39. A 0.25-kg block oscillates on the end of the spring with a spring constant of 200 N/m. If
the oscillation is started by elongating the spring 0.15 m and giving the block a speed of 3.0 m/s,
then the maximum speed of the block is:
A) 0.13 m/s
B) 0.18 m/s
C) 3.7 m/s
D) 5.2 m/s
E) 13 m/s
40. A 0.25-kg block oscillates on the end of the spring with a spring constant of 200 N/m. If
the oscillation is started by elongating the spring 0.15 m and giving the block a speed of 3.0 m/s,
then the amplitude of the oscillation is:
A) 0.13 m
B) 0.18 m
C) 3.7 m
D) 5.2 m
E) 13 m
41. An object on the end of a spring is set into oscillation by giving it an initial velocity while it
is at its equilibrium position. In the first trial the initial velocity is v0 and in the second it is 4v0.
In the second trial:
A) the amplitude is half as great and the maximum acceleration is twice as great
B) the amplitude is twice as great and the maximum acceleration is half as great
C) both the amplitude and the maximum acceleration are twice as great
D) both the amplitude and the maximum acceleration are four times as great
E) the amplitude is four times as great and the maximum acceleration is twice as great
42. A block attached to a spring undergoes simple harmonic motion on a horizontal frictionless
surface. Its total energy is 50 J. When the displacement is half the amplitude, the kinetic energy
is:
A) 0 J
B) 12.5 J
C) 25 J
D) 37.5 J
E) 50 J
43. A mass-spring system is oscillating with amplitude A. The kinetic energy will equal the
potential energy only when the displacement is
A) 0
B) A/4
C) ±𝐴/√2
D) A/2
E) anywhere between –A and +A
44. A particle is in simple harmonic motion along the x axis. The amplitude of the motion is xm.
When it is at x = x1, its kinetic energy is K = 5 J and its potential energy (measured with U = 0 at
x = 0) is U = 3 J. When it is at x = –1/2 xm, its total energy is:
A) 0 J
B) 3 J
C) 4 J
D) 5 J
E) 8 J
45. A particle is in simple harmonic motion along the x axis. The amplitude of the motion is xm.
When it is at x = x1, its kinetic energy is K = 5 J and its potential energy (measured with U = 0 at
x = 0) is U = 3 J. When its kinetic energy is 8 J, it is at:
A) x = 0
B) x = x1
C) x = xm/2
D) x = xm/
2
E) x = xm
46. A particle is in simple harmonic motion along the x axis. The amplitude of the motion is xm.
When it is at x = x1, its kinetic energy is K = 5 J and its potential energy (measured with U = 0 at
x = 0) is U = 3 J. When its potential energy is 8 J, it is at:
A) x = 0
B) x = x1
C) x = xm/2
D) x = xm/
2
E) x = xm
47. An angular simple harmonic oscillator:
A) oscillates at an angle to the x axis
B) oscillates along the y axis
C) involves an angular displacement and a restoring torque
D) involves an angular displacement and a restoring force
E) involves a linear displacement and a restoring torque
48. An angular simple harmonic oscillator is displaced 5.2 x 10-2 rad from its equilibrium
position. If the torsion constant is 1200 N∙m/rad, what is the torque?
A) 12 N∙m
B) 23 N∙m
C) 43 N∙m
D) 52 N∙m
E) 62 N∙m
49. A disk whose rotational inertia is 450 kg∙m2 hangs from a wire whose torsion constant is
2300 N∙m/rad. What is the angular frequency of its torsional oscillations?
A) 0.20 rad/s
B) 0.44 rad/s
C) 1.0 rad/s
D) 2.3 rad/s
E) 5.1 rad/s
50. A disk whose rotational inertia is 450 kg∙m2 hangs from a wire whose torsion constant is
2300 N∙m/rad. When its angular displacement is −0.23 rad, what is its angular acceleration?
A) 1.0 x 10-2 rad/s2
B) 4.5 x 10-2 rad/s2
C) 0.23 rad/s2
D) 0.52 rad/s2
E) 1.2 rad/s2
51. The amplitude of oscillation of a simple pendulum is increased from 1 to 4. Its maximum
acceleration changes by a factor of:
A) 1/4
B) 1/2
C) 2
D) 4
E) 16
52. A simple pendulum is suspended from the ceiling of an elevator. The elevator is
accelerating upwards with acceleration a. The period of this pendulum, in terms of its length L, g
and a is:
A) 2𝜋√𝐿/𝑔
B) 2𝜋√𝐿/(𝑔 + 𝑎)
C) 2𝜋√𝐿/(𝑔 − 𝑎)
D) 2𝜋√𝐿/𝑎
E) 1
2𝜋 √𝑔/𝐿
53. A simple pendulum consists of a small ball tied to a string and set in oscillation. As the
pendulum swings the tension in the string is:
A) constant
B) a sinusoidal function of time
C) the square of a sinusoidal function of time
D) the reciprocal of a sinusoidal function of time
E) none of the above
54. Three physical pendulums, with masses m1, m2 = 2m1, and m3 = 3m1, have the same shape
and size and are suspended at the same point. Rank them according to their periods, from
shortest to longest.
A) 1, 2, 3
B) 3, 2, 1
C) 2, 3, 1
D) 2, 1, 3
E) All three are the same
55. Five hoops are each pivoted at a point on the rim and allowed to swing as physical
pendulums. The masses and radii are
hoop 1: M = 150g and R = 50 cm
hoop 2: M = 200g and R = 40 cm
hoop 3: M = 250g and R = 30 cm
hoop 4: M = 300g and R = 20 cm
hoop 5: M = 350g and R = 10 cm
Order the hoops according to the periods of their motions, smallest to largest.
A) 1, 2, 3, 4, 5
B) 5, 4, 3, 2, 1
C) 1, 2, 3, 5, 4
D) 1, 2, 5, 4, 3
E) 5, 4, 1, 2, 3
56. Which of the following is NOT required for a simple pendulum undergoing simple harmonic
oscillation?
A) a point mass
B) a massless string
C) a small amplitude
D) gravitational force
E) a large spring constant
57. If the length of a simple pendulum is doubled, its period will:
A) halve
B) increase by a factor of √2
C) decrease by a factor of √2
D) double
E) remain the same
58. A simple pendulum of length L and mass M has frequency f. To increase its frequency to
2f:
A) increase its length to 4L
B) increase its length to 2L
C) decrease its length to L/2
D) decrease its length to L/4
E) decrease its mass to < M/4
59. A simple pendulum has length L and period T. As it passes through its equilibrium position,
the string is suddenly clamped at its mid-point. The period then becomes:
A) 2T
B) T
C) T/2
D) T/4
E) none of the above
60. Which of the following is the difference between a simple pendulum and a physical
pendulum?
A) The physical pendulum does not rotate around its center of mass.
B) The physical pendulum does not depend on the acceleration of gravity.
C) The physical pendulum has an extended mass.
D) The simple pendulum has a small amplitude.
E) The physical pendulum rotates around its center of mass.
61. A meter stick is pivoted at a point a distance a from its center and swings as a physical
pendulum. Of the following values for a, which results in the shortest period of oscillation?
A) a = 0.1 m
B) a = 0.2 m
C) a = 0.3 m
D) a = 0.4 m
E) a = 0.5 m
62. Two uniform spheres are pivoted on horizontal axes that are tangent to their surfaces. The
one with the longer period of oscillation is the one with:
A) the larger mass
B) the smaller mass
C) the larger rotational inertia
D) the smaller rotational inertia
E) the larger radius
63. At the instant its angular displacement is 0.32 rad, the angular acceleration of a physical
pendulum is -630 rad/s2. What is its angular frequency of oscillation?
A) 6.6 rad/s
B) 14 rad/s
C) 20 rad/s
D) 44 rad/s
E) 200 rad/s
64. The angular frequency of a simple pendulum depends on its length and on the local
acceleration due to gravity. The rate at which the angular displacement of the pendulum changes,
dθ/dt, is:
A) √𝑚𝑔𝐿/𝐼
B) √𝑔/𝐿
C) 2π√𝐿/𝑔
D) √𝑘/𝑚
E) none of the above
65. The angular displacement of a simple pendulum is given by θ(t) = θm cos (ωt + φ). If the
pendulum is 45 cm in length, and is given an angular speed dθ/dt = 3.4 rad/s at time t = 0, when
it is hanging vertically, what is θm?
A) 4.6 rad
B) 3.4 rad
C) 1.4 rad
D) 0.73 rad
E) 0.45 rad
66. The period of a simple pendulum is 1 s on Earth. When brought to a planet where g is
one-tenth that on Earth, its period becomes:
A) 1 s
B) 1/√10 s
C) 1/10 s
D) √10 s
E) 10 s
67. The rotational inertia of a uniform thin rod about its end is ML2/3, where M is the mass and
L is the length. Such a rod is hung vertically from one end and set into small amplitude
oscillation. If L = 1.0 m this rod will have the same period as a simple pendulum of length:
A) 33 cm
B) 50 cm
C) 67 cm
D) 100 cm
E) 150 cm
68. Both the x and y coordinates of a point execute simple harmonic motion. The result might
be a circular orbit if:
A) the amplitudes are the same but the frequencies are different
B) the amplitudes and frequencies are both the same
C) the amplitudes and frequencies are both different
D) the phase constants are the same but the amplitudes are different
E) the amplitudes and the phase constants are both different
69. Both the x and y coordinates of a point execute simple harmonic motion. The frequencies
are the same but the amplitudes are different. The resulting orbit might be:
A) an ellipse
B) a circle
C) a parabola
D) a hyperbola
E) a square
70. For an oscillator subjected to a damping force proportional to its velocity:
A) the displacement is a sinusoidal function of time
B) the velocity is a sinusoidal function of time
C) the frequency is a decreasing function of time
D) the mechanical energy is constant
E) none of the above is true
71. A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping
constant is 8.0 x 10-3 kg∙m/s, and the mass is 0.050 kg. If the particle starts at its maximum
displacement, x = 1.5 m, at time t = 0, what is the particle’s position at t = 5.0 s?
A) -1.5 m
B) -0.73 m
C) 0 m
D) 0.73 m
E) 1.5 m
72. A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping
constant is 8.0 x 10-3 kg∙m/s, and the mass is 0.050 kg. If the particle starts at its maximum
displacement, x = 1.5 m, at time t = 0, what is the amplitude of the motion at t = 5.0 s?
A) 1.5 m
B) 1.3 m
C) 1.0 m
D) 0.67 m
E) 0.24 m
73. A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping
constant is 8.0 x 10-3 kg∙m/s, and the mass is 0.050 kg. If the particle starts at its maximum
displacement, x = 1.5 m, at time t = 0, what is the angular frequency of the oscillations?
A) 4.0 rad/s
B) 8.0 rad/s
C) 12 rad/s
D) 23 rad/s
E) 45 rad/s
74. Five particles undergo damped harmonic motion. Values for the spring constant k, the
damping constant b, and the mass m are given below. Which leads to the smallest rate of loss of
mechanical energy?
A) k = 100 N/m, m = 50 g, b = 8 g∙m/s
B) k = 150 N/m, m = 50 g, b = 5 g∙m/s
C) k = 150 N/m, m = 10 g, b = 8 g∙m/s
D) k = 200 N/m, m = 8 g, b = 6 g∙m/s
E) k = 100 N/m, m = 2 g, b = 4 g∙m/s
75. Below are sets of values for the spring constant k, damping constant b, and mass m for a
particle in damped harmonic motion. Which of the sets takes the longest time for its mechanical
energy to decrease to one-fourth of its initial value?
k
b
m
A)
k0
b0
m0
B)
3k0
2b0
m0
C)
k0/2
6b0
2m0
D)
4k0
b0
2m0
E)
k0
b0
10m0
76. An oscillator is subjected to a damping force that is proportional to its velocity. A
sinusoidal force is applied to it. After a long time:
A) its amplitude is an increasing function of time
B) its amplitude is a decreasing function of time
C) its amplitude is constant
D) its amplitude is a decreasing function of time only if the damping constant is large
E) its amplitude increases over some portions of a cycle and decreases over other portions
77. A block on a spring is subjected to an applied sinusoidal force AND to a damping force
that is proportional to its velocity. The energy dissipated by damping is supplied by:
A) the potential energy of the spring
B) the kinetic energy of the mass
C) gravity
D) friction
E) the applied force
78. An oscillator is driven by a sinusoidal force. The frequency of the applied force:
A) must be equal to the natural frequency of the oscillator
B) becomes the natural frequency of the oscillator
C) must be less than the natural frequency of the oscillator
D) must be greater than the natural frequency of the oscillator
E) is independent of the natural frequency of the oscillator
79. A sinusoidal force with a given amplitude is applied to an oscillator. At resonance the
amplitude of the oscillation is limited by:
A) the damping force
B) the initial amplitude
C) the initial velocity
D) the force of gravity
E) none of the above
80. A sinusoidal force with a given amplitude is applied to an oscillator. To maintain the
largest amplitude oscillation the frequency of the applied force should be:
A) half the natural frequency of the oscillator
B) the same as the natural frequency of the oscillator
C) twice the natural frequency of the oscillator
D) unrelated to the natural frequency of the oscillator
E) determined from the maximum speed desired