Both are taken to the moon in a lunar landing module. While they are inside the module on the surface of the moon, A) the pendulum has a period longer than 1 s. B) the mass spring system has a period longer than 1 s. C) both a and b are true. D) the periods of both are unchanged. E) one of them has a period shorter than 1 s. 44. 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 45. 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. Page 9

10 46. 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 perce ntage increase in the length of the pendulum. 47. What must be the length of a simple pendulum with a period of 2.0 s if g = 9.8 m/s 2? A) 99 cm B) 97 m C) 6.2 cm D) 3.1 m E) 2.0 m 48. 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. 49. If the length of a simple pendulum is increased by 4% and the mass is decreased by 4%, the period is A) not changed. D) increased by 4%. B) increased by 2%. E) decreased by 2%. C) decreased by 4%. 50. Which of the following statements concerning the motion of a simple pendulum is incorrect? A) The kinetic energy is a maximum when the displacement is a maximum. B) The acceleration is a maximum when the displacement is a maximum. C) The period is changed if the mass of the bob is doubled and the length of the pendulum is halved. D) The time interval between conditions of maximum potential energy is one period. E) The velocity is a maximum when the acceleration is a minimum. 51. A pendulum is oscillating with a total mechanical energy E 0. When the pendulum is at its maximum displacement, the kinetic energy K and the potential energy U are A) K = ½E 0 ; U = ½E 0 D) K = E 0 ; U = 0 B) K = 0; U = E 0 E) K = E 0 ; U = ½E 0 C) K = E 0 ; U = E 0 Page 10

11 52. The graph shows the period squared versus the length for a simple pendulum. The slope of the graph corresponds to A) 1/g B) g/(4π 2 ) C) g D) 4π 2 /g E) 4π 2 g 53. An oscillator has a quality factor of 300. By what percentage does its energy decrease in each cycle? A) 0.33% B) 1% C) 2% D) 3% E) 4% 54. The energy of an oscillator decreases by 3% each cycle. The quality factor of this oscillator is approximately A) 209 B) 157 C) 87 D) 63 E) Which of the following statements is true for a simple harmonic oscillator that is not subject to dissipative forces? A) The potential energy of the system attains a maximum value when the displacement is one-half the amplitude. B) The kinetic energy of the system attains a maximum value when the acceleration has the greatest absolute value. C) The total mechanical energy of the system decreases as the mass slows down and increases as the mass speeds up. D) The total mechanical energy of the system is equal to the maximum value of the kinetic energy of the system. Use the following to answer questions Page 11

12 56. The graph shows the average power delivered to an oscillating system as a function of the driving frequency. According to these data A) the resonant frequency is greater than ω o. B) the system corresponding to curve 1 has the largest quality factor. C) the system corresponding to curve 4 has the largest quality factor. D) the resonant frequency is less than ω o. E) None of these is correct. 57. The graph shows the average power delivered to an oscillating system as a function of the driving frequency. According to these data A) the resonant frequency is greater than ωo. B) the system corresponding to curve 1 has the smallest quality factor. C) the system corresponding to curve 4 has the smallest quality factor. D) the resonant frequency is less than ω o. E) None of these is correct. 58. The graph shows the average power delivered to an oscillating system as a function of the driving frequency. According to these data, the damping is greatest for system(s) A) 1 B) 2 C) 3 D) 4 E) 1 and The graph shows the average power delivered to an oscillating system as a function of the driving frequency. According to these data, the damping is least for system(s) A) 1 B) 2 C) 3 D) 4 E) 3 and The differential equation for a damped oscillator is If the damping is not too large, the time constant for the motion of this oscillator is determined by the A) spring constant k and the mass m of the system. B) spring constant k and the damping coefficient b of the system. C) mass m and the damping coefficient b of the system. D) initial displacement of the system. E) initial velocity of the system. Page 12

13 61. The solution to the differential equation of a damped oscillator, for the case in which the damping is small, is x = A 0 e (b/2m)t cos(ω't + δ) The phase constant δ is determined by the A) spring constant k and the mass m of the system. B) spring cosntant k and the damping coefficient b of the system. C) initial velocity of the system. D) initial displacement of the system. E) c and d. Use the following to answer questions The graph shows the displacement of an oscillator as a function of time. The oscillator that is critically damped is A) 1 B) 2 C) 3 D) 4 E) 1, 2, 3, and The graph shows the displacement of an oscillator as a function of time. The oscillator that is undamped is A) 1 B) 2 C) 3 D) 4 E) 1, 2, 3, and The graph shows the displacement of an oscillator as a function of time. The oscillator that is overdamped is A) 1 B) 2 C) 3 D) 4 E) 1, 2, 3, and 4 Page 13

14 65. When you push a child in a swing, you most likely A) push with as large a force as possible. B) push with a periodic force as often as possible. C) push with a periodic force, with a period that depends on the weight of the child. D) push with a periodic force, with a period that depends on the length of the ropes on the swing. E) push with a force equal to the weight of the child. 66. The shattering of a crystal glass by an intense sound is an example of A) resonance. D) an exponential decrease. B) a Q factor. E) overdamping. C) critical damping. 67. When a body capable of oscillating is acted on by a periodic series of impulses having a frequency equal to one of the natural frequencies of oscillation of the body, the body is set in vibration with relatively large amplitude. This phenomenon is known as A) beats. D) resonance. B) harmonics. E) pressure amplitude. C) overtones. 68. Near resonance, if the damping is small (large Q), the oscillator A) absorbs less energy from the driving force than it does at other frequencies. B) absorbs more energy from the driving force than it does at other frequencies. C) absorbs the same amount of energy from the driving force than it does at other frequencies. D) moves with a small amplitude. E) is described by none of these. Page 14

15 69. The width of a resonance curve is an indication of A) the damping of the system. B) the Q-value of the system. C) the extent to which the system is oscillatory. D) the rate at which energy is being dissipated each cycle. E) all of the above. Page 15

16 Answers 1. C 2. B 3. D 4. D 5. E 6. D 7. D 8. D 9. E 10. C 11. B 12. D 13. E 14. C 15. B 16. E 17. E 18. D 19. B 20. D 21. B 22. D 23. B 24. A 25. E 26. C 27. B 28. C 29. E 30. E 31. D 32. C 33. A 34. B 35. D 36. E 37. E 38. D 39. C 40. C 41. B 42. A 43. A 44. B 45. D 46. A 47. A 48. C 49. B 50. D 51. B 52. D 53. C 54. A 55. D 56. B 57. C 58. D 59. A 60. C 61. E 62. A 63. D 64. B 65. D 66. A 67. D 68. B 69. E Page 16

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